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sha256:371bff03fd24c70b431317a2e8458a2095ce5e74107b93a798a09440f05b1213 +size 43138 diff --git a/parse/train/9MdLwggYa02/9MdLwggYa02.md b/parse/train/9MdLwggYa02/9MdLwggYa02.md new file mode 100644 index 0000000000000000000000000000000000000000..cceb9ef23e1b307abf26ba163a391b9b6b21b3f0 --- /dev/null +++ b/parse/train/9MdLwggYa02/9MdLwggYa02.md @@ -0,0 +1,222 @@ +# ROMUL: SCALE ADAPTATIVE POPULATION BASED TRAINING + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +In most pragmatic settings, data augmentation and regularization are essential, and require hyperparameter search. Population based training (PBT) is an effective tool for efficiently finding them as well as schedules over hyperparameters. In this paper, we compare existing PBT algorithms and contribute a new one: ROMUL, for RObust MULtistep search, which adapts its stepsize over the course of training. We report competitive results with standard models on CIFAR (image classification) as well as Penn Tree Bank (language modeling), which both depend on heavy regularization. We also open-source hoptim, a PBT library agnostic to the training framework, which is simple to use, reentrant, and provides good defaults with ROMUL. + +# 1 INTRODUCTION + +Hyperparameter tuning is essential for good performance in most machine learning tasks, and poses numerous challenges. First, optimal hyperparameter values can change over the course of training (schedules), e.g. for learning rate, fine tuning phases, data augmentation. Hyperparameters values are also rarely independent from each other (e.g. the magnitude of individual data augmentations depends on the number of data augmentations applied), and the search space grows exponentially with the number of hyperparameters. All of that search has to be performed within a computational budget, and sometimes even within a wall-clock time budget (e.g. models that are frequently retrained on new data), requiring efficient parallelization. In practice, competitive existing methods range from random search (Bergstra & Bengio, 2012) to more advanced methods (that aim at being more compute-efficient) like sequential search (Bergstra et al., 2011; 2013; Li et al., 2018), population based training (PBT, e.g. Jaderberg et al. (2017); Ho et al. (2019)) and search structured by the space of the hyperparameters (Liu et al., 2018; Cubuk et al., 2019b). + +A major drawback of advanced hyperparameter optimization methods is that they themselves require attention from the user to reliably outperform random search. In this work, we empirically study the different training dynamics of data augmentation and regularization hyperparameters across vision and language modeling tasks, in particular for multistep (sequential) hyperparameter search. A common failure mode (i) is due to hyperparameters that have a different effect on the validation loss in the short and long terms, for instance using a smaller dropout often leads to faster but worse convergence. Another common problem (ii) is that successful searches are constrained on adequate “hyper-hyperparameters” (such as value ranges or the search policy used, which in current methods are non-adaptative mutation steps). Our contributions can be summarized as follows: + +• We present a robust algorithm for leveraging population based training for hyperparameter search: ROMUL (RObust MULtistep) search, which addresses (i) and (ii). We empirically study its benefits and limitations, and show that it provides good defaults that compare favorably to existing methods. + +• We open-source hoptim, a simple library for sequential hyperparameter search, that provides multiple optimizers (including ROMUL), as well as toy benchmarks showcasing hyperparameter optimization problems we identified empirically and standard datasets. + +# 2 HYPERPARAMETER OPTIMIZATION WITH POPULATION-BASED TRAINING + +In this article, we refer to the family of algorithms that continuously tunes hyperparameters of a set of models over the course of their training as “PBT algorithms” or “PBT optimizers”. Hyperparameter optimization is thus a zero order optimization performed at a slower frequency than the (often first order, e.g. SGD) optimization of the model. A PBT step happens typically after a fixed number of epochs or updates of the model, often optimizing the loss from the validation set, continuing from an already produced “parent” checkpoint, and producing and evaluating a new checkpoint. At every PBT step, hyperparameters can be updated (mutated), incremented or decremented by some number (step size), or sampled. + +There are multiple aspects to consider when designing a PBT algorithm. Technical constraints: how the optimization is distributed, with a centralized or decentralized algorithm, workers to run the trainings, how failed workers are handled. They are solved in a unified manner in the experiments we performed, by the hoptim library to implement and compare multiple algorithms. It is decoupled from the scheduling of the jobs and designed to accommodate adding more workers to scale up the training, or fewer when some are killed, for example through preemption or time-out on a shared cluster. Optimization method: how the hyper-parameters are modified throughout the training, for instance through mutations. Selection process: which individual of the population are kept, both in term of hyper-parameters and state of the neural network (checkpoint). For those last two points, some solutions are described below. + +# 2.1 CHALLENGES + +In order to have a clearer understanding of our proposed methods, we show below the main concerns we have observed in PBT: + +Anisotropy: by definition, the optimal value of the hyperparameters considered is unknown, and oftentimes the range (or mutation scheme) provided to the algorithm is a loose estimate only. As modifying two hyperparameters with the same step size can produce effects with very different magnitudes, the user is required to to normalize the search space. But pre-tuning the hyperparameter tuner itself can be cumbersome as dynamics evolve during training. Section 3.1 provides an example based on the Rosenbrock function which illustrates this issue and highlights the interest of adaptative mutations. + +Checkpoint vs. hyperparameters: comparing individuals in the population is extremely hard as improvements can be due to better hyperparameters, or better checkpoints (including potentially better batches). Better performance through better checkpoints is an optimization phenomenon (e.g. random restarts), that can bias the hyperparameter selection. We will detail this aspect in Section 4.2. + +Short-term-long-term discordance: we observed empirically that hyperparameters which induce better performance in the short term are not always optimal in the longer term. This is a challenge that does not exist in classical static optimization, but is crucial for PBT since local minima are easy to reach and pose a danger for greedy algorithms. An example of such a parameter is the learning rate. Dropping the learning rate often induces a drop in the validation loss, even early in the training, and increasing it has the opposite effect, causing greedy PBT algorithms to reduce it to the minimum value too early, without being able to recover. We will detail this aspect in Section 4.1. + +# 2.2 DIFFERENTIAL EVOLUTION AND ROMUL + +Differential Evolution Storn & Price (1997) (DE) is a standard black-box optimization method, for minimizing $f : \mathbb { R } ^ { n } \mathbb { R }$ . It operates on a population $x ^ { i } \in \mathbb { R } ^ { n }$ for all $i \stackrel { \textstyle \dag } { \in } \{ 1 , . . . , M \}$ , $M \geq 4$ , and indefinitely repeats the following steps for each individual $x ^ { \mathrm { b a s e } }$ in the population to generate another individual called mutated vector that could replace $x ^ { \mathrm { b a s e } }$ if better: + +1. given the best individual $x ^ { \mathrm { b e s t } }$ which minimizes $f$ in the population, as well as two randomly selected ones $x ^ { a }$ and $x ^ { b }$ , compute the donor $d$ , which will give part of its coefficients to the mutated vector. In the current-to-best/1 scheme we use, these are the base coefficients plus a term attracting to the best set of coefficients from the current population, and an additional random variation (a standard value for $F _ { i }$ is $F _ { 1 } = F _ { 2 } = 0 . 8$ ): + +$$ +d = x ^ { \mathrm { b a s e } } + F _ { 1 } ( x ^ { \mathrm { b e s t } } - x ^ { \mathrm { b a s e } } ) + F _ { 2 } ( x ^ { b } - x ^ { a } ) +$$ + +2. create the new mutated vector $\widetilde { x } ^ { b a s e }$ by randomly selecting each component $j \in \{ 1 , . . . , n \}$ of the base $x ^ { \mathrm { b a s e } }$ eor the donor $d$ through the binary crossover operator: $\begin{array} { r l } { \widetilde { x } _ { j } ^ { \mathrm { b a s e } } } & { { } = } \end{array}$ $\mathrm { C H O I C E } ( x _ { j } ^ { \mathrm { b a s e } } , d _ { j } )$ e. This non-linear operation lets the optimization leave the vector span of the population. + +3. compute $f ( \widetilde { x } ^ { \mathrm { b a s e } } )$ and replace $x ^ { \mathrm { b a s e } }$ by $\widetilde { x } ^ { \mathrm { b a s e } }$ within the population if and only if $f ( \widetilde { x } ^ { \mathrm { b a s e } } ) \leq$ $f ( x ^ { \mathrm { b a s e } } )$ . + +This method is interesting as it is already based on a population and adapts well to parameters with different dynamics while being simple and fully parallelizable. In particular, it does not rely on mutation ranges or step sizes - Equation 1 samples new parameters close to the current population, and as the population individuals go through selection this sampling is refined and becomes sharper around optimal values. In practice, if a parameter’s bounds are too loose or wrong, DE will eventually adapt after iterations of selection by removing individuals too far from the optimal value, and concentrate its computation budget on relevant values for this parameter. + +In order to use it for PBT, the set of hyper-parameters is converted to a vector in $\mathbb { R } ^ { n }$ using nevergrad parametrization system (Rapin & Teytaud, 2018). However, this basic version of differential evolution (also implemented in nevergrad) is not adapted to PBT. Indeed the training function $f$ changes with the checkpoint as we are updating the parameters (not the hyperparameters) with a stochastic gradient from the task loss. The trend of $f$ is therefore typically downwards during the training, younger generations/later epochs tending to have a lower loss than their parents’, biasing the hyperperameter selection process in favor of those of the children (later steps of SGD updates) instead of in favor of better hyperparameters. + +ROMUL We therefore propose an adaptation: a population of $n$ individuals is trained, after finishing their step, individuals are compared to the rest of the population. If they have one of the $n / k$ best loss (we use $k \ = \ 2$ throughout), the training continues without changing the hyperparameters, otherwise, the hyperparameters are mutated. If the hyperparameters of an individual are mutated $m$ times in a row (we use $m \ : = \ : 3$ throughout), its checkpoint is killed and replaced by one of the $n / k$ best individuals. The values of $k$ and $m$ are hyperparameters, although we did not vary them in any experiments: $k = 2$ allows to have, on average, one alternative (mutated) version to each of the ones we keep training without hyperparameter change, and $m = 3$ proved to be robust across our experiments, to select when to discard a checkpoint. If using lower $m$ values, one should consider increasing the number of epochs per PBT step to prevent culling checkpoints too early (see Section 4.1 and 4.2). + +The mutation scheme is adapted to fit this use case. In Eq. 1, $x ^ { \mathrm { b a s e } }$ and $x ^ { \mathrm { b e s t } }$ are both replaced by a randomly selected set of hyperparameters $x ^ { \mathrm { { c } } }$ and $x ^ { \mathrm { d } }$ from the best $n / 2$ individuals (“rand-to-rand/1” scheme following (Storn & Price, 1997; Das & Suganthan, 2011) notations). Replacing $x ^ { \mathrm { b a s e } }$ aims at keeping the path through checkpoints unimodal, since keeping several modes with corresponding checkpoints is unnecessary. Replacing $x ^ { \mathrm { b e s t } }$ by any other ”good” (top $50 \%$ ) set of hyperparameters aims at avoiding early convergence, which we observed as one of the main problems during trainings. This also avoids a strong bias by a good checkpoint (more on this in 4.2). To avoid duplication of hyperparameters, we opt for making $F _ { 1 }$ and $F _ { 2 }$ random vectors instead of using the binary crossover non-linearity. In order to keep the initial scaling of DE, we chose ${ \bf F _ { 1 } } [ i ]$ uniformly distributed between 0 and $2 F$ (we use the common value for $F$ from vanilla DE: $F = 0 . 8$ ), and $\mathbf { F _ { 2 } } [ i ] = 2 F - \mathbf { F _ { 1 } } [ i ]$ , $\forall i$ . This ensures that the sum $\mathbf { F _ { 1 } } [ i ] + \mathbf { F _ { 2 } } [ i ] = 2 F $ $\forall i$ , as in vanilla DE. With $\odot$ the elementwise multiplication, this yields $d = x ^ { c } + \mathbf { \bar { F _ { 1 } } } \odot ( x ^ { d } - x ^ { c } ) + \mathbf { F _ { 2 } } \odot ( x ^ { b } - x ^ { a } )$ . + +# 2.3 OTHER ALGORITHMS + +In our experiments, we compare several algorithms briefly presented below. We aim to compare how effective they can be for practical use-cases of hyperparameter tuning, where the user does not want to tune the hyperparameters of its hyperparameter tuner, and desires meaningful defaults. The only input they take are the number of parallel trainings, the range of hyperparameters, and a hint for an initial value (e.g. the same value 0 for dropout values and data augmentation magnitudes). + +Initiator PBT: We reimplemented Initiator Based Evolution, presented in Li et al. (2019). New hyper-parameters are sampled from parent hyper-parameters by adding/removing a mutation constant (for instance d $r o p o u t C h i l d = d r o p o u t P a r e n t \pm 0 . 1 )$ . A newly created checkpoint is compared to a randomly sampled checkpoint in the population: if the latter is better, the new checkpoint is discarded and the latter is forked with its hyperparameters - this ensures that only the best performing models remain eventually, and allows it to run asynchronously. For each parameter, we specify a range, and use $( h i - l o ) / 3 0$ as a mutation constant unless specified otherwise. + +Truncation Selection: $N$ models are trained in parallel. Regularly, the $M$ worst performing models are stopped and replaced with clones of the $M$ bests, and hyperparameters are randomly perturbated. This scheme was first introduced in Jaderberg et al. (2017). In our experiments, we use $M = N / 4$ . For hyperparameter perturbation, we generalize the mutation scheme introduced in Ho et al. (2019): each parameter is sampled uniformly in its range $[ l o , h i ]$ with $20 \%$ probability, or incremented by random.choice([-3, -2, $^ { - 1 }$ , 0, 0, 1, 2, 3]) $\star$ (hi - lo) / 10 and then clipped to stay within $[ l o , h i ]$ . + +ASHA: This is not a PBT algorithm, but a strong hyperparameter search algorithm that we compare to. In the Asynchronous Successive Halving Algorithm (ASHA, Li et al. (2018)), hyperparameters are sampled uniformly like in Random Search, but models are evaluated early and stopped if not in the top $1 / \eta$ percentile. For a given model, the first evaluation can happen after $1 , \eta , \eta ^ { 2 }$ , .. steps, making this algorithm robust to hyperparameters whose optimal value does not perform well until late in the training (Section 4.1). Unlike Initiator PBT or Truncation Selection, ASHA finds constant values for hyperparameters rather than schedules. We set the reduction factor $\eta$ to 3. + +# 3 EXPERIMENTS + +We ran experiments on a toy optimization problem (the Rosenbrock function), CIFAR, and Penn Tree Bank, all with the same ROMUL hyperparameters to test its robustness. Each of these experiments train in around 100 to 300 epochs, and we used 1 step per epoch, so that they all have similar time scales. + +# 3.1 EXAMPLE ON A TOY OPTIMIZATION PROBLEM + +Current PBT mutation schemes have fixed steps and therefore do not automatically adapt to the landscape of the optimized function. This means that they are not well-suited for anisotropic problems, which often arise in real life applications since some hyperparameters may be very important to tune finely, while other do not require the same precision. To highlight this issue, we experiment below on the Rosenbrock function: $\mathcal { R } _ { a , b } ( x , y ) = ( a - x ) ^ { 2 } + b ( y - x ^ { 2 } ) ^ { 2 }$ + +We will aim at minimizing $\mathcal { R } _ { 1 , 1 0 0 }$ through the surrogate $\mathcal { R } _ { \hat { a } , \hat { b } }$ , with $\hat { a }$ and $\hat { b }$ two hyperparameters handled with PBT. We initialize both parameters at 20 and bound them by -12.12 and 212.12 (using integers would be a special case since actual $a$ and $b$ values are integers). This experiment can be reproduced using the hoptim toolbox with the command: hop bench rosenbrock. + +While standard PBT with random steps wastes mutations on $\hat { a }$ , DE is able to adapt its step-size to large steps on $\hat { a }$ until getting close, then smaller steps on $\hat { a }$ to tune $\hat { a }$ and $\hat { b }$ more finely. This is visible in Fig. 1a with ROMUL values of $\hat { a }$ converging quickly to around 1. The mutations then become sharper, while the ones for Initiator-PBT (small steps) are still too large and oscillate around the optimal value. Arguably, the mutation step could have been even smaller, but that would have slowed down the convergence, and these steps would be painful meta-parameters to tune at scale. Initiator-PBT with larger steps and Truncation selection are not displayed in this figure because their variations are too large. + +The impact on the loss $\hat { \mathcal { R } }$ is then visible in Fig. 1b: Initiator PBT can’t decrease past 0.2 with large steps, and 0.048 with smaller steps, since it is trapped trying to optimize $\hat { a }$ while DE is able to reach better values. Fig. 1c and 1d show the trajectory of $( x , y )$ for Truncation selection and ROMUL, with the same number of training steps. Truncation selection is hampered by more random mutations. On the other hand, ROMUL is able to reach a much lower value after exhibiting a more chaotic behavior when it initially adapts to the scale of the problem. The trajectory for both versions of Initiator-PBT can be found in Fig. 2 of the appendix. Initiator-PBT with large steps (Fig. 2a) moves + +![](images/3d5d48c5c340c00fcf8b3d8f4759575d093be9629d0fa63e3af03d559266f1b5.jpg) + +![](images/3ba09c208b2a0a1eb0ad236dae6d70897fd8ce52659c2eb40bad71ba2576af7e.jpg) + +(a) Rosenbrock parameter $\hat { a }$ with respect to the number of steps (optimal at 1). After ${ \approx } 5 0 $ steps, ROMUL’s distribution for $\hat { a }$ gets sharper around 1, Initiator always uses an hardcoded mutation step size. + +(b) Loss $\mathcal { R } _ { 1 , 1 0 0 }$ with respect to the number of steps (lower is better, minimum value is 0). ROMUL keeps adapting and decreasing while other optimizers are locked to higher levels depending on their step sizes. + +![](images/34a9a888f74479fa60039e297e33e410ba7750db6eac0ea6f9cd1d5621e931e2.jpg) +Figure 1: Training on the Rosenbrock benchmark. ROMUL outperforms initiator and truncation selection because it can adapt its step size. Bottom plots: Trajectories of 100 PBT training steps (16 jobs per step) on the Rosenbrock function with $a = 1$ and $b = 1 0 0$ (minimum at the red cross $( 1 , 1 )$ , trajectories go from blue to green) + +very slowly to the minimum because of big extra oscillations. With smaller steps (Fig. 2b), it reaches better values through a slow and non-direct path. Tab. 3 in the appendix provides quantitative results by averaging over 20 runs, including a version of Initiator PBT in which updates are performed through multiplications by 0.8 or 1.2. In particular, ROMUL performs statically better than all over optimizers on this testbed $( p \textless 1 . 1 e \_ 5$ with a two sample Welch’s t-test). In Fig. 3 in Appendix, we also show the behavior with more variables by performing optimization on an average of Rosenbrocks functions, each with independent $a$ and $b$ variable to be estimated by PBT. Overall ROMUL performs consistently well across the board for a wide range of number of variables. + +# 3.2 APPLICATION TO CIFAR (IMAGE CLASSIFICATION) + +In this section, we compare various algorithms for tuning hyperparameters for image classification on CIFAR (Krizhevsky et al., 2009). We reproduce the population based augmentation (PBA) setup from (Ho et al., 2019) with their original implementation. Our algorithms train a Wide-ResNet-28- 10 model on Reduced CIFAR-10 (using $10 \%$ , i.e. 4000 images, of the training set for actual training, and the remainder as a validation set), and optimize the same 60 hyperparameters as in Ho et al. (2019): 2 magnitudes and 2 probabilities for each of the 15 possible data augmentations. For each algorithm, we take the best model in the validation set at epoch 200, and use its hyperparameters schedules to train another Wide-ResNet-28-10 model on CIFAR-10 and CIFAR-100 and finally report test accuracies at epoch 200 in Table 1. Trainings can be reproduced with the hoptim package and its benchmarking counterpart hoptim benchmarks in the cifar folder. ROMUL recovers most of the gains on CIFAR-10 $2 . 8 \%$ error vs. $2 . 6 \%$ for the SOTA and $3 . 9 \%$ for the baseline), and is a bit further away on CIFAR-100 ( $1 7 . 1 \%$ vs. $1 6 . 7 \%$ for the SOTA and $1 8 . 8 \%$ for the baseline). PBA, which yields state-of-the-art results on CIFAR, used Truncation selection PBT introduced in (Jaderberg et al., 2017), which we implemented and compared to. We adopted all the PBA hyperparameters and observe $2 . 7 \%$ on CIFAR-10 (ROMUL: $2 . 8 \%$ ) and $1 7 . 7 \%$ on CIFAR-100 (ROMUL: $1 7 . 1 \%$ ). The differences in the job and population management in hoptim may explain the difference between our implementation and theirs, which is particularly marked on the training set reduced CIFAR-10: $12 . 8 \%$ for their vs. $1 3 . 9 \%$ for our implementation. + +Table 1: Classification error (lower is better) on CIFAR-10 and CIFAR-100 test sets for a WideResNet-28-10 (36M params). The algorithms run with 16 workers in parallel with the same compute budget (except when stated otherwise) on reduced CIFAR-10. After that, the schedule found is used for training the same model from scratch on CIFAR-10 and CIFAR-100 + +
AlgorithmReduced CIFAR-10 (10%)CIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-10n/a3.918.8
RandAugment (Cubuk et al., 2019b)n/a2.716.7
PBA (3 epochs/step) (Ho et al., 2019)12.82.616.7
ASHA14.72.817.6
ASHA (running for double the time)14.12.717.2
Truncation Selection (PBA, ours)13.92.717.7
Initiator PBT (Li et al. (2019), ours)14.72.917.9
ROMUL14.02.817.1
+ +As PBA waits for more than 1 epoch/step to evaluate a set of hyperparameters, we compared 1 epoch/step and 3 epochs/step (their setting), as it could help thwarting short-term/long-term discrepancy effects (see 4.1) and noise as explained above, but we could not identify a sufficiently generic scheme for all applications. In general, this is part of the PBT hyperparameters that are tuned in PBA, that we try to completely remove as hyperparameters in ROMUL, by being insensitive to it (in this case it does not seem to affect Truncation Selection either). For this hyperparameter, the constraint is to do PBT steps slow enough so that the number of updates is sufficiently large for the model to adapt to new mutated hyperparameters, and high enough so that PBT has enough steps to optimize the hyperparameters. In practice, trainings are long enough (regarding the number of SGD updates of the model) for a wide range of PBT steps frequencies to work. + +# 3.3 APPLICATION TO THE PENN TREEBANK DATASET (LANGUAGE MODELING) + +We experiment with the TransformerXL model (Dai et al., 2019) on the PTB dataset (Marcus, 1993). TranformerXL’s code is open-source and is the state-of-the-art for tranformer models on this dataset when using proper regularization, making it an interesting challenge for PBT. It comes with several dropout hyperparameters: we search for optimal values for five different dropout hyperparameters, that we describe in Table 4 in appendix. They are all initialized to 0 with standard deviation of 0.1 for ROMUL (negative values are reflected to positive values), hence not at the baseline values. + +Results are reported in Table 2. Trainings can be reproduced (up to random variance) with the hoptim package and its benchmarking counterpart hoptim benchmarks in the ptb folder. The baseline TransformerXL was obtained with the author’s code and is close to the one reported in the initial paper. Noticeably, ASHA and Random Search (with a uniform prior) are not able to come close to the baseline, with more than 4 points difference in both validation and test perplexity (PPL). Truncation selection and Initiator PBT on the other hand are able to reach the baseline although they were not able to excel it in test perplexity. Only ROMUL is able to reach (marginally) better results than the reproduced baseline in test PPL with both 16 and 32 workers. A found dropout schedule is displayed in the appendix (Fig. 4) and show dropouts rapidly increasing in the beginning and stabilizing to different levels. Using 16 and 32 workers provided similar results up to noise for ROMUL (in this very case, 32 workers does not actually perform better than with 16 workers). + +However, using 8 workers results in a notable drop in performance for all optimizers (Test PPL ROMUL 56.39, TruncSel 57.98, Initiator 56.33). + +Table 2: Perplexity (lower is better) on PTB for a Transformer-XL with 16 layers and 24M parameters, best validation PPL before iteration 175 and corresponding test PPL, given the resources needed these values are not averaged, numbers excelling our training baseline are in bold. + +
TrainingworkersValidation PPLTest PPL
TransformerXL SOTA (Dai et al. (2019))1/54.52
TransformerXL SOTA (our training, their code)159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMULPBT1657.8355.16
Random Search3263.8460.90
ASHA3264.3161.63
Truncation Selection PBT3258.4555.93
Initiator PBT3259.3655.73
ROMUL PBT3258.6355.28
+ +# 4 DISCUSSION + +# 4.1 SHORT TERM - LONG TERM DISCORDANCE + +We have observed on PTB and other applications that some hyperparameters were never contributing positively to the model’s performance in the short term (eg: 1 step) but could become better on a longer term (eg: 5 steps or more), hence checkpoints need time to adapt to a new parameter set (e.g.: building more redundancy). In an experiment on PTB, we used one epoch per step for half the training and then modified it to 10 epochs per step. We observed that increasing one of the dropout contributed negatively for small steps (1 epoch), but positively for long steps (10th epochs). This is a major roadblock for PBT-based approaches since two models with different hyperparameters can’t be straightforwardly compared at every step, but only after an unknown delay. This is partially handled by being conservative on models to keep: keeping the best $50 \%$ unchanged in ROMUL, or the random tournament scheme that allows bad models to continue in Initiator PBT when assigned an even worse opponent. Fig. 5 in appendix shows such an example in another domain: a large dropout seems very detrimental early on, but very beneficial in the longer term. While this is expected for regularizations, we observe that a straightforward schedule increasing the dropout in steps (in red) is not able to compensate this - we observed the same effect with a continuous schedule. This behavior adds complexity to the task of PBT algorithms, because bad early choices can’t be compensated later. Arguably, it can be due to interactions with the learning rate scheduler used and a more appropriate schedule could help solve this issue (although it is not clear what such a schedule should be). + +# 4.2 CHECKPOINTS VS HYPERPARAMETERS - SELECTION BIASES + +For PBT optimizers, one critical question is how much of the loss difference between two individuals is caused by different hyperparameters, and how much about different checkpoints. Both contributions are tightly entwined making it harder to identify which hyperparameters are best. + +A naive initial option to counter this is to always start a PBT step from the best checkpoint in the population. In our experiments this performed worse, at least because of the noise in the evaluation metric, but also because of the short-term long-term discrepancy detailed above (Section 4.1). We also expect that doing so could make optimizers less robust by getting trapped in local minima too easily, or aggressively discarding more promising models in the longer term. + +On the opposite side of the spectrum, we experimented with never culling checkpoints, effectively performing $n$ full trainings in parallel. Conceptually, keeping checkpoints is attractive since it should add robustness to the optimizer: selected hyperparameters have to work well for more than one particular checkpoint. Indeed, this way the performance can be attributed to actual parameter schedules fitting different trainings, instead of being biased by checkpoints culling/random restarts. It also adds more variability which could be beneficial especially with respect to the short-term long-term discrepancy. That being said, we have neither observed significant improvement nor deterioration when keeping checkpoints, as long as the mutation schemes were not biased towards the best set of hyperparameters (e.g.: removing the $x ^ { \mathrm { b e s t } }$ term in Eq. 1), because doing so can make all hyperparameters converge towards the best checkpoint, making the optimization process early converge to values which are not necessarily adapted to other checkpoints. + +Still, even with ROMUL’s loss-agnostic mutation scheme, some checkpoints were observed to fall behind and waste resources if not culled, so we expect that a trade-off like the one we implemented (killing checkpoints after 3 failed mutations in a row) is necessary. + +Another source of selection bias is noise. While the trainings are well behaved in PTB because the shuffling of the training set is synchronized by epoch, the trainings in CIFAR are much noisier because of the randomness introduced by data augmentation and the very small training set size. PBT optimizers based on more noise-robust blackbox optimization methods could be beneficial, but it is not clear how to adapt them. + +# 5 RELATED WORK + +Several families of methods exist for tuning hyperparameters of neural networks. Methods closest to grid search like random search (Bergstra & Bengio, 2012) and ASHA (Li et al., 2018) are based on minimal constraints and can be parallelized extensively. Methods striving for more data-efficient search (Bergstra et al., 2011; 2013; Feurer & Hutter, 2019) are more sequential in nature, requiring convergence of some trainings before launching new ones. Population-based training approaches (Jaderberg et al., 2017; Ho et al., 2019; Li et al., 2019) loosen the requirements of training different models, as hyperparameters are changed on-the-fly during training, which also makes the search for schedules easier and less structured, i.e. not based on a predefined function. + +Recent advances in automatic discovery of data augmentation policies include Population Based Augmentation (Ho et al., 2019) which we compared to in this paper (denoted Truncation Selection). Another line of work on structuring the hyperparameter space for data augmentation policy search is AutoAugment (Cubuk et al., 2019a), FastAutoAugment Lim et al. (2019) and RandAugment Cubuk et al. (2019b), the later being faster and reaching top performance on CIFAR. + +PBT is used successfully in reinforcement learning (Jaderberg et al., 2017), providing diversity in self-play and progressive difficulty, so other experimental comparisons that we did include Initiator PBT from (Li et al., 2019), which presented a generic PBT setup that inspired hoptim. For nonPBT baselines we used random search (Bergstra & Bengio, 2012), and ASHA (Li et al., 2018), which is an update on HyperBand (Li et al., 2017). + +# 6 CONCLUSION + +We introduced ROMUL, a robust PBT algorithm that we benchmarked on standard datasets with multiple regularization and data augmentation hyperparameters. Its main strength comes from its robustness to hyperparameters definitions by automatically adapting to the scale of each parameter. Although it did not show better performance on CIFAR than PBA – that was tuned for this benchmark – we demonstrated that it is more robust to domain changes. More importantly for the practical use-cases, it constitutes a good default that does not require extensive tuning to work well. We open-sourced its implementation as well as a simple and broadly compatible PBT library. + +The main difficulties we observed for PBT-based optimizers came from short-term vs. long-term effects: parameters can have a positive impact in the short term but a negative one in the longer term which may not be rectifiable. Learning rate falls in this category, since decreasing it often provides quick gains at the risk of being trapped in a local minimum. Studying how to deal with such behaviors is in our opinion the main challenge of future work. + +# REFERENCES + +James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. The Journal of Machine Learning Research, 13(1):281–305, 2012. + +James Bergstra, Dan Yamins, and David D Cox. Hyperopt: A python library for optimizing the hyperparameters of machine learning algorithms. In Proceedings of the 12th Python in science conference, volume 13, pp. 20. Citeseer, 2013. + +James S Bergstra, Remi Bardenet, Yoshua Bengio, and Bal ´ azs K ´ egl. Algorithms for hyper-parameter ´ optimization. In Advances in neural information processing systems, pp. 2546–2554, 2011. + +Ekin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V Le. Autoaugment: Learning augmentation strategies from data. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 113–123, 2019a. + +Ekin D Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V Le. Randaugment: Practical automated data augmentation with a reduced search space. arXiv preprint arXiv:1909.13719, 2019b. + +Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V. Le, and Ruslan Salakhutdinov. Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context. arXiv:1901.02860 [cs, stat], June 2019. + +Swagatam Das and Ponnuthurai Nagaratnam Suganthan. Differential Evolution: A Survey of the State-of-the-Art. IEEE Transactions on Evolutionary Computation, 15(1):4–31, February 2011. ISSN 1089-778X, 1941-0026. doi: 10.1109/TEVC.2010.2059031. + +Angela Fan, Edouard Grave, and Armand Joulin. Reducing transformer depth on demand with structured dropout. arXiv preprint arXiv:1909.11556, 2019. + +Matthias Feurer and Frank Hutter. Hyperparameter optimization. In Automated Machine Learning, pp. 3–33. Springer, Cham, 2019. + +Daniel Ho, Eric Liang, Ion Stoica, Pieter Abbeel, and Xi Chen. Population based augmentation: Efficient learning of augmentation policy schedules. arXiv preprint arXiv:1905.05393, 2019. + +Max Jaderberg, Valentin Dalibard, Simon Osindero, Wojciech M. Czarnecki, Jeff Donahue, Ali Razavi, Oriol Vinyals, Tim Green, Iain Dunning, Karen Simonyan, Chrisantha Fernando, and Koray Kavukcuoglu. Population Based Training of Neural Networks. arXiv:1711.09846 [cs], November 2017. arXiv: 1711.09846. + +Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009. + +Ang Li, Ola Spyra, Sagi Perel, Valentin Dalibard, Max Jaderberg, Chenjie Gu, David Budden, Tim Harley, and Pramod Gupta. A generalized framework for population based training, 2019. + +Liam Li, Kevin Jamieson, Afshin Rostamizadeh, Ekaterina Gonina, Moritz Hardt, Benjamin Recht, and Ameet Talwalkar. Massively parallel hyperparameter tuning. arXiv preprint arXiv:1810.05934, 2018. + +Lisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar. Hyperband: A novel bandit-based approach to hyperparameter optimization. The Journal of Machine Learning Research, 18(1):6765–6816, 2017. + +Sungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, and Sungwoong Kim. Fast autoaugment. In Advances in Neural Information Processing Systems, pp. 6662–6672, 2019. + +Hanxiao Liu, Karen Simonyan, and Yiming Yang. Darts: Differentiable architecture search. arXiv preprint arXiv:1806.09055, 2018. + +Mitch Marcus. Building a Large Annotated Corpus of English: The Penn Treebank:. Technical report, Defense Technical Information Center, Fort Belvoir, VA, April 1993. + +J. Rapin and O. Teytaud. Nevergrad - A gradient-free optimization platform. https://GitHub. com/FacebookResearch/Nevergrad, 2018. + +Rainer Storn and Kenneth Price. Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. Journal of Global Optimization, 11(4):341–359, December 1997. ISSN 1573-2916. doi: 10.1023/A:1008202821328. + +A APPENDIX + +A.1 ROSENBROCK EXPERIMENTS + +![](images/0d4070f821f96f59c4fb0727906a39d95055383043b96afb6151f9feaaf12148.jpg) +Figure 2: Trajectories of 100 Initiator PBT training steps (16 jobs per step) on the Rosenbrock function with $a = 1$ and $b = 1 0 0$ (minimum at the red cross $( 1 , 1 )$ , trajectories go from blue to green) + +
Trainingmean ( (log10)std (log10)
Initiator (0.8/1.2 mult. steps)-1.180.045
Initiator (big steps)-0.7070.069
Initiator (small steps)-0.9920.143
ROMUL-2.1010.678
Truncation Selection-0.8340.327
+ +![](images/a79afcffc16883a48c9c4c8102cf27b411ab7049370779458a2e86bb3b4580d2.jpg) +Table 3: Final loss (in log10) mean and standard deviation for independent runs on the Rosenbrock testbed, computed over 20 runs (two-sample Welsh’s test provides $p < 1 . 1 e - 5$ when comparing each algorithm with ROMUL). +Figure 3: Score on the multi-variate Rosenbrock benchmark (explained in 3.1) over 20 experiments for each point. Lower is better, standard deviations are indicated. ROMUL performs well across the board for a wide range of number of variables, being surpassed only by Truncated selection in some regime $\mathbf { \bar { \rho } } n \in [ [ 8 \dots 1 \bar { 4 } ] ] ,$ ). + +Table 4: The dropouts from Transformer-XL that we tune through PBT. + +
dropouta dropoute dropoutf dropouti dropoutoapplied to multi-head attention layers to remove words from embedding layer applied to positionwise ff layers for input embedding vectors applied to the output (before the logit)
+ +![](images/9d8f17d06528e3d7b011ed10a35b47166d7a1caabadcf5ea2941b47b1110b562.jpg) +Figure 4: Dropout schedule of the best run of ROMUL 32 workers on PTB + +A.2 LANGUAGE MODELING ON PENN TREE BANK + +![](images/1b2176a1a603c7a8451dac079474aa220be4550453ab1a8255906c1942111af9.jpg) +Figure 5: Lower dropout values are better early, but are outperformed by more strongly regularized models later (red, orange and blue lines) - here on wikitext103 with a 247M parameters language model from Fan et al. (2019) (Adaptive Inputs $^ +$ LayerDrop). PBT algorithms would tend to reduce dropout aggressively early on: after that, even if the dropout is increased later, the performance remains worse than training with a high dropout from the beginning (red line). Perhaps counterintuitively, this hints against increasing regularization over the course of the training - in the opposite, we observe that fine-tuning the model without dropout significatively improves test performance (purple line reaches 17.98 test perplexity) compared to the baseline (green: 18.42 test perplexity) + +A.4 SLIDES FOR INTERNAL PRESENTATION +Table 5: Perplexity (lower is better) on PTB for a Transformer-XL with 16 layers and 24M parameters + +
TrainingParallelismValidation PPLTest PPL
TransformerXL SOTA159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMUL PBT1657.8355.16
+ +Table 6: Classification error (lower is better) on CIFAR-10 and CIFAR-100 test sets for a WideResNet-28-10 (36M params) + +
AlgorithmCIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-103.918.8
RandAugment (Cubuk et al., 2019b)2.716.7
PBA (3 epochs/step) (Ho et al., 2019)2.616.7
ASHA2.817.6
Truncation Selection (PBA, ours)2.717.7
Initiator PBT (Li et al. (2019), ours)2.917.9
ROMUL2.817.1
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Population based training (PBT) is an effective tool for efficiently finding them as well as schedules over hyperparameters. In this paper, we compare existing PBT algorithms and contribute a new one: ROMUL, for RObust MULtistep search, which adapts its stepsize over the course of training. We report competitive results with standard models on CIFAR (image classification) as well as Penn Tree Bank (language modeling), which both depend on heavy regularization. We also open-source hoptim, a PBT library agnostic to the training framework, which is simple to use, reentrant, and provides good defaults with ROMUL. ", + "bbox": [ + 233, + 273, + 764, + 412 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 458, + 336, + 473 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hyperparameter tuning is essential for good performance in most machine learning tasks, and poses numerous challenges. First, optimal hyperparameter values can change over the course of training (schedules), e.g. for learning rate, fine tuning phases, data augmentation. Hyperparameters values are also rarely independent from each other (e.g. the magnitude of individual data augmentations depends on the number of data augmentations applied), and the search space grows exponentially with the number of hyperparameters. All of that search has to be performed within a computational budget, and sometimes even within a wall-clock time budget (e.g. models that are frequently retrained on new data), requiring efficient parallelization. In practice, competitive existing methods range from random search (Bergstra & Bengio, 2012) to more advanced methods (that aim at being more compute-efficient) like sequential search (Bergstra et al., 2011; 2013; Li et al., 2018), population based training (PBT, e.g. Jaderberg et al. (2017); Ho et al. (2019)) and search structured by the space of the hyperparameters (Liu et al., 2018; Cubuk et al., 2019b). ", + "bbox": [ + 174, + 496, + 825, + 662 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A major drawback of advanced hyperparameter optimization methods is that they themselves require attention from the user to reliably outperform random search. In this work, we empirically study the different training dynamics of data augmentation and regularization hyperparameters across vision and language modeling tasks, in particular for multistep (sequential) hyperparameter search. A common failure mode (i) is due to hyperparameters that have a different effect on the validation loss in the short and long terms, for instance using a smaller dropout often leads to faster but worse convergence. Another common problem (ii) is that successful searches are constrained on adequate “hyper-hyperparameters” (such as value ranges or the search policy used, which in current methods are non-adaptative mutation steps). Our contributions can be summarized as follows: ", + "bbox": [ + 174, + 670, + 825, + 794 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• We present a robust algorithm for leveraging population based training for hyperparameter search: ROMUL (RObust MULtistep) search, which addresses (i) and (ii). We empirically study its benefits and limitations, and show that it provides good defaults that compare favorably to existing methods. ", + "bbox": [ + 215, + 811, + 823, + 867 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• We open-source hoptim, a simple library for sequential hyperparameter search, that provides multiple optimizers (including ROMUL), as well as toy benchmarks showcasing hyperparameter optimization problems we identified empirically and standard datasets. ", + "bbox": [ + 217, + 882, + 821, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2 HYPERPARAMETER OPTIMIZATION WITH POPULATION-BASED TRAINING ", + "text_level": 1, + "bbox": [ + 169, + 103, + 805, + 117 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this article, we refer to the family of algorithms that continuously tunes hyperparameters of a set of models over the course of their training as “PBT algorithms” or “PBT optimizers”. Hyperparameter optimization is thus a zero order optimization performed at a slower frequency than the (often first order, e.g. SGD) optimization of the model. A PBT step happens typically after a fixed number of epochs or updates of the model, often optimizing the loss from the validation set, continuing from an already produced “parent” checkpoint, and producing and evaluating a new checkpoint. At every PBT step, hyperparameters can be updated (mutated), incremented or decremented by some number (step size), or sampled. ", + "bbox": [ + 174, + 137, + 825, + 248 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "There are multiple aspects to consider when designing a PBT algorithm. Technical constraints: how the optimization is distributed, with a centralized or decentralized algorithm, workers to run the trainings, how failed workers are handled. They are solved in a unified manner in the experiments we performed, by the hoptim library to implement and compare multiple algorithms. It is decoupled from the scheduling of the jobs and designed to accommodate adding more workers to scale up the training, or fewer when some are killed, for example through preemption or time-out on a shared cluster. Optimization method: how the hyper-parameters are modified throughout the training, for instance through mutations. Selection process: which individual of the population are kept, both in term of hyper-parameters and state of the neural network (checkpoint). For those last two points, some solutions are described below. ", + "bbox": [ + 174, + 256, + 825, + 393 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 CHALLENGES ", + "text_level": 1, + "bbox": [ + 174, + 417, + 310, + 431 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In order to have a clearer understanding of our proposed methods, we show below the main concerns we have observed in PBT: ", + "bbox": [ + 176, + 445, + 821, + 473 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Anisotropy: by definition, the optimal value of the hyperparameters considered is unknown, and oftentimes the range (or mutation scheme) provided to the algorithm is a loose estimate only. As modifying two hyperparameters with the same step size can produce effects with very different magnitudes, the user is required to to normalize the search space. But pre-tuning the hyperparameter tuner itself can be cumbersome as dynamics evolve during training. Section 3.1 provides an example based on the Rosenbrock function which illustrates this issue and highlights the interest of adaptative mutations. ", + "bbox": [ + 174, + 481, + 825, + 578 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Checkpoint vs. hyperparameters: comparing individuals in the population is extremely hard as improvements can be due to better hyperparameters, or better checkpoints (including potentially better batches). Better performance through better checkpoints is an optimization phenomenon (e.g. random restarts), that can bias the hyperparameter selection. We will detail this aspect in Section 4.2. ", + "bbox": [ + 176, + 585, + 825, + 641 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Short-term-long-term discordance: we observed empirically that hyperparameters which induce better performance in the short term are not always optimal in the longer term. This is a challenge that does not exist in classical static optimization, but is crucial for PBT since local minima are easy to reach and pose a danger for greedy algorithms. An example of such a parameter is the learning rate. Dropping the learning rate often induces a drop in the validation loss, even early in the training, and increasing it has the opposite effect, causing greedy PBT algorithms to reduce it to the minimum value too early, without being able to recover. We will detail this aspect in Section 4.1. ", + "bbox": [ + 174, + 647, + 825, + 746 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 DIFFERENTIAL EVOLUTION AND ROMUL ", + "text_level": 1, + "bbox": [ + 174, + 768, + 506, + 782 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Differential Evolution Storn & Price (1997) (DE) is a standard black-box optimization method, for minimizing $f : \\mathbb { R } ^ { n } \\mathbb { R }$ . It operates on a population $x ^ { i } \\in \\mathbb { R } ^ { n }$ for all $i \\stackrel { \\textstyle \\dag } { \\in } \\{ 1 , . . . , M \\}$ , $M \\geq 4$ , and indefinitely repeats the following steps for each individual $x ^ { \\mathrm { b a s e } }$ in the population to generate another individual called mutated vector that could replace $x ^ { \\mathrm { b a s e } }$ if better: ", + "bbox": [ + 176, + 796, + 825, + 853 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. given the best individual $x ^ { \\mathrm { b e s t } }$ which minimizes $f$ in the population, as well as two randomly selected ones $x ^ { a }$ and $x ^ { b }$ , compute the donor $d$ , which will give part of its coefficients to the mutated vector. In the current-to-best/1 scheme we use, these are the base coefficients plus a term attracting to the best set of coefficients from the current population, and an additional random variation (a standard value for $F _ { i }$ is $F _ { 1 } = F _ { 2 } = 0 . 8$ ): ", + "bbox": [ + 215, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 230, + 103, + 702, + 118 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b2c147b7a7b64b2d94c13ad0de1f303e04798fcfc96a3167c2ccd4363a258814.jpg", + "text": "$$\nd = x ^ { \\mathrm { b a s e } } + F _ { 1 } ( x ^ { \\mathrm { b e s t } } - x ^ { \\mathrm { b a s e } } ) + F _ { 2 } ( x ^ { b } - x ^ { a } )\n$$", + "text_format": "latex", + "bbox": [ + 379, + 125, + 676, + 143 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2. create the new mutated vector $\\widetilde { x } ^ { b a s e }$ by randomly selecting each component $j \\in \\{ 1 , . . . , n \\}$ of the base $x ^ { \\mathrm { b a s e } }$ eor the donor $d$ through the binary crossover operator: $\\begin{array} { r l } { \\widetilde { x } _ { j } ^ { \\mathrm { b a s e } } } & { { } = } \\end{array}$ $\\mathrm { C H O I C E } ( x _ { j } ^ { \\mathrm { b a s e } } , d _ { j } )$ e. This non-linear operation lets the optimization leave the vector span of the population. ", + "bbox": [ + 214, + 154, + 825, + 214 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3. compute $f ( \\widetilde { x } ^ { \\mathrm { b a s e } } )$ and replace $x ^ { \\mathrm { b a s e } }$ by $\\widetilde { x } ^ { \\mathrm { b a s e } }$ within the population if and only if $f ( \\widetilde { x } ^ { \\mathrm { b a s e } } ) \\leq$ $f ( x ^ { \\mathrm { b a s e } } )$ . ", + "bbox": [ + 205, + 218, + 823, + 247 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This method is interesting as it is already based on a population and adapts well to parameters with different dynamics while being simple and fully parallelizable. In particular, it does not rely on mutation ranges or step sizes - Equation 1 samples new parameters close to the current population, and as the population individuals go through selection this sampling is refined and becomes sharper around optimal values. In practice, if a parameter’s bounds are too loose or wrong, DE will eventually adapt after iterations of selection by removing individuals too far from the optimal value, and concentrate its computation budget on relevant values for this parameter. ", + "bbox": [ + 174, + 260, + 825, + 358 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order to use it for PBT, the set of hyper-parameters is converted to a vector in $\\mathbb { R } ^ { n }$ using nevergrad parametrization system (Rapin & Teytaud, 2018). However, this basic version of differential evolution (also implemented in nevergrad) is not adapted to PBT. Indeed the training function $f$ changes with the checkpoint as we are updating the parameters (not the hyperparameters) with a stochastic gradient from the task loss. The trend of $f$ is therefore typically downwards during the training, younger generations/later epochs tending to have a lower loss than their parents’, biasing the hyperperameter selection process in favor of those of the children (later steps of SGD updates) instead of in favor of better hyperparameters. ", + "bbox": [ + 174, + 364, + 825, + 476 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "ROMUL We therefore propose an adaptation: a population of $n$ individuals is trained, after finishing their step, individuals are compared to the rest of the population. If they have one of the $n / k$ best loss (we use $k \\ = \\ 2$ throughout), the training continues without changing the hyperparameters, otherwise, the hyperparameters are mutated. If the hyperparameters of an individual are mutated $m$ times in a row (we use $m \\ : = \\ : 3$ throughout), its checkpoint is killed and replaced by one of the $n / k$ best individuals. The values of $k$ and $m$ are hyperparameters, although we did not vary them in any experiments: $k = 2$ allows to have, on average, one alternative (mutated) version to each of the ones we keep training without hyperparameter change, and $m = 3$ proved to be robust across our experiments, to select when to discard a checkpoint. If using lower $m$ values, one should consider increasing the number of epochs per PBT step to prevent culling checkpoints too early (see Section 4.1 and 4.2). ", + "bbox": [ + 174, + 483, + 825, + 635 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The mutation scheme is adapted to fit this use case. In Eq. 1, $x ^ { \\mathrm { b a s e } }$ and $x ^ { \\mathrm { b e s t } }$ are both replaced by a randomly selected set of hyperparameters $x ^ { \\mathrm { { c } } }$ and $x ^ { \\mathrm { d } }$ from the best $n / 2$ individuals (“rand-to-rand/1” scheme following (Storn & Price, 1997; Das & Suganthan, 2011) notations). Replacing $x ^ { \\mathrm { b a s e } }$ aims at keeping the path through checkpoints unimodal, since keeping several modes with corresponding checkpoints is unnecessary. Replacing $x ^ { \\mathrm { b e s t } }$ by any other ”good” (top $50 \\%$ ) set of hyperparameters aims at avoiding early convergence, which we observed as one of the main problems during trainings. This also avoids a strong bias by a good checkpoint (more on this in 4.2). To avoid duplication of hyperparameters, we opt for making $F _ { 1 }$ and $F _ { 2 }$ random vectors instead of using the binary crossover non-linearity. In order to keep the initial scaling of DE, we chose ${ \\bf F _ { 1 } } [ i ]$ uniformly distributed between 0 and $2 F$ (we use the common value for $F$ from vanilla DE: $F = 0 . 8$ ), and $\\mathbf { F _ { 2 } } [ i ] = 2 F - \\mathbf { F _ { 1 } } [ i ]$ , $\\forall i$ . This ensures that the sum $\\mathbf { F _ { 1 } } [ i ] + \\mathbf { F _ { 2 } } [ i ] = 2 F $ $\\forall i$ , as in vanilla DE. With $\\odot$ the elementwise multiplication, this yields $d = x ^ { c } + \\mathbf { \\bar { F _ { 1 } } } \\odot ( x ^ { d } - x ^ { c } ) + \\mathbf { F _ { 2 } } \\odot ( x ^ { b } - x ^ { a } )$ . ", + "bbox": [ + 173, + 642, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 OTHER ALGORITHMS ", + "text_level": 1, + "bbox": [ + 176, + 827, + 361, + 842 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In our experiments, we compare several algorithms briefly presented below. We aim to compare how effective they can be for practical use-cases of hyperparameter tuning, where the user does not want to tune the hyperparameters of its hyperparameter tuner, and desires meaningful defaults. The only input they take are the number of parallel trainings, the range of hyperparameters, and a hint for an initial value (e.g. the same value 0 for dropout values and data augmentation magnitudes). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Initiator PBT: We reimplemented Initiator Based Evolution, presented in Li et al. (2019). New hyper-parameters are sampled from parent hyper-parameters by adding/removing a mutation constant (for instance d $r o p o u t C h i l d = d r o p o u t P a r e n t \\pm 0 . 1 )$ . A newly created checkpoint is compared to a randomly sampled checkpoint in the population: if the latter is better, the new checkpoint is discarded and the latter is forked with its hyperparameters - this ensures that only the best performing models remain eventually, and allows it to run asynchronously. For each parameter, we specify a range, and use $( h i - l o ) / 3 0$ as a mutation constant unless specified otherwise. ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Truncation Selection: $N$ models are trained in parallel. Regularly, the $M$ worst performing models are stopped and replaced with clones of the $M$ bests, and hyperparameters are randomly perturbated. This scheme was first introduced in Jaderberg et al. (2017). In our experiments, we use $M = N / 4$ . For hyperparameter perturbation, we generalize the mutation scheme introduced in Ho et al. (2019): each parameter is sampled uniformly in its range $[ l o , h i ]$ with $20 \\%$ probability, or incremented by random.choice([-3, -2, $^ { - 1 }$ , 0, 0, 1, 2, 3]) $\\star$ (hi - lo) / 10 and then clipped to stay within $[ l o , h i ]$ . ", + "bbox": [ + 174, + 208, + 825, + 306 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "ASHA: This is not a PBT algorithm, but a strong hyperparameter search algorithm that we compare to. In the Asynchronous Successive Halving Algorithm (ASHA, Li et al. (2018)), hyperparameters are sampled uniformly like in Random Search, but models are evaluated early and stopped if not in the top $1 / \\eta$ percentile. For a given model, the first evaluation can happen after $1 , \\eta , \\eta ^ { 2 }$ , .. steps, making this algorithm robust to hyperparameters whose optimal value does not perform well until late in the training (Section 4.1). Unlike Initiator PBT or Truncation Selection, ASHA finds constant values for hyperparameters rather than schedules. We set the reduction factor $\\eta$ to 3. ", + "bbox": [ + 174, + 313, + 825, + 410 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 429, + 326, + 445 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We ran experiments on a toy optimization problem (the Rosenbrock function), CIFAR, and Penn Tree Bank, all with the same ROMUL hyperparameters to test its robustness. Each of these experiments train in around 100 to 300 epochs, and we used 1 step per epoch, so that they all have similar time scales. ", + "bbox": [ + 174, + 460, + 825, + 516 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 EXAMPLE ON A TOY OPTIMIZATION PROBLEM ", + "text_level": 1, + "bbox": [ + 174, + 532, + 539, + 547 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Current PBT mutation schemes have fixed steps and therefore do not automatically adapt to the landscape of the optimized function. This means that they are not well-suited for anisotropic problems, which often arise in real life applications since some hyperparameters may be very important to tune finely, while other do not require the same precision. To highlight this issue, we experiment below on the Rosenbrock function: $\\mathcal { R } _ { a , b } ( x , y ) = ( a - x ) ^ { 2 } + b ( y - x ^ { 2 } ) ^ { 2 }$ ", + "bbox": [ + 174, + 559, + 825, + 630 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We will aim at minimizing $\\mathcal { R } _ { 1 , 1 0 0 }$ through the surrogate $\\mathcal { R } _ { \\hat { a } , \\hat { b } }$ , with $\\hat { a }$ and $\\hat { b }$ two hyperparameters handled with PBT. We initialize both parameters at 20 and bound them by -12.12 and 212.12 (using integers would be a special case since actual $a$ and $b$ values are integers). This experiment can be reproduced using the hoptim toolbox with the command: hop bench rosenbrock. ", + "bbox": [ + 174, + 638, + 825, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While standard PBT with random steps wastes mutations on $\\hat { a }$ , DE is able to adapt its step-size to large steps on $\\hat { a }$ until getting close, then smaller steps on $\\hat { a }$ to tune $\\hat { a }$ and $\\hat { b }$ more finely. This is visible in Fig. 1a with ROMUL values of $\\hat { a }$ converging quickly to around 1. The mutations then become sharper, while the ones for Initiator-PBT (small steps) are still too large and oscillate around the optimal value. Arguably, the mutation step could have been even smaller, but that would have slowed down the convergence, and these steps would be painful meta-parameters to tune at scale. Initiator-PBT with larger steps and Truncation selection are not displayed in this figure because their variations are too large. ", + "bbox": [ + 174, + 703, + 825, + 818 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The impact on the loss $\\hat { \\mathcal { R } }$ is then visible in Fig. 1b: Initiator PBT can’t decrease past 0.2 with large steps, and 0.048 with smaller steps, since it is trapped trying to optimize $\\hat { a }$ while DE is able to reach better values. Fig. 1c and 1d show the trajectory of $( x , y )$ for Truncation selection and ROMUL, with the same number of training steps. Truncation selection is hampered by more random mutations. On the other hand, ROMUL is able to reach a much lower value after exhibiting a more chaotic behavior when it initially adapts to the scale of the problem. The trajectory for both versions of Initiator-PBT can be found in Fig. 2 of the appendix. Initiator-PBT with large steps (Fig. 2a) moves ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/3d5d48c5c340c00fcf8b3d8f4759575d093be9629d0fa63e3af03d559266f1b5.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 197, + 119, + 470, + 276 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/3ba09c208b2a0a1eb0ad236dae6d70897fd8ce52659c2eb40bad71ba2576af7e.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 531, + 119, + 807, + 276 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(a) Rosenbrock parameter $\\hat { a }$ with respect to the number of steps (optimal at 1). After ${ \\approx } 5 0 $ steps, ROMUL’s distribution for $\\hat { a }$ gets sharper around 1, Initiator always uses an hardcoded mutation step size. ", + "bbox": [ + 173, + 292, + 493, + 343 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(b) Loss $\\mathcal { R } _ { 1 , 1 0 0 }$ with respect to the number of steps (lower is better, minimum value is 0). ROMUL keeps adapting and decreasing while other optimizers are locked to higher levels depending on their step sizes. ", + "bbox": [ + 513, + 292, + 833, + 344 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/34a9a888f74479fa60039e297e33e410ba7750db6eac0ea6f9cd1d5621e931e2.jpg", + "image_caption": [ + "Figure 1: Training on the Rosenbrock benchmark. ROMUL outperforms initiator and truncation selection because it can adapt its step size. Bottom plots: Trajectories of 100 PBT training steps (16 jobs per step) on the Rosenbrock function with $a = 1$ and $b = 1 0 0$ (minimum at the red cross $( 1 , 1 )$ , trajectories go from blue to green) " + ], + "image_footnote": [], + "bbox": [ + 184, + 363, + 805, + 549 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "very slowly to the minimum because of big extra oscillations. With smaller steps (Fig. 2b), it reaches better values through a slow and non-direct path. Tab. 3 in the appendix provides quantitative results by averaging over 20 runs, including a version of Initiator PBT in which updates are performed through multiplications by 0.8 or 1.2. In particular, ROMUL performs statically better than all over optimizers on this testbed $( p \\textless 1 . 1 e \\_ 5$ with a two sample Welch’s t-test). In Fig. 3 in Appendix, we also show the behavior with more variables by performing optimization on an average of Rosenbrocks functions, each with independent $a$ and $b$ variable to be estimated by PBT. Overall ROMUL performs consistently well across the board for a wide range of number of variables. ", + "bbox": [ + 174, + 643, + 825, + 755 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 APPLICATION TO CIFAR (IMAGE CLASSIFICATION) ", + "text_level": 1, + "bbox": [ + 174, + 772, + 570, + 787 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we compare various algorithms for tuning hyperparameters for image classification on CIFAR (Krizhevsky et al., 2009). We reproduce the population based augmentation (PBA) setup from (Ho et al., 2019) with their original implementation. Our algorithms train a Wide-ResNet-28- 10 model on Reduced CIFAR-10 (using $10 \\%$ , i.e. 4000 images, of the training set for actual training, and the remainder as a validation set), and optimize the same 60 hyperparameters as in Ho et al. (2019): 2 magnitudes and 2 probabilities for each of the 15 possible data augmentations. For each algorithm, we take the best model in the validation set at epoch 200, and use its hyperparameters schedules to train another Wide-ResNet-28-10 model on CIFAR-10 and CIFAR-100 and finally report test accuracies at epoch 200 in Table 1. Trainings can be reproduced with the hoptim package and its benchmarking counterpart hoptim benchmarks in the cifar folder. ROMUL recovers most of the gains on CIFAR-10 $2 . 8 \\%$ error vs. $2 . 6 \\%$ for the SOTA and $3 . 9 \\%$ for the baseline), and is a bit further away on CIFAR-100 ( $1 7 . 1 \\%$ vs. $1 6 . 7 \\%$ for the SOTA and $1 8 . 8 \\%$ for the baseline). PBA, which yields state-of-the-art results on CIFAR, used Truncation selection PBT introduced in (Jaderberg et al., 2017), which we implemented and compared to. We adopted all the PBA hyperparameters and observe $2 . 7 \\%$ on CIFAR-10 (ROMUL: $2 . 8 \\%$ ) and $1 7 . 7 \\%$ on CIFAR-100 (ROMUL: $1 7 . 1 \\%$ ). The differences in the job and population management in hoptim may explain the difference between our implementation and theirs, which is particularly marked on the training set reduced CIFAR-10: $12 . 8 \\%$ for their vs. $1 3 . 9 \\%$ for our implementation. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/c119733ab102874dc37e20adabfbaad72ffd703d45fcb30ce782b017b9a34c8c.jpg", + "table_caption": [ + "Table 1: Classification error (lower is better) on CIFAR-10 and CIFAR-100 test sets for a WideResNet-28-10 (36M params). The algorithms run with 16 workers in parallel with the same compute budget (except when stated otherwise) on reduced CIFAR-10. After that, the schedule found is used for training the same model from scratch on CIFAR-10 and CIFAR-100 " + ], + "table_footnote": [], + "table_body": "
AlgorithmReduced CIFAR-10 (10%)CIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-10n/a3.918.8
RandAugment (Cubuk et al., 2019b)n/a2.716.7
PBA (3 epochs/step) (Ho et al., 2019)12.82.616.7
ASHA14.72.817.6
ASHA (running for double the time)14.12.717.2
Truncation Selection (PBA, ours)13.92.717.7
Initiator PBT (Li et al. (2019), ours)14.72.917.9
ROMUL14.02.817.1
", + "bbox": [ + 174, + 320, + 823, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As PBA waits for more than 1 epoch/step to evaluate a set of hyperparameters, we compared 1 epoch/step and 3 epochs/step (their setting), as it could help thwarting short-term/long-term discrepancy effects (see 4.1) and noise as explained above, but we could not identify a sufficiently generic scheme for all applications. In general, this is part of the PBT hyperparameters that are tuned in PBA, that we try to completely remove as hyperparameters in ROMUL, by being insensitive to it (in this case it does not seem to affect Truncation Selection either). For this hyperparameter, the constraint is to do PBT steps slow enough so that the number of updates is sufficiently large for the model to adapt to new mutated hyperparameters, and high enough so that PBT has enough steps to optimize the hyperparameters. In practice, trainings are long enough (regarding the number of SGD updates of the model) for a wide range of PBT steps frequencies to work. ", + "bbox": [ + 173, + 491, + 825, + 631 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 APPLICATION TO THE PENN TREEBANK DATASET (LANGUAGE MODELING) ", + "text_level": 1, + "bbox": [ + 176, + 652, + 728, + 666 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We experiment with the TransformerXL model (Dai et al., 2019) on the PTB dataset (Marcus, 1993). TranformerXL’s code is open-source and is the state-of-the-art for tranformer models on this dataset when using proper regularization, making it an interesting challenge for PBT. It comes with several dropout hyperparameters: we search for optimal values for five different dropout hyperparameters, that we describe in Table 4 in appendix. They are all initialized to 0 with standard deviation of 0.1 for ROMUL (negative values are reflected to positive values), hence not at the baseline values. ", + "bbox": [ + 174, + 680, + 825, + 763 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results are reported in Table 2. Trainings can be reproduced (up to random variance) with the hoptim package and its benchmarking counterpart hoptim benchmarks in the ptb folder. The baseline TransformerXL was obtained with the author’s code and is close to the one reported in the initial paper. Noticeably, ASHA and Random Search (with a uniform prior) are not able to come close to the baseline, with more than 4 points difference in both validation and test perplexity (PPL). Truncation selection and Initiator PBT on the other hand are able to reach the baseline although they were not able to excel it in test perplexity. Only ROMUL is able to reach (marginally) better results than the reproduced baseline in test PPL with both 16 and 32 workers. A found dropout schedule is displayed in the appendix (Fig. 4) and show dropouts rapidly increasing in the beginning and stabilizing to different levels. Using 16 and 32 workers provided similar results up to noise for ROMUL (in this very case, 32 workers does not actually perform better than with 16 workers). ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, using 8 workers results in a notable drop in performance for all optimizers (Test PPL ROMUL 56.39, TruncSel 57.98, Initiator 56.33). ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/1282b189082be409934a10dd9d69d5aa0728707999c599d118e0fe531361a48d.jpg", + "table_caption": [ + "Table 2: Perplexity (lower is better) on PTB for a Transformer-XL with 16 layers and 24M parameters, best validation PPL before iteration 175 and corresponding test PPL, given the resources needed these values are not averaged, numbers excelling our training baseline are in bold. " + ], + "table_footnote": [], + "table_body": "
TrainingworkersValidation PPLTest PPL
TransformerXL SOTA (Dai et al. (2019))1/54.52
TransformerXL SOTA (our training, their code)159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMULPBT1657.8355.16
Random Search3263.8460.90
ASHA3264.3161.63
Truncation Selection PBT3258.4555.93
Initiator PBT3259.3655.73
ROMUL PBT3258.6355.28
", + "bbox": [ + 196, + 199, + 800, + 395 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 431, + 310, + 446 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 SHORT TERM - LONG TERM DISCORDANCE", + "text_level": 1, + "bbox": [ + 174, + 463, + 511, + 477 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We have observed on PTB and other applications that some hyperparameters were never contributing positively to the model’s performance in the short term (eg: 1 step) but could become better on a longer term (eg: 5 steps or more), hence checkpoints need time to adapt to a new parameter set (e.g.: building more redundancy). In an experiment on PTB, we used one epoch per step for half the training and then modified it to 10 epochs per step. We observed that increasing one of the dropout contributed negatively for small steps (1 epoch), but positively for long steps (10th epochs). This is a major roadblock for PBT-based approaches since two models with different hyperparameters can’t be straightforwardly compared at every step, but only after an unknown delay. This is partially handled by being conservative on models to keep: keeping the best $50 \\%$ unchanged in ROMUL, or the random tournament scheme that allows bad models to continue in Initiator PBT when assigned an even worse opponent. Fig. 5 in appendix shows such an example in another domain: a large dropout seems very detrimental early on, but very beneficial in the longer term. While this is expected for regularizations, we observe that a straightforward schedule increasing the dropout in steps (in red) is not able to compensate this - we observed the same effect with a continuous schedule. This behavior adds complexity to the task of PBT algorithms, because bad early choices can’t be compensated later. Arguably, it can be due to interactions with the learning rate scheduler used and a more appropriate schedule could help solve this issue (although it is not clear what such a schedule should be). ", + "bbox": [ + 173, + 488, + 825, + 724 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 CHECKPOINTS VS HYPERPARAMETERS - SELECTION BIASES ", + "text_level": 1, + "bbox": [ + 174, + 743, + 632, + 757 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For PBT optimizers, one critical question is how much of the loss difference between two individuals is caused by different hyperparameters, and how much about different checkpoints. Both contributions are tightly entwined making it harder to identify which hyperparameters are best. ", + "bbox": [ + 174, + 770, + 823, + 811 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A naive initial option to counter this is to always start a PBT step from the best checkpoint in the population. In our experiments this performed worse, at least because of the noise in the evaluation metric, but also because of the short-term long-term discrepancy detailed above (Section 4.1). We also expect that doing so could make optimizers less robust by getting trapped in local minima too easily, or aggressively discarding more promising models in the longer term. ", + "bbox": [ + 174, + 819, + 823, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "On the opposite side of the spectrum, we experimented with never culling checkpoints, effectively performing $n$ full trainings in parallel. Conceptually, keeping checkpoints is attractive since it should add robustness to the optimizer: selected hyperparameters have to work well for more than one particular checkpoint. Indeed, this way the performance can be attributed to actual parameter schedules fitting different trainings, instead of being biased by checkpoints culling/random restarts. It also adds more variability which could be beneficial especially with respect to the short-term long-term discrepancy. That being said, we have neither observed significant improvement nor deterioration when keeping checkpoints, as long as the mutation schemes were not biased towards the best set of hyperparameters (e.g.: removing the $x ^ { \\mathrm { b e s t } }$ term in Eq. 1), because doing so can make all hyperparameters converge towards the best checkpoint, making the optimization process early converge to values which are not necessarily adapted to other checkpoints. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Still, even with ROMUL’s loss-agnostic mutation scheme, some checkpoints were observed to fall behind and waste resources if not culled, so we expect that a trade-off like the one we implemented (killing checkpoints after 3 failed mutations in a row) is necessary. ", + "bbox": [ + 176, + 236, + 825, + 277 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Another source of selection bias is noise. While the trainings are well behaved in PTB because the shuffling of the training set is synchronized by epoch, the trainings in CIFAR are much noisier because of the randomness introduced by data augmentation and the very small training set size. PBT optimizers based on more noise-robust blackbox optimization methods could be beneficial, but it is not clear how to adapt them. ", + "bbox": [ + 176, + 285, + 825, + 354 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 383, + 339, + 398 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Several families of methods exist for tuning hyperparameters of neural networks. Methods closest to grid search like random search (Bergstra & Bengio, 2012) and ASHA (Li et al., 2018) are based on minimal constraints and can be parallelized extensively. Methods striving for more data-efficient search (Bergstra et al., 2011; 2013; Feurer & Hutter, 2019) are more sequential in nature, requiring convergence of some trainings before launching new ones. Population-based training approaches (Jaderberg et al., 2017; Ho et al., 2019; Li et al., 2019) loosen the requirements of training different models, as hyperparameters are changed on-the-fly during training, which also makes the search for schedules easier and less structured, i.e. not based on a predefined function. ", + "bbox": [ + 174, + 419, + 825, + 531 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recent advances in automatic discovery of data augmentation policies include Population Based Augmentation (Ho et al., 2019) which we compared to in this paper (denoted Truncation Selection). Another line of work on structuring the hyperparameter space for data augmentation policy search is AutoAugment (Cubuk et al., 2019a), FastAutoAugment Lim et al. (2019) and RandAugment Cubuk et al. (2019b), the later being faster and reaching top performance on CIFAR. ", + "bbox": [ + 174, + 537, + 823, + 608 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "PBT is used successfully in reinforcement learning (Jaderberg et al., 2017), providing diversity in self-play and progressive difficulty, so other experimental comparisons that we did include Initiator PBT from (Li et al., 2019), which presented a generic PBT setup that inspired hoptim. For nonPBT baselines we used random search (Bergstra & Bengio, 2012), and ASHA (Li et al., 2018), which is an update on HyperBand (Li et al., 2017). ", + "bbox": [ + 174, + 614, + 823, + 684 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 713, + 318, + 728 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We introduced ROMUL, a robust PBT algorithm that we benchmarked on standard datasets with multiple regularization and data augmentation hyperparameters. Its main strength comes from its robustness to hyperparameters definitions by automatically adapting to the scale of each parameter. Although it did not show better performance on CIFAR than PBA – that was tuned for this benchmark – we demonstrated that it is more robust to domain changes. More importantly for the practical use-cases, it constitutes a good default that does not require extensive tuning to work well. We open-sourced its implementation as well as a simple and broadly compatible PBT library. ", + "bbox": [ + 174, + 750, + 823, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The main difficulties we observed for PBT-based optimizers came from short-term vs. long-term effects: parameters can have a positive impact in the short term but a negative one in the longer term which may not be rectifiable. Learning rate falls in this category, since decreasing it often provides quick gains at the risk of being trapped in a local minimum. Studying how to deal with such behaviors is in our opinion the main challenge of future work. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 118 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. The Journal of Machine Learning Research, 13(1):281–305, 2012. ", + "bbox": [ + 171, + 126, + 823, + 155 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "James Bergstra, Dan Yamins, and David D Cox. Hyperopt: A python library for optimizing the hyperparameters of machine learning algorithms. In Proceedings of the 12th Python in science conference, volume 13, pp. 20. Citeseer, 2013. ", + "bbox": [ + 173, + 164, + 823, + 207 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "James S Bergstra, Remi Bardenet, Yoshua Bengio, and Bal ´ azs K ´ egl. 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", + "bbox": [ + 178, + 141, + 823, + 183 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A APPENDIX ", + "bbox": [ + 176, + 102, + 299, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.1 ROSENBROCK EXPERIMENTS ", + "bbox": [ + 174, + 136, + 419, + 150 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/0d4070f821f96f59c4fb0727906a39d95055383043b96afb6151f9feaaf12148.jpg", + "image_caption": [ + "Figure 2: Trajectories of 100 Initiator PBT training steps (16 jobs per step) on the Rosenbrock function with $a = 1$ and $b = 1 0 0$ (minimum at the red cross $( 1 , 1 )$ , trajectories go from blue to green) " + ], + "image_footnote": [], + "bbox": [ + 196, + 188, + 790, + 372 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/3de502b855bdeaf82c55d0334ef979627288af3b4bb9f4e75aa3d8d4d16a4ae1.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Trainingmean ( (log10)std (log10)
Initiator (0.8/1.2 mult. steps)-1.180.045
Initiator (big steps)-0.7070.069
Initiator (small steps)-0.9920.143
ROMUL-2.1010.678
Truncation Selection-0.8340.327
", + "bbox": [ + 294, + 454, + 702, + 553 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/a79afcffc16883a48c9c4c8102cf27b411ab7049370779458a2e86bb3b4580d2.jpg", + "image_caption": [ + "Table 3: Final loss (in log10) mean and standard deviation for independent runs on the Rosenbrock testbed, computed over 20 runs (two-sample Welsh’s test provides $p < 1 . 1 e - 5$ when comparing each algorithm with ROMUL). ", + "Figure 3: Score on the multi-variate Rosenbrock benchmark (explained in 3.1) over 20 experiments for each point. Lower is better, standard deviations are indicated. ROMUL performs well across the board for a wide range of number of variables, being surpassed only by Truncated selection in some regime $\\mathbf { \\bar { \\rho } } n \\in [ [ 8 \\dots 1 \\bar { 4 } ] ] ,$ ). " + ], + "image_footnote": [], + "bbox": [ + 271, + 640, + 725, + 840 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/e7fbc781722893e27ff18a176de83cf5d72c1aaa293d369d044fcd4224829de1.jpg", + "table_caption": [ + "Table 4: The dropouts from Transformer-XL that we tune through PBT. " + ], + "table_footnote": [], + "table_body": "
dropouta dropoute dropoutf dropouti dropoutoapplied to multi-head attention layers to remove words from embedding layer applied to positionwise ff layers for input embedding vectors applied to the output (before the logit)
", + "bbox": [ + 320, + 133, + 678, + 213 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/9d8f17d06528e3d7b011ed10a35b47166d7a1caabadcf5ea2941b47b1110b562.jpg", + "image_caption": [ + "Figure 4: Dropout schedule of the best run of ROMUL 32 workers on PTB " + ], + "image_footnote": [], + "bbox": [ + 217, + 246, + 746, + 441 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 LANGUAGE MODELING ON PENN TREE BANK ", + "bbox": [ + 174, + 493, + 534, + 508 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/1b2176a1a603c7a8451dac079474aa220be4550453ab1a8255906c1942111af9.jpg", + "image_caption": [ + "Figure 5: Lower dropout values are better early, but are outperformed by more strongly regularized models later (red, orange and blue lines) - here on wikitext103 with a 247M parameters language model from Fan et al. (2019) (Adaptive Inputs $^ +$ LayerDrop). PBT algorithms would tend to reduce dropout aggressively early on: after that, even if the dropout is increased later, the performance remains worse than training with a high dropout from the beginning (red line). Perhaps counterintuitively, this hints against increasing regularization over the course of the training - in the opposite, we observe that fine-tuning the model without dropout significatively improves test performance (purple line reaches 17.98 test perplexity) compared to the baseline (green: 18.42 test perplexity) " + ], + "image_footnote": [], + "bbox": [ + 264, + 555, + 728, + 791 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/81ae4a2291474674391b03e335eb90715b11b91785529413e0b73732db9154f7.jpg", + "table_caption": [ + "A.4 SLIDES FOR INTERNAL PRESENTATION ", + "Table 5: Perplexity (lower is better) on PTB for a Transformer-XL with 16 layers and 24M parameters " + ], + "table_footnote": [], + "table_body": "
TrainingParallelismValidation PPLTest PPL
TransformerXL SOTA159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMUL PBT1657.8355.16
", + "bbox": [ + 256, + 132, + 740, + 239 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/9e708030827ae3a94bc4e81b412b902088a0bee9734868a30255d12d75c7c32a.jpg", + "table_caption": [ + "Table 6: Classification error (lower is better) on CIFAR-10 and CIFAR-100 test sets for a WideResNet-28-10 (36M params) " + ], + "table_footnote": [], + "table_body": "
AlgorithmCIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-103.918.8
RandAugment (Cubuk et al., 2019b)2.716.7
PBA (3 epochs/step) (Ho et al., 2019)2.616.7
ASHA2.817.6
Truncation Selection (PBA, ours)2.717.7
Initiator PBT (Li et al. (2019), ours)2.917.9
ROMUL2.817.1
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We empirically", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 141, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "study its benefits and limitations, and show that it provides good defaults that compare", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 677, + 265, + 688 + ], + "spans": [ + { + "bbox": [ + 142, + 677, + 265, + 688 + ], + "score": 1.0, + "content": "favorably to existing methods.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 133, + 699, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 698, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 132, + 698, + 504, + 712 + ], + "score": 1.0, + "content": "• We open-source hoptim, a simple library for sequential hyperparameter search, that pro-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "vides multiple optimizers (including ROMUL), as well as toy benchmarks showcasing hy-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 721, + 482, + 733 + ], + "spans": [ + { + "bbox": [ + 141, + 721, + 482, + 733 + ], + "score": 1.0, + "content": "perparameter optimization problems we identified empirically and standard datasets.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 97 + ], + "score": 1.0, + "content": "ROMUL: SCALE ADAPTATIVE POPULATION BASED", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 98, + 185, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 98, + 185, + 118 + ], + "score": 1.0, + "content": "TRAINING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 112, + 136, + 245, + 159 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 201 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 201 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 217, + 468, + 327 + ], + "lines": [ + { + "bbox": [ + 142, + 218, + 469, + 230 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 469, + 230 + ], + "score": 1.0, + "content": "In most pragmatic settings, data augmentation and regularization are essential,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 228, + 469, + 241 + ], + "spans": [ + { + "bbox": [ + 141, + 228, + 469, + 241 + ], + "score": 1.0, + "content": "and require hyperparameter search. Population based training (PBT) is an effec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 239, + 469, + 252 + ], + "spans": [ + { + "bbox": [ + 141, + 239, + 469, + 252 + ], + "score": 1.0, + "content": "tive tool for efficiently finding them as well as schedules over hyperparameters. In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 250, + 470, + 263 + ], + "spans": [ + { + "bbox": [ + 141, + 250, + 470, + 263 + ], + "score": 1.0, + "content": "this paper, we compare existing PBT algorithms and contribute a new one: RO-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 261, + 469, + 274 + ], + "spans": [ + { + "bbox": [ + 141, + 261, + 469, + 274 + ], + "score": 1.0, + "content": "MUL, for RObust MULtistep search, which adapts its stepsize over the course", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 272, + 470, + 285 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 470, + 285 + ], + "score": 1.0, + "content": "of training. 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(2019)) and search structured by the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 514, + 381, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 381, + 527 + ], + "score": 1.0, + "content": "space of the hyperparameters (Liu et al., 2018; Cubuk et al., 2019b).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 393, + 505, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "A major drawback of advanced hyperparameter optimization methods is that they themselves require", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "attention from the user to reliably outperform random search. 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A", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "common failure mode (i) is due to hyperparameters that have a different effect on the validation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "loss in the short and long terms, for instance using a smaller dropout often leads to faster but worse", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 609 + ], + "score": 1.0, + "content": "convergence. Another common problem (ii) is that successful searches are constrained on adequate", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "“hyper-hyperparameters” (such as value ranges or the search policy used, which in current methods", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 619, + 447, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 447, + 630 + ], + "score": 1.0, + "content": "are non-adaptative mutation steps). Our contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 531, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 643, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 132, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 132, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "• We present a robust algorithm for leveraging population based training for hyperparameter", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 141, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "search: ROMUL (RObust MULtistep) search, which addresses (i) and (ii). We empirically", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 141, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "study its benefits and limitations, and show that it provides good defaults that compare", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 677, + 265, + 688 + ], + "spans": [ + { + "bbox": [ + 142, + 677, + 265, + 688 + ], + "score": 1.0, + "content": "favorably to existing methods.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 132, + 643, + 505, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 699, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 698, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 132, + 698, + 504, + 712 + ], + "score": 1.0, + "content": "• We open-source hoptim, a simple library for sequential hyperparameter search, that pro-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "vides multiple optimizers (including ROMUL), as well as toy benchmarks showcasing hy-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 721, + 482, + 733 + ], + "spans": [ + { + "bbox": [ + 141, + 721, + 482, + 733 + ], + "score": 1.0, + "content": "perparameter optimization problems we identified empirically and standard datasets.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 132, + 698, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 104, + 82, + 493, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 495, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 495, + 95 + ], + "score": 1.0, + "content": "2 HYPERPARAMETER OPTIMIZATION WITH POPULATION-BASED TRAINING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 109, + 505, + 197 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 122 + ], + "score": 1.0, + "content": "In this article, we refer to the family of algorithms that continuously tunes hyperparameters of a set of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 104, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "models over the course of their training as “PBT algorithms” or “PBT optimizers”. Hyperparameter", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "optimization is thus a zero order optimization performed at a slower frequency than the (often first", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 154 + ], + "score": 1.0, + "content": "order, e.g. SGD) optimization of the model. A PBT step happens typically after a fixed number of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "epochs or updates of the model, often optimizing the loss from the validation set, continuing from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 178 + ], + "score": 1.0, + "content": "an already produced “parent” checkpoint, and producing and evaluating a new checkpoint. At every", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "score": 1.0, + "content": "PBT step, hyperparameters can be updated (mutated), incremented or decremented by some number", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 200, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 200, + 198 + ], + "score": 1.0, + "content": "(step size), or sampled.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "There are multiple aspects to consider when designing a PBT algorithm. Technical constraints:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 504, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 504, + 225 + ], + "score": 1.0, + "content": "how the optimization is distributed, with a centralized or decentralized algorithm, workers to run the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "trainings, how failed workers are handled. They are solved in a unified manner in the experiments we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "performed, by the hoptim library to implement and compare multiple algorithms. It is decoupled", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "from the scheduling of the jobs and designed to accommodate adding more workers to scale up the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "training, or fewer when some are killed, for example through preemption or time-out on a shared", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "cluster. Optimization method: how the hyper-parameters are modified throughout the training, for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "instance through mutations. Selection process: which individual of the population are kept, both in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "term of hyper-parameters and state of the neural network (checkpoint). For those last two points,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 252, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 252, + 313 + ], + "score": 1.0, + "content": "some solutions are described below.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 331, + 190, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 192, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 192, + 344 + ], + "score": 1.0, + "content": "2.1 CHALLENGES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 353, + 503, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "In order to have a clearer understanding of our proposed methods, we show below the main concerns", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 365, + 213, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 213, + 375 + ], + "score": 1.0, + "content": "we have observed in PBT:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Anisotropy: by definition, the optimal value of the hyperparameters considered is unknown, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "oftentimes the range (or mutation scheme) provided to the algorithm is a loose estimate only. As", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "modifying two hyperparameters with the same step size can produce effects with very different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "magnitudes, the user is required to to normalize the search space. But pre-tuning the hyperparameter", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 426, + 504, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 504, + 437 + ], + "score": 1.0, + "content": "tuner itself can be cumbersome as dynamics evolve during training. Section 3.1 provides an example", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "based on the Rosenbrock function which illustrates this issue and highlights the interest of adaptative", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 151, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 151, + 459 + ], + "score": 1.0, + "content": "mutations.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 108, + 464, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "Checkpoint vs. hyperparameters: comparing individuals in the population is extremely hard as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "improvements can be due to better hyperparameters, or better checkpoints (including potentially", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 484, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 505, + 501 + ], + "score": 1.0, + "content": "better batches). Better performance through better checkpoints is an optimization phenomenon (e.g.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "random restarts), that can bias the hyperparameter selection. We will detail this aspect in Section 4.2.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "Short-term-long-term discordance: we observed empirically that hyperparameters which induce", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "better performance in the short term are not always optimal in the longer term. This is a challenge", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 534, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 550 + ], + "score": 1.0, + "content": "that does not exist in classical static optimization, but is crucial for PBT since local minima are easy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "to reach and pose a danger for greedy algorithms. An example of such a parameter is the learning", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "score": 1.0, + "content": "rate. Dropping the learning rate often induces a drop in the validation loss, even early in the training,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "and increasing it has the opposite effect, causing greedy PBT algorithms to reduce it to the minimum", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 580, + 453, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 453, + 592 + ], + "score": 1.0, + "content": "value too early, without being able to recover. We will detail this aspect in Section 4.1.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 609, + 310, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 312, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 312, + 622 + ], + "score": 1.0, + "content": "2.2 DIFFERENTIAL EVOLUTION AND ROMUL", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Differential Evolution Storn & Price (1997) (DE) is a standard black-box optimization method, for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 156, + 655 + ], + "score": 1.0, + "content": "minimizing", + "type": "text" + }, + { + "bbox": [ + 156, + 643, + 213, + 654 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 642, + 332, + 655 + ], + "score": 1.0, + "content": ". It operates on a population", + "type": "text" + }, + { + "bbox": [ + 332, + 642, + 370, + 653 + ], + "score": 0.92, + "content": "x ^ { i } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 642, + 400, + 655 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 401, + 642, + 463, + 655 + ], + "score": 0.86, + "content": "i \\stackrel { \\textstyle \\dag } { \\in } \\{ 1 , . . . , M \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 642, + 468, + 655 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 468, + 643, + 501, + 654 + ], + "score": 0.8, + "content": "M \\geq 4", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 642, + 506, + 655 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 652, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 363, + 667 + ], + "score": 1.0, + "content": "and indefinitely repeats the following steps for each individual", + "type": "text" + }, + { + "bbox": [ + 364, + 654, + 383, + 664 + ], + "score": 0.88, + "content": "x ^ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 652, + 506, + 667 + ], + "score": 1.0, + "content": "in the population to generate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 663, + 402, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 343, + 677 + ], + "score": 1.0, + "content": "another individual called mutated vector that could replace", + "type": "text" + }, + { + "bbox": [ + 343, + 664, + 363, + 675 + ], + "score": 0.88, + "content": "x ^ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 663, + 402, + 677 + ], + "score": 1.0, + "content": "if better:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 132, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 685, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 129, + 685, + 240, + 701 + ], + "score": 1.0, + "content": "1. given the best individual", + "type": "text" + }, + { + "bbox": [ + 240, + 687, + 259, + 698 + ], + "score": 0.88, + "content": "x ^ { \\mathrm { b e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 685, + 329, + 701 + ], + "score": 1.0, + "content": "which minimizes", + "type": "text" + }, + { + "bbox": [ + 330, + 688, + 336, + 699 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 685, + 506, + 701 + ], + "score": 1.0, + "content": "in the population, as well as two randomly", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 141, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 141, + 698, + 199, + 711 + ], + "score": 1.0, + "content": "selected ones", + "type": "text" + }, + { + "bbox": [ + 199, + 699, + 211, + 709 + ], + "score": 0.87, + "content": "x ^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 698, + 230, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 231, + 698, + 241, + 709 + ], + "score": 0.87, + "content": "x ^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 698, + 325, + 711 + ], + "score": 1.0, + "content": ", compute the donor", + "type": "text" + }, + { + "bbox": [ + 326, + 700, + 332, + 709 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 698, + 506, + 711 + ], + "score": 1.0, + "content": ", which will give part of its coefficients to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 142, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the mutated vector. 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Hyperparameter", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "optimization is thus a zero order optimization performed at a slower frequency than the (often first", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 154 + ], + "score": 1.0, + "content": "order, e.g. SGD) optimization of the model. A PBT step happens typically after a fixed number of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "epochs or updates of the model, often optimizing the loss from the validation set, continuing from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 178 + ], + "score": 1.0, + "content": "an already produced “parent” checkpoint, and producing and evaluating a new checkpoint. At every", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "score": 1.0, + "content": "PBT step, hyperparameters can be updated (mutated), incremented or decremented by some number", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 200, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 200, + 198 + ], + "score": 1.0, + "content": "(step size), or sampled.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 109, + 506, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "There are multiple aspects to consider when designing a PBT algorithm. Technical constraints:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 504, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 504, + 225 + ], + "score": 1.0, + "content": "how the optimization is distributed, with a centralized or decentralized algorithm, workers to run the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "trainings, how failed workers are handled. They are solved in a unified manner in the experiments we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "performed, by the hoptim library to implement and compare multiple algorithms. It is decoupled", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "from the scheduling of the jobs and designed to accommodate adding more workers to scale up the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "training, or fewer when some are killed, for example through preemption or time-out on a shared", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "cluster. Optimization method: how the hyper-parameters are modified throughout the training, for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "instance through mutations. Selection process: which individual of the population are kept, both in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "term of hyper-parameters and state of the neural network (checkpoint). For those last two points,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 252, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 252, + 313 + ], + "score": 1.0, + "content": "some solutions are described below.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 202, + 506, + 313 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 331, + 190, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 192, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 192, + 344 + ], + "score": 1.0, + "content": "2.1 CHALLENGES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 353, + 503, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "In order to have a clearer understanding of our proposed methods, we show below the main concerns", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 365, + 213, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 213, + 375 + ], + "score": 1.0, + "content": "we have observed in PBT:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 352, + 505, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Anisotropy: by definition, the optimal value of the hyperparameters considered is unknown, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "oftentimes the range (or mutation scheme) provided to the algorithm is a loose estimate only. As", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "modifying two hyperparameters with the same step size can produce effects with very different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "magnitudes, the user is required to to normalize the search space. But pre-tuning the hyperparameter", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 426, + 504, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 504, + 437 + ], + "score": 1.0, + "content": "tuner itself can be cumbersome as dynamics evolve during training. Section 3.1 provides an example", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "based on the Rosenbrock function which illustrates this issue and highlights the interest of adaptative", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 151, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 151, + 459 + ], + "score": 1.0, + "content": "mutations.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 381, + 505, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 464, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "Checkpoint vs. hyperparameters: comparing individuals in the population is extremely hard as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "improvements can be due to better hyperparameters, or better checkpoints (including potentially", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 484, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 505, + 501 + ], + "score": 1.0, + "content": "better batches). Better performance through better checkpoints is an optimization phenomenon (e.g.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "random restarts), that can bias the hyperparameter selection. We will detail this aspect in Section 4.2.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 464, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "Short-term-long-term discordance: we observed empirically that hyperparameters which induce", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "better performance in the short term are not always optimal in the longer term. This is a challenge", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 534, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 550 + ], + "score": 1.0, + "content": "that does not exist in classical static optimization, but is crucial for PBT since local minima are easy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "to reach and pose a danger for greedy algorithms. An example of such a parameter is the learning", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "score": 1.0, + "content": "rate. 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In particular, it does not rely on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "mutation ranges or step sizes - Equation 1 samples new parameters close to the current population,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "and as the population individuals go through selection this sampling is refined and becomes sharper", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 504, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 504, + 262 + ], + "score": 1.0, + "content": "around optimal values. In practice, if a parameter’s bounds are too loose or wrong, DE will eventu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "ally adapt after iterations of selection by removing individuals too far from the optimal value, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 398, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 398, + 285 + ], + "score": 1.0, + "content": "concentrate its computation budget on relevant values for this parameter.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 437, + 301 + ], + "score": 1.0, + "content": "In order to use it for PBT, the set of hyper-parameters is converted to a vector in", + "type": "text" + }, + { + "bbox": [ + 437, + 290, + 451, + 299 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "using never-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 312 + ], + "score": 1.0, + "content": "grad parametrization system (Rapin & Teytaud, 2018). However, this basic version of differential", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 311, + 504, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 496, + 323 + ], + "score": 1.0, + "content": "evolution (also implemented in nevergrad) is not adapted to PBT. Indeed the training function", + "type": "text" + }, + { + "bbox": [ + 497, + 311, + 504, + 322 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "changes with the checkpoint as we are updating the parameters (not the hyperparameters) with a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 317, + 345 + ], + "score": 1.0, + "content": "stochastic gradient from the task loss. The trend of", + "type": "text" + }, + { + "bbox": [ + 317, + 333, + 325, + 344 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "is therefore typically downwards during the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 343, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 506, + 358 + ], + "score": 1.0, + "content": "training, younger generations/later epochs tending to have a lower loss than their parents’, biasing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "the hyperperameter selection process in favor of those of the children (later steps of SGD updates)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 289, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 289, + 378 + ], + "score": 1.0, + "content": "instead of in favor of better hyperparameters.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 352, + 396 + ], + "score": 1.0, + "content": "ROMUL We therefore propose an adaptation: a population of", + "type": "text" + }, + { + "bbox": [ + 353, + 385, + 360, + 393 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "individuals is trained, after finishing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 466, + 406 + ], + "score": 1.0, + "content": "their step, individuals are compared to the rest of the population. 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In order to keep the initial scaling of DE, we chose", + "type": "text" + }, + { + "bbox": [ + 439, + 597, + 461, + 608 + ], + "score": 0.89, + "content": "{ \\bf F _ { 1 } } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "uniformly", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 608, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 214, + 620 + ], + "score": 1.0, + "content": "distributed between 0 and", + "type": "text" + }, + { + "bbox": [ + 215, + 608, + 229, + 618 + ], + "score": 0.85, + "content": "2 F", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 608, + 358, + 620 + ], + "score": 1.0, + "content": "(we use the common value for", + "type": "text" + }, + { + "bbox": [ + 359, + 608, + 368, + 618 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 608, + 442, + 620 + ], + "score": 1.0, + "content": "from vanilla DE:", + "type": "text" + }, + { + "bbox": [ + 442, + 608, + 481, + 618 + ], + "score": 0.86, + "content": "F = 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 608, + 506, + 620 + ], + "score": 1.0, + "content": "), and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 618, + 188, + 631 + ], + "score": 0.81, + "content": "\\mathbf { F _ { 2 } } [ i ] = 2 F - \\mathbf { F _ { 1 } } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 618, + 193, + 632 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 194, + 619, + 205, + 630 + ], + "score": 0.6, + "content": "\\forall i", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 618, + 313, + 632 + ], + "score": 1.0, + "content": ". This ensures that the sum", + "type": "text" + }, + { + "bbox": [ + 313, + 619, + 395, + 630 + ], + "score": 0.86, + "content": "\\mathbf { F _ { 1 } } [ i ] + \\mathbf { F _ { 2 } } [ i ] = 2 F ", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 619, + 410, + 629 + ], + "score": 0.51, + "content": "\\forall i", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 618, + 506, + 632 + ], + "score": 1.0, + "content": ", as in vanilla DE. With", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 629, + 474, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 631, + 116, + 640 + ], + "score": 0.82, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 629, + 288, + 643 + ], + "score": 1.0, + "content": "the elementwise multiplication, this yields", + "type": "text" + }, + { + "bbox": [ + 288, + 630, + 470, + 642 + ], + "score": 0.87, + "content": "d = x ^ { c } + \\mathbf { \\bar { F _ { 1 } } } \\odot ( x ^ { d } - x ^ { c } ) + \\mathbf { F _ { 2 } } \\odot ( x ^ { b } - x ^ { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 629, + 474, + 643 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 506, + 507, + 643 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 221, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 223, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 223, + 668 + ], + "score": 1.0, + "content": "2.3 OTHER ALGORITHMS", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "In our experiments, we compare several algorithms briefly presented below. We aim to compare", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "how effective they can be for practical use-cases of hyperparameter tuning, where the user does not", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "want to tune the hyperparameters of its hyperparameter tuner, and desires meaningful defaults. The", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "only input they take are the number of parallel trainings, the range of hyperparameters, and a hint", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 494, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 494, + 733 + ], + "score": 1.0, + "content": "for an initial value (e.g. the same value 0 for dropout values and data augmentation magnitudes).", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Initiator PBT: We reimplemented Initiator Based Evolution, presented in Li et al. (2019). New", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "hyper-parameters are sampled from parent hyper-parameters by adding/removing a mutation con-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 188, + 117 + ], + "score": 1.0, + "content": "stant (for instance d", + "type": "text" + }, + { + "bbox": [ + 189, + 105, + 351, + 116 + ], + "score": 0.32, + "content": "r o p o u t C h i l d = d r o p o u t P a r e n t \\pm 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 105, + 505, + 117 + ], + "score": 1.0, + "content": ". A newly created checkpoint is com-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "pared to a randomly sampled checkpoint in the population: if the latter is better, the new checkpoint", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "is discarded and the latter is forked with its hyperparameters - this ensures that only the best per-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "forming models remain eventually, and allows it to run asynchronously. For each parameter, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 459, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 205, + 162 + ], + "score": 1.0, + "content": "specify a range, and use", + "type": "text" + }, + { + "bbox": [ + 205, + 148, + 258, + 160 + ], + "score": 0.91, + "content": "( h i - l o ) / 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 148, + 459, + 162 + ], + "score": 1.0, + "content": "as a mutation constant unless specified otherwise.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 200, + 178 + ], + "score": 1.0, + "content": "Truncation Selection:", + "type": "text" + }, + { + "bbox": [ + 200, + 166, + 210, + 175 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 165, + 389, + 178 + ], + "score": 1.0, + "content": "models are trained in parallel. Regularly, the", + "type": "text" + }, + { + "bbox": [ + 390, + 166, + 402, + 176 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "worst performing models", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 290, + 190 + ], + "score": 1.0, + "content": "are stopped and replaced with clones of the", + "type": "text" + }, + { + "bbox": [ + 290, + 177, + 302, + 186 + ], + "score": 0.73, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 176, + 505, + 190 + ], + "score": 1.0, + "content": "bests, and hyperparameters are randomly pertur-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "bated. This scheme was first introduced in Jaderberg et al. (2017). In our experiments, we use", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 151, + 210 + ], + "score": 0.91, + "content": "M = N / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 197, + 506, + 212 + ], + "score": 1.0, + "content": ". For hyperparameter perturbation, we generalize the mutation scheme introduced in Ho", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 352, + 222 + ], + "score": 1.0, + "content": "et al. (2019): each parameter is sampled uniformly in its range", + "type": "text" + }, + { + "bbox": [ + 353, + 209, + 380, + 221 + ], + "score": 0.92, + "content": "[ l o , h i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 209, + 401, + 222 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 401, + 209, + 420, + 220 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "probability, or incre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 287, + 233 + ], + "score": 1.0, + "content": "mented by random.choice([-3, -2,", + "type": "text" + }, + { + "bbox": [ + 287, + 221, + 300, + 231 + ], + "score": 0.31, + "content": "^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 220, + 406, + 233 + ], + "score": 1.0, + "content": ", 0, 0, 1, 2, 3])", + "type": "text" + }, + { + "bbox": [ + 407, + 222, + 415, + 230 + ], + "score": 0.74, + "content": "\\star", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "(hi - lo) / 10", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 264, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 232, + 244 + ], + "score": 1.0, + "content": "and then clipped to stay within", + "type": "text" + }, + { + "bbox": [ + 232, + 231, + 260, + 243 + ], + "score": 0.91, + "content": "[ l o , h i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 230, + 264, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "ASHA: This is not a PBT algorithm, but a strong hyperparameter search algorithm that we compare", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "to. In the Asynchronous Successive Halving Algorithm (ASHA, Li et al. (2018)), hyperparameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "are sampled uniformly like in Random Search, but models are evaluated early and stopped if not in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 137, + 294 + ], + "score": 1.0, + "content": "the top", + "type": "text" + }, + { + "bbox": [ + 138, + 281, + 155, + 293 + ], + "score": 0.89, + "content": "1 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 281, + 432, + 294 + ], + "score": 1.0, + "content": "percentile. For a given model, the first evaluation can happen after", + "type": "text" + }, + { + "bbox": [ + 432, + 280, + 465, + 293 + ], + "score": 0.45, + "content": "1 , \\eta , \\eta ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 281, + 505, + 294 + ], + "score": 1.0, + "content": ", .. steps,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "making this algorithm robust to hyperparameters whose optimal value does not perform well until", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "late in the training (Section 4.1). Unlike Initiator PBT or Truncation Selection, ASHA finds constant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 445, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 416, + 326 + ], + "score": 1.0, + "content": "values for hyperparameters rather than schedules. We set the reduction factor", + "type": "text" + }, + { + "bbox": [ + 416, + 316, + 423, + 325 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 314, + 445, + 326 + ], + "score": 1.0, + "content": "to 3.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 340, + 200, + 353 + ], + "lines": [ + { + "bbox": [ + 104, + 339, + 202, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 202, + 356 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "We ran experiments on a toy optimization problem (the Rosenbrock function), CIFAR, and Penn", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "Tree Bank, all with the same ROMUL hyperparameters to test its robustness. Each of these experi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "ments train in around 100 to 300 epochs, and we used 1 step per epoch, so that they all have similar", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 156, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 156, + 410 + ], + "score": 1.0, + "content": "time scales.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 422, + 330, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 330, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 330, + 435 + ], + "score": 1.0, + "content": "3.1 EXAMPLE ON A TOY OPTIMIZATION PROBLEM", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "Current PBT mutation schemes have fixed steps and therefore do not automatically adapt to the land-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "score": 1.0, + "content": "scape of the optimized function. This means that they are not well-suited for anisotropic problems,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "which often arise in real life applications since some hyperparameters may be very important to tune", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "finely, while other do not require the same precision. To highlight this issue, we experiment below", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 486, + 371, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 223, + 500 + ], + "score": 1.0, + "content": "on the Rosenbrock function:", + "type": "text" + }, + { + "bbox": [ + 223, + 487, + 371, + 500 + ], + "score": 0.92, + "content": "\\mathcal { R } _ { a , b } ( x , y ) = ( a - x ) ^ { 2 } + b ( y - x ^ { 2 } ) ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 218, + 520 + ], + "score": 1.0, + "content": "We will aim at minimizing", + "type": "text" + }, + { + "bbox": [ + 219, + 507, + 247, + 519 + ], + "score": 0.92, + "content": "\\mathcal { R } _ { 1 , 1 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 506, + 339, + 520 + ], + "score": 1.0, + "content": "through the surrogate", + "type": "text" + }, + { + "bbox": [ + 339, + 507, + 360, + 520 + ], + "score": 0.91, + "content": "\\mathcal { R } _ { \\hat { a } , \\hat { b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 506, + 384, + 520 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 385, + 507, + 392, + 517 + ], + "score": 0.76, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 506, + 410, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 411, + 505, + 417, + 516 + ], + "score": 0.81, + "content": "\\hat { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "two hyperparameters", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 533 + ], + "score": 1.0, + "content": "handled with PBT. We initialize both parameters at 20 and bound them by -12.12 and 212.12 (using", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 289, + 542 + ], + "score": 1.0, + "content": "integers would be a special case since actual", + "type": "text" + }, + { + "bbox": [ + 290, + 532, + 297, + 540 + ], + "score": 0.75, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 530, + 315, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 315, + 531, + 321, + 540 + ], + "score": 0.64, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "values are integers). This experiment can be", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 468, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 468, + 553 + ], + "score": 1.0, + "content": "reproduced using the hoptim toolbox with the command: hop bench rosenbrock.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 353, + 570 + ], + "score": 1.0, + "content": "While standard PBT with random steps wastes mutations on", + "type": "text" + }, + { + "bbox": [ + 353, + 558, + 360, + 568 + ], + "score": 0.74, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 557, + 505, + 570 + ], + "score": 1.0, + "content": ", DE is able to adapt its step-size to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 166, + 583 + ], + "score": 1.0, + "content": "large steps on", + "type": "text" + }, + { + "bbox": [ + 167, + 571, + 173, + 581 + ], + "score": 0.71, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 570, + 345, + 583 + ], + "score": 1.0, + "content": "until getting close, then smaller steps on", + "type": "text" + }, + { + "bbox": [ + 345, + 571, + 352, + 581 + ], + "score": 0.79, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 570, + 385, + 583 + ], + "score": 1.0, + "content": "to tune", + "type": "text" + }, + { + "bbox": [ + 385, + 571, + 393, + 581 + ], + "score": 0.76, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 570, + 412, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 412, + 569, + 418, + 581 + ], + "score": 0.74, + "content": "\\hat { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "more finely. This is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 280, + 594 + ], + "score": 1.0, + "content": "visible in Fig. 1a with ROMUL values of", + "type": "text" + }, + { + "bbox": [ + 281, + 582, + 287, + 591 + ], + "score": 0.76, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "converging quickly to around 1. The mutations then", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 593, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 604 + ], + "score": 1.0, + "content": "become sharper, while the ones for Initiator-PBT (small steps) are still too large and oscillate around", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 604, + 504, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 504, + 615 + ], + "score": 1.0, + "content": "the optimal value. Arguably, the mutation step could have been even smaller, but that would have", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "slowed down the convergence, and these steps would be painful meta-parameters to tune at scale.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "Initiator-PBT with larger steps and Truncation selection are not displayed in this figure because their", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 635, + 203, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 203, + 649 + ], + "score": 1.0, + "content": "variations are too large.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 199, + 667 + ], + "score": 1.0, + "content": "The impact on the loss", + "type": "text" + }, + { + "bbox": [ + 200, + 653, + 210, + 665 + ], + "score": 0.86, + "content": "\\hat { \\mathcal { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "is then visible in Fig. 1b: Initiator PBT can’t decrease past 0.2 with large", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 395, + 678 + ], + "score": 1.0, + "content": "steps, and 0.048 with smaller steps, since it is trapped trying to optimize", + "type": "text" + }, + { + "bbox": [ + 395, + 666, + 402, + 676 + ], + "score": 0.74, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "while DE is able to reach", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 308, + 690 + ], + "score": 1.0, + "content": "better values. Fig. 1c and 1d show the trajectory of", + "type": "text" + }, + { + "bbox": [ + 308, + 677, + 331, + 689 + ], + "score": 0.92, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "for Truncation selection and ROMUL, with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the same number of training steps. Truncation selection is hampered by more random mutations.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "On the other hand, ROMUL is able to reach a much lower value after exhibiting a more chaotic", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "behavior when it initially adapts to the scale of the problem. The trajectory for both versions of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "Initiator-PBT can be found in Fig. 2 of the appendix. Initiator-PBT with large steps (Fig. 2a) moves", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Initiator PBT: We reimplemented Initiator Based Evolution, presented in Li et al. (2019). New", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "hyper-parameters are sampled from parent hyper-parameters by adding/removing a mutation con-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 188, + 117 + ], + "score": 1.0, + "content": "stant (for instance d", + "type": "text" + }, + { + "bbox": [ + 189, + 105, + 351, + 116 + ], + "score": 0.32, + "content": "r o p o u t C h i l d = d r o p o u t P a r e n t \\pm 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 105, + 505, + 117 + ], + "score": 1.0, + "content": ". A newly created checkpoint is com-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "pared to a randomly sampled checkpoint in the population: if the latter is better, the new checkpoint", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "is discarded and the latter is forked with its hyperparameters - this ensures that only the best per-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "forming models remain eventually, and allows it to run asynchronously. For each parameter, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 459, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 205, + 162 + ], + "score": 1.0, + "content": "specify a range, and use", + "type": "text" + }, + { + "bbox": [ + 205, + 148, + 258, + 160 + ], + "score": 0.91, + "content": "( h i - l o ) / 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 148, + 459, + 162 + ], + "score": 1.0, + "content": "as a mutation constant unless specified otherwise.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 82, + 506, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 200, + 178 + ], + "score": 1.0, + "content": "Truncation Selection:", + "type": "text" + }, + { + "bbox": [ + 200, + 166, + 210, + 175 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 165, + 389, + 178 + ], + "score": 1.0, + "content": "models are trained in parallel. Regularly, the", + "type": "text" + }, + { + "bbox": [ + 390, + 166, + 402, + 176 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "worst performing models", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 290, + 190 + ], + "score": 1.0, + "content": "are stopped and replaced with clones of the", + "type": "text" + }, + { + "bbox": [ + 290, + 177, + 302, + 186 + ], + "score": 0.73, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 176, + 505, + 190 + ], + "score": 1.0, + "content": "bests, and hyperparameters are randomly pertur-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "bated. This scheme was first introduced in Jaderberg et al. (2017). In our experiments, we use", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 151, + 210 + ], + "score": 0.91, + "content": "M = N / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 197, + 506, + 212 + ], + "score": 1.0, + "content": ". For hyperparameter perturbation, we generalize the mutation scheme introduced in Ho", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 352, + 222 + ], + "score": 1.0, + "content": "et al. (2019): each parameter is sampled uniformly in its range", + "type": "text" + }, + { + "bbox": [ + 353, + 209, + 380, + 221 + ], + "score": 0.92, + "content": "[ l o , h i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 209, + 401, + 222 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 401, + 209, + 420, + 220 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "probability, or incre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 287, + 233 + ], + "score": 1.0, + "content": "mented by random.choice([-3, -2,", + "type": "text" + }, + { + "bbox": [ + 287, + 221, + 300, + 231 + ], + "score": 0.31, + "content": "^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 220, + 406, + 233 + ], + "score": 1.0, + "content": ", 0, 0, 1, 2, 3])", + "type": "text" + }, + { + "bbox": [ + 407, + 222, + 415, + 230 + ], + "score": 0.74, + "content": "\\star", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "(hi - lo) / 10", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 264, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 232, + 244 + ], + "score": 1.0, + "content": "and then clipped to stay within", + "type": "text" + }, + { + "bbox": [ + 232, + 231, + 260, + 243 + ], + "score": 0.91, + "content": "[ l o , h i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 230, + 264, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 165, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "ASHA: This is not a PBT algorithm, but a strong hyperparameter search algorithm that we compare", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "to. In the Asynchronous Successive Halving Algorithm (ASHA, Li et al. (2018)), hyperparameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "are sampled uniformly like in Random Search, but models are evaluated early and stopped if not in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 137, + 294 + ], + "score": 1.0, + "content": "the top", + "type": "text" + }, + { + "bbox": [ + 138, + 281, + 155, + 293 + ], + "score": 0.89, + "content": "1 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 281, + 432, + 294 + ], + "score": 1.0, + "content": "percentile. For a given model, the first evaluation can happen after", + "type": "text" + }, + { + "bbox": [ + 432, + 280, + 465, + 293 + ], + "score": 0.45, + "content": "1 , \\eta , \\eta ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 281, + 505, + 294 + ], + "score": 1.0, + "content": ", .. steps,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "making this algorithm robust to hyperparameters whose optimal value does not perform well until", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "late in the training (Section 4.1). Unlike Initiator PBT or Truncation Selection, ASHA finds constant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 445, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 416, + 326 + ], + "score": 1.0, + "content": "values for hyperparameters rather than schedules. We set the reduction factor", + "type": "text" + }, + { + "bbox": [ + 416, + 316, + 423, + 325 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 314, + 445, + 326 + ], + "score": 1.0, + "content": "to 3.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 247, + 505, + 326 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 340, + 200, + 353 + ], + "lines": [ + { + "bbox": [ + 104, + 339, + 202, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 202, + 356 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "We ran experiments on a toy optimization problem (the Rosenbrock function), CIFAR, and Penn", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "Tree Bank, all with the same ROMUL hyperparameters to test its robustness. Each of these experi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "ments train in around 100 to 300 epochs, and we used 1 step per epoch, so that they all have similar", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 156, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 156, + 410 + ], + "score": 1.0, + "content": "time scales.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 365, + 505, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 422, + 330, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 330, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 330, + 435 + ], + "score": 1.0, + "content": "3.1 EXAMPLE ON A TOY OPTIMIZATION PROBLEM", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 499 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "Current PBT mutation schemes have fixed steps and therefore do not automatically adapt to the land-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "score": 1.0, + "content": "scape of the optimized function. This means that they are not well-suited for anisotropic problems,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "which often arise in real life applications since some hyperparameters may be very important to tune", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "finely, while other do not require the same precision. To highlight this issue, we experiment below", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 486, + 371, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 223, + 500 + ], + "score": 1.0, + "content": "on the Rosenbrock function:", + "type": "text" + }, + { + "bbox": [ + 223, + 487, + 371, + 500 + ], + "score": 0.92, + "content": "\\mathcal { R } _ { a , b } ( x , y ) = ( a - x ) ^ { 2 } + b ( y - x ^ { 2 } ) ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 443, + 505, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 218, + 520 + ], + "score": 1.0, + "content": "We will aim at minimizing", + "type": "text" + }, + { + "bbox": [ + 219, + 507, + 247, + 519 + ], + "score": 0.92, + "content": "\\mathcal { R } _ { 1 , 1 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 506, + 339, + 520 + ], + "score": 1.0, + "content": "through the surrogate", + "type": "text" + }, + { + "bbox": [ + 339, + 507, + 360, + 520 + ], + "score": 0.91, + "content": "\\mathcal { R } _ { \\hat { a } , \\hat { b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 506, + 384, + 520 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 385, + 507, + 392, + 517 + ], + "score": 0.76, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 506, + 410, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 411, + 505, + 417, + 516 + ], + "score": 0.81, + "content": "\\hat { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "two hyperparameters", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 533 + ], + "score": 1.0, + "content": "handled with PBT. We initialize both parameters at 20 and bound them by -12.12 and 212.12 (using", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 289, + 542 + ], + "score": 1.0, + "content": "integers would be a special case since actual", + "type": "text" + }, + { + "bbox": [ + 290, + 532, + 297, + 540 + ], + "score": 0.75, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 530, + 315, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 315, + 531, + 321, + 540 + ], + "score": 0.64, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "values are integers). This experiment can be", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 468, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 468, + 553 + ], + "score": 1.0, + "content": "reproduced using the hoptim toolbox with the command: hop bench rosenbrock.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 505, + 505, + 553 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 353, + 570 + ], + "score": 1.0, + "content": "While standard PBT with random steps wastes mutations on", + "type": "text" + }, + { + "bbox": [ + 353, + 558, + 360, + 568 + ], + "score": 0.74, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 557, + 505, + 570 + ], + "score": 1.0, + "content": ", DE is able to adapt its step-size to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 166, + 583 + ], + "score": 1.0, + "content": "large steps on", + "type": "text" + }, + { + "bbox": [ + 167, + 571, + 173, + 581 + ], + "score": 0.71, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 570, + 345, + 583 + ], + "score": 1.0, + "content": "until getting close, then smaller steps on", + "type": "text" + }, + { + "bbox": [ + 345, + 571, + 352, + 581 + ], + "score": 0.79, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 570, + 385, + 583 + ], + "score": 1.0, + "content": "to tune", + "type": "text" + }, + { + "bbox": [ + 385, + 571, + 393, + 581 + ], + "score": 0.76, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 570, + 412, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 412, + 569, + 418, + 581 + ], + "score": 0.74, + "content": "\\hat { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "more finely. This is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 280, + 594 + ], + "score": 1.0, + "content": "visible in Fig. 1a with ROMUL values of", + "type": "text" + }, + { + "bbox": [ + 281, + 582, + 287, + 591 + ], + "score": 0.76, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "converging quickly to around 1. The mutations then", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 593, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 604 + ], + "score": 1.0, + "content": "become sharper, while the ones for Initiator-PBT (small steps) are still too large and oscillate around", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 604, + 504, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 504, + 615 + ], + "score": 1.0, + "content": "the optimal value. Arguably, the mutation step could have been even smaller, but that would have", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "slowed down the convergence, and these steps would be painful meta-parameters to tune at scale.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "Initiator-PBT with larger steps and Truncation selection are not displayed in this figure because their", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 635, + 203, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 203, + 649 + ], + "score": 1.0, + "content": "variations are too large.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 557, + 505, + 649 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 199, + 667 + ], + "score": 1.0, + "content": "The impact on the loss", + "type": "text" + }, + { + "bbox": [ + 200, + 653, + 210, + 665 + ], + "score": 0.86, + "content": "\\hat { \\mathcal { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "is then visible in Fig. 1b: Initiator PBT can’t decrease past 0.2 with large", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 395, + 678 + ], + "score": 1.0, + "content": "steps, and 0.048 with smaller steps, since it is trapped trying to optimize", + "type": "text" + }, + { + "bbox": [ + 395, + 666, + 402, + 676 + ], + "score": 0.74, + "content": "\\hat { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "while DE is able to reach", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 308, + 690 + ], + "score": 1.0, + "content": "better values. Fig. 1c and 1d show the trajectory of", + "type": "text" + }, + { + "bbox": [ + 308, + 677, + 331, + 689 + ], + "score": 0.92, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "for Truncation selection and ROMUL, with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the same number of training steps. Truncation selection is hampered by more random mutations.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "On the other hand, ROMUL is able to reach a much lower value after exhibiting a more chaotic", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "behavior when it initially adapts to the scale of the problem. The trajectory for both versions of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "Initiator-PBT can be found in Fig. 2 of the appendix. 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In Fig. 3 in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "Appendix, we also show the behavior with more variables by performing optimization on an average", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 307, + 588 + ], + "score": 1.0, + "content": "of Rosenbrocks functions, each with independent", + "type": "text" + }, + { + "bbox": [ + 307, + 578, + 314, + 586 + ], + "score": 0.68, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 577, + 332, + 588 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 577, + 338, + 586 + ], + "score": 0.67, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "variable to be estimated by PBT. 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In Fig. 3 in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "Appendix, we also show the behavior with more variables by performing optimization on an average", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 307, + 588 + ], + "score": 1.0, + "content": "of Rosenbrocks functions, each with independent", + "type": "text" + }, + { + "bbox": [ + 307, + 578, + 314, + 586 + ], + "score": 0.68, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 577, + 332, + 588 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 577, + 338, + 586 + ], + "score": 0.67, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "variable to be estimated by PBT. 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AlgorithmReduced CIFAR-10 (10%)CIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-10n/a3.918.8
RandAugment (Cubuk et al., 2019b)n/a2.716.7
PBA (3 epochs/step) (Ho et al., 2019)12.82.616.7
ASHA14.72.817.6
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Initiator PBT (Li et al. (2019), ours)14.72.917.9
ROMUL14.02.817.1
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Only ROMUL is able to reach (marginally) better results", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "than the reproduced baseline in test PPL with both 16 and 32 workers. A found dropout schedule", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "is displayed in the appendix (Fig. 4) and show dropouts rapidly increasing in the beginning and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "stabilizing to different levels. 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The algorithms run with 16 workers in parallel with the same compute", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "budget (except when stated otherwise) on reduced CIFAR-10. After that, the schedule found is used", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 396, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 396, + 241 + ], + "score": 1.0, + "content": "for training the same model from scratch on CIFAR-10 and CIFAR-100", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 254, + 504, + 370 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 254, + 504, + 370 + ], + "spans": [ + { + "bbox": [ + 107, + 254, + 504, + 370 + ], + "score": 0.985, + "html": "
AlgorithmReduced CIFAR-10 (10%)CIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-10n/a3.918.8
RandAugment (Cubuk et al., 2019b)n/a2.716.7
PBA (3 epochs/step) (Ho et al., 2019)12.82.616.7
ASHA14.72.817.6
ASHA (running for double the time)14.12.717.2
Truncation Selection (PBA, ours)13.92.717.7
Initiator PBT (Li et al. (2019), ours)14.72.917.9
ROMUL14.02.817.1
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In general, this is part of the PBT hyperparameters that are tuned in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "PBA, that we try to completely remove as hyperparameters in ROMUL, by being insensitive to it", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "(in this case it does not seem to affect Truncation Selection either). For this hyperparameter, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "constraint is to do PBT steps slow enough so that the number of updates is sufficiently large for the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "model to adapt to new mutated hyperparameters, and high enough so that PBT has enough steps to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "optimize the hyperparameters. In practice, trainings are long enough (regarding the number of SGD", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 489, + 401, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 401, + 501 + ], + "score": 1.0, + "content": "updates of the model) for a wide range of PBT steps frequencies to work.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 389, + 506, + 501 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 517, + 446, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 448, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 448, + 531 + ], + "score": 1.0, + "content": "3.3 APPLICATION TO THE PENN TREEBANK DATASET (LANGUAGE MODELING)", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 107, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "We experiment with the TransformerXL model (Dai et al., 2019) on the PTB dataset (Marcus, 1993).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "TranformerXL’s code is open-source and is the state-of-the-art for tranformer models on this dataset", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "when using proper regularization, making it an interesting challenge for PBT. It comes with several", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "dropout hyperparameters: we search for optimal values for five different dropout hyperparameters,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "that we describe in Table 4 in appendix. They are all initialized to 0 with standard deviation of 0.1", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 595, + 484, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 484, + 606 + ], + "score": 1.0, + "content": "for ROMUL (negative values are reflected to positive values), hence not at the baseline values.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 540, + 505, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "Results are reported in Table 2. Trainings can be reproduced (up to random variance) with the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "hoptim package and its benchmarking counterpart hoptim benchmarks in the ptb folder.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "The baseline TransformerXL was obtained with the author’s code and is close to the one reported in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the initial paper. 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Using 16 and 32 workers provided similar results up to noise for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "ROMUL (in this very case, 32 workers does not actually perform better than with 16 workers).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 610, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "score": 1.0, + "content": "However, using 8 workers results in a notable drop in performance for all optimizers (Test PPL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 302, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 302, + 105 + ], + "score": 1.0, + "content": "ROMUL 56.39, TruncSel 57.98, Initiator 56.33).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table", + "bbox": [ + 120, + 158, + 490, + 313 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 115, + 505, + 149 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "Table 2: Perplexity (lower is better) on PTB for a Transformer-XL with 16 layers and 24M pa-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "rameters, best validation PPL before iteration 175 and corresponding test PPL, given the resources", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 466, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 466, + 150 + ], + "score": 1.0, + "content": "needed these values are not averaged, numbers excelling our training baseline are in bold.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "table_body", + "bbox": [ + 120, + 158, + 490, + 313 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 158, + 490, + 313 + ], + "spans": [ + { + "bbox": [ + 120, + 158, + 490, + 313 + ], + "score": 0.985, + "html": "
TrainingworkersValidation PPLTest PPL
TransformerXL SOTA (Dai et al. (2019))1/54.52
TransformerXL SOTA (our training, their code)159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMULPBT1657.8355.16
Random Search3263.8460.90
ASHA3264.3161.63
Truncation Selection PBT3258.4555.93
Initiator PBT3259.3655.73
ROMUL PBT3258.6355.28
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In an experiment on PTB, we used one epoch per step for half the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "training and then modified it to 10 epochs per step. We observed that increasing one of the dropout", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "contributed negatively for small steps (1 epoch), but positively for long steps (10th epochs). 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This behavior", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 541, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 504, + 554 + ], + "score": 1.0, + "content": "adds complexity to the task of PBT algorithms, because bad early choices can’t be compensated later.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "Arguably, it can be due to interactions with the learning rate scheduler used and a more appropriate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 564, + 479, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 479, + 576 + ], + "score": 1.0, + "content": "schedule could help solve this issue (although it is not clear what such a schedule should be).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 589, + 387, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 389, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 389, + 602 + ], + "score": 1.0, + "content": "4.2 CHECKPOINTS VS HYPERPARAMETERS - SELECTION BIASES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 504, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "For PBT optimizers, one critical question is how much of the loss difference between two indi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "viduals is caused by different hyperparameters, and how much about different checkpoints. Both", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 632, + 487, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 487, + 644 + ], + "score": 1.0, + "content": "contributions are tightly entwined making it harder to identify which hyperparameters are best.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "A naive initial option to counter this is to always start a PBT step from the best checkpoint in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "population. 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In an experiment on PTB, we used one epoch per step for half the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "training and then modified it to 10 epochs per step. We observed that increasing one of the dropout", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "contributed negatively for small steps (1 epoch), but positively for long steps (10th epochs). This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "is a major roadblock for PBT-based approaches since two models with different hyperparameters", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "can’t be straightforwardly compared at every step, but only after an unknown delay. This is partially", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 376, + 489 + ], + "score": 1.0, + "content": "handled by being conservative on models to keep: keeping the best", + "type": "text" + }, + { + "bbox": [ + 376, + 476, + 396, + 486 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "unchanged in ROMUL, or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "the random tournament scheme that allows bad models to continue in Initiator PBT when assigned an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "even worse opponent. Fig. 5 in appendix shows such an example in another domain: a large dropout", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "seems very detrimental early on, but very beneficial in the longer term. While this is expected for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "regularizations, we observe that a straightforward schedule increasing the dropout in steps (in red) is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "not able to compensate this - we observed the same effect with a continuous schedule. This behavior", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 541, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 504, + 554 + ], + "score": 1.0, + "content": "adds complexity to the task of PBT algorithms, because bad early choices can’t be compensated later.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "Arguably, it can be due to interactions with the learning rate scheduler used and a more appropriate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 564, + 479, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 479, + 576 + ], + "score": 1.0, + "content": "schedule could help solve this issue (although it is not clear what such a schedule should be).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 387, + 507, + 576 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 589, + 387, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 389, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 389, + 602 + ], + "score": 1.0, + "content": "4.2 CHECKPOINTS VS HYPERPARAMETERS - SELECTION BIASES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 504, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "For PBT optimizers, one critical question is how much of the loss difference between two indi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "viduals is caused by different hyperparameters, and how much about different checkpoints. Both", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 632, + 487, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 487, + 644 + ], + "score": 1.0, + "content": "contributions are tightly entwined making it harder to identify which hyperparameters are best.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 610, + 505, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "A naive initial option to counter this is to always start a PBT step from the best checkpoint in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "population. 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We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "also expect that doing so could make optimizers less robust by getting trapped in local minima too", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 694, + 412, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 412, + 705 + ], + "score": 1.0, + "content": "easily, or aggressively discarding more promising models in the longer term.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 648, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "On the opposite side of the spectrum, we experimented with never culling checkpoints, effectively", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 153, + 733 + ], + "score": 1.0, + "content": "performing", + "type": "text" + }, + { + "bbox": [ + 153, + 722, + 161, + 730 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "full trainings in parallel. 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It also", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "adds more variability which could be beneficial especially with respect to the short-term long-term", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "score": 1.0, + "content": "discrepancy. That being said, we have neither observed significant improvement nor deterioration", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "when keeping checkpoints, as long as the mutation schemes were not biased towards the best set of", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 256, + 162 + ], + "score": 1.0, + "content": "hyperparameters (e.g.: removing the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 257, + 148, + 275, + 159 + ], + "score": 0.86, + "content": "x ^ { \\mathrm { b e s t } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 276, + 146, + 506, + 162 + ], + "score": 1.0, + "content": "term in Eq. 1), because doing so can make all hyperpa-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "rameters converge towards the best checkpoint, making the optimization process early converge to", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 357, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 357, + 183 + ], + "score": 1.0, + "content": "values which are not necessarily adapted to other checkpoints.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "add robustness to the optimizer: selected hyperparameters have to work well for more than one par-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "ticular checkpoint. Indeed, this way the performance can be attributed to actual parameter schedules", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "fitting different trainings, instead of being biased by checkpoints culling/random restarts. It also", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "adds more variability which could be beneficial especially with respect to the short-term long-term", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 141 + ], + "score": 1.0, + "content": "discrepancy. That being said, we have neither observed significant improvement nor deterioration", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "when keeping checkpoints, as long as the mutation schemes were not biased towards the best set of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 256, + 162 + ], + "score": 1.0, + "content": "hyperparameters (e.g.: removing the", + "type": "text" + }, + { + "bbox": [ + 257, + 148, + 275, + 159 + ], + "score": 0.86, + "content": "x ^ { \\mathrm { b e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 146, + 506, + 162 + ], + "score": 1.0, + "content": "term in Eq. 1), because doing so can make all hyperpa-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "rameters converge towards the best checkpoint, making the optimization process early converge to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 357, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 357, + 183 + ], + "score": 1.0, + "content": "values which are not necessarily adapted to other checkpoints.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 108, + 187, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Still, even with ROMUL’s loss-agnostic mutation scheme, some checkpoints were observed to fall", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "behind and waste resources if not culled, so we expect that a trade-off like the one we implemented", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 374, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 374, + 223 + ], + "score": 1.0, + "content": "(killing checkpoints after 3 failed mutations in a row) is necessary.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 226, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 107, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 107, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "Another source of selection bias is noise. While the trainings are well behaved in PTB because", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "the shuffling of the training set is synchronized by epoch, the trainings in CIFAR are much noisier", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "because of the randomness introduced by data augmentation and the very small training set size.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "PBT optimizers based on more noise-robust blackbox optimization methods could be beneficial, but", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 238, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 238, + 281 + ], + "score": 1.0, + "content": "it is not clear how to adapt them.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 304, + 208, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 211, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 211, + 319 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "Several families of methods exist for tuning hyperparameters of neural networks. Methods closest", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 344, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 355 + ], + "score": 1.0, + "content": "to grid search like random search (Bergstra & Bengio, 2012) and ASHA (Li et al., 2018) are based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "on minimal constraints and can be parallelized extensively. Methods striving for more data-efficient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "search (Bergstra et al., 2011; 2013; Feurer & Hutter, 2019) are more sequential in nature, requiring", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "convergence of some trainings before launching new ones. Population-based training approaches", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "(Jaderberg et al., 2017; Ho et al., 2019; Li et al., 2019) loosen the requirements of training different", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "models, as hyperparameters are changed on-the-fly during training, which also makes the search for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 409, + 410, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 410, + 421 + ], + "score": 1.0, + "content": "schedules easier and less structured, i.e. not based on a predefined function.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "Recent advances in automatic discovery of data augmentation policies include Population Based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "score": 1.0, + "content": "Augmentation (Ho et al., 2019) which we compared to in this paper (denoted Truncation Selection).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "Another line of work on structuring the hyperparameter space for data augmentation policy search is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "AutoAugment (Cubuk et al., 2019a), FastAutoAugment Lim et al. (2019) and RandAugment Cubuk", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 470, + 416, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 416, + 483 + ], + "score": 1.0, + "content": "et al. (2019b), the later being faster and reaching top performance on CIFAR.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 504, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "PBT is used successfully in reinforcement learning (Jaderberg et al., 2017), providing diversity in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "self-play and progressive difficulty, so other experimental comparisons that we did include Initiator", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "PBT from (Li et al., 2019), which presented a generic PBT setup that inspired hoptim. For non-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "PBT baselines we used random search (Bergstra & Bengio, 2012), and ASHA (Li et al., 2018),", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 531, + 311, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 311, + 544 + ], + "score": 1.0, + "content": "which is an update on HyperBand (Li et al., 2017).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 565, + 195, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 197, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 197, + 581 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "We introduced ROMUL, a robust PBT algorithm that we benchmarked on standard datasets with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "multiple regularization and data augmentation hyperparameters. Its main strength comes from its", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "robustness to hyperparameters definitions by automatically adapting to the scale of each parame-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "ter. Although it did not show better performance on CIFAR than PBA – that was tuned for this", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "benchmark – we demonstrated that it is more robust to domain changes. More importantly for the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "practical use-cases, it constitutes a good default that does not require extensive tuning to work well.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 659, + 480, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 480, + 673 + ], + "score": 1.0, + "content": "We open-sourced its implementation as well as a simple and broadly compatible PBT library.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "The main difficulties we observed for PBT-based optimizers came from short-term vs. long-term", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "effects: parameters can have a positive impact in the short term but a negative one in the longer", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "term which may not be rectifiable. Learning rate falls in this category, since decreasing it often", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "provides quick gains at the risk of being trapped in a local minimum. 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Methods closest", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 344, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 355 + ], + "score": 1.0, + "content": "to grid search like random search (Bergstra & Bengio, 2012) and ASHA (Li et al., 2018) are based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "on minimal constraints and can be parallelized extensively. Methods striving for more data-efficient", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "search (Bergstra et al., 2011; 2013; Feurer & Hutter, 2019) are more sequential in nature, requiring", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "convergence of some trainings before launching new ones. Population-based training approaches", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "(Jaderberg et al., 2017; Ho et al., 2019; Li et al., 2019) loosen the requirements of training different", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "models, as hyperparameters are changed on-the-fly during training, which also makes the search for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 409, + 410, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 410, + 421 + ], + "score": 1.0, + "content": "schedules easier and less structured, i.e. not based on a predefined function.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 333, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "Recent advances in automatic discovery of data augmentation policies include Population Based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "score": 1.0, + "content": "Augmentation (Ho et al., 2019) which we compared to in this paper (denoted Truncation Selection).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "Another line of work on structuring the hyperparameter space for data augmentation policy search is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "AutoAugment (Cubuk et al., 2019a), FastAutoAugment Lim et al. (2019) and RandAugment Cubuk", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 470, + 416, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 416, + 483 + ], + "score": 1.0, + "content": "et al. (2019b), the later being faster and reaching top performance on CIFAR.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 425, + 506, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 504, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "PBT is used successfully in reinforcement learning (Jaderberg et al., 2017), providing diversity in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "self-play and progressive difficulty, so other experimental comparisons that we did include Initiator", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "PBT from (Li et al., 2019), which presented a generic PBT setup that inspired hoptim. 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Its main strength comes from its", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "robustness to hyperparameters definitions by automatically adapting to the scale of each parame-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "ter. Although it did not show better performance on CIFAR than PBA – that was tuned for this", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "benchmark – we demonstrated that it is more robust to domain changes. More importantly for the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "practical use-cases, it constitutes a good default that does not require extensive tuning to work well.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 659, + 480, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 480, + 673 + ], + "score": 1.0, + "content": "We open-sourced its implementation as well as a simple and broadly compatible PBT library.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 594, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "The main difficulties we observed for PBT-based optimizers came from short-term vs. long-term", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "effects: parameters can have a positive impact in the short term but a negative one in the longer", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "term which may not be rectifiable. Learning rate falls in this category, since decreasing it often", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "provides quick gains at the risk of being trapped in a local minimum. 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Trainingmean ( (log10)std (log10)
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Trainingmean ( (log10)std (log10)
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ROMUL-2.1010.678
Truncation Selection-0.8340.327
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TrainingParallelismValidation PPLTest PPL
TransformerXL SOTA159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMUL PBT1657.8355.16
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AlgorithmCIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-103.918.8
RandAugment (Cubuk et al., 2019b)2.716.7
PBA (3 epochs/step) (Ho et al., 2019)2.616.7
ASHA2.817.6
Truncation Selection (PBA, ours)2.717.7
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TrainingParallelismValidation PPLTest PPL
TransformerXL SOTA159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMUL PBT1657.8355.16
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AlgorithmCIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-103.918.8
RandAugment (Cubuk et al., 2019b)2.716.7
PBA (3 epochs/step) (Ho et al., 2019)2.616.7
ASHA2.817.6
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Initiator PBT (Li et al. (2019), ours)2.917.9
ROMUL2.817.1
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AlgorithmReduced CIFAR-10 (10%)CIFAR-10CIFAR-100
Baseline:Wide-ResNet-28-10n/a3.918.8
RandAugment (Cubuk et al., 2019b)n/a2.716.7
PBA (3 epochs/step) (Ho et al., 2019)12.82.616.7
ASHA14.72.817.6
ASHA (running for double the time)14.12.717.2
Truncation Selection (PBA, ours)13.92.717.7
Initiator PBT (Li et al. (2019), ours)14.72.917.9
ROMUL14.02.817.1
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TrainingworkersValidation PPLTest PPL
TransformerXL SOTA (Dai et al. (2019))1/54.52
TransformerXL SOTA (our training, their code)159.6555.43
ASHA1663.2058.35
Truncation Selection PBT1660.2457.29
Initiator PBT1659.4255.80
ROMULPBT1657.8355.16
Random Search3263.8460.90
ASHA3264.3161.63
Truncation Selection PBT3258.4555.93
Initiator PBT3259.3655.73
ROMUL PBT3258.6355.28
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The weights of these neural networks are currently determined by data-driven “black-box” training. In this work, we propose Analytic LISTA (ALISTA), where the weight matrix in LISTA is computed as the solution to a data-free optimization problem, leaving only the stepsize and threshold parameters to data-driven learning. This significantly simplifies the training. Specifically, the data-free optimization problem is based on coherence minimization. We show our ALISTA retains the optimal linear convergence proved in (Chen et al., 2018) and has a performance comparable to LISTA. Furthermore, we extend ALISTA to convolutional linear operators, again determined in a data-free manner. We also propose a feed-forward framework that combines the data-free optimization and ALISTA networks from end to end, one that can be jointly trained to gain robustness to small perturbations in the encoding model. + +# 1 INTRODUCTION + +Sparse vector recovery, or sparse coding, is a classical problem in source coding, signal reconstruction, pattern recognition and feature selection. There is an unknown sparse vector $\mathbf { x } ^ { * } =$ $[ x _ { 1 } ^ { * } , \cdot \cdot \cdot , x _ { M } ^ { * } ] ^ { T } \in \mathbb { R } ^ { M }$ . We observe its noisy linear measurements: + +$$ +{ \bf b } = \sum _ { m = 1 } ^ { M } { \bf d } _ { m } x _ { m } ^ { * } + \varepsilon = { \bf D } { \bf x } ^ { * } + \varepsilon , +$$ + +where $\mathbf { b } \in \mathbb { R } ^ { N }$ , $\mathbf { D } = [ \mathbf { d } _ { 1 } , \therefore \cdot \cdot , \mathbf { d } _ { M } ] \in \mathbb { R } ^ { N \times M }$ is the dictionary, and $ { \varepsilon } \in \mathbb { R } ^ { N }$ is additive Gaussian white noise. For simplicity, each column of $\mathbf { D }$ , named as a dictionary kernel, is normalized, that is, $\lVert \mathbf { d } _ { m } \rVert _ { 2 } = \lVert \mathbf { D } _ { : , m } \rVert _ { 2 } \stackrel { - } { = } 1$ , $m = 1 , 2 , \cdots , M$ . Typically, we have $N \ll M$ , so Equation (1) is an under-determined system. + +However, when $\mathbf { x } ^ { * }$ is sufficiently sparse, it can be recovered faithfully. A popular approach is to solve the LASSO problem below (where $\lambda$ is a scalar): + +$$ +\underset { \mathbf { x } } { \mathrm { m i n i m i z e } } \frac { 1 } { 2 } \| \mathbf { b } - \mathbf { D x } \| _ { 2 } ^ { 2 } + \lambda \| \mathbf { x } \| _ { 1 } +$$ + +using iterative algorithms such as the iterative shrinkage thresholding algorithm (ISTA): + +$$ +\mathbf { x } ^ { ( k + 1 ) } = \eta _ { \lambda / L } \Big ( \mathbf { x } ^ { ( k ) } + \frac { 1 } { L } \mathbf { D } ^ { T } ( \mathbf { b } - \mathbf { D x } ^ { ( k ) } ) \Big ) , \quad k = 0 , 1 , 2 , . . . +$$ + +where $\eta _ { \theta }$ is the soft-thresholding function1 and $L$ is usually taken as the largest eigenvalue of $\mathbf { D } ^ { T } \mathbf { D }$ . + +Inspired by ISTA, the authors of (Gregor $\&$ LeCun, 2010) proposed to learn the weights in the matrices in ISTA rather than fixing them. Their methods is called Learned ISTA (LISTA) and resembles a recurrent neural network (RNN). If the iteration is truncated to $K$ iterations, LISTA becomes a $K$ -layer feed-forward neural network with side connections. Specifically, LISTA is: + +$$ +\mathbf { x } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } ( \mathbf { W } _ { 1 } ^ { ( k ) } \mathbf { b } + \mathbf { W } _ { 2 } ^ { ( k ) } \mathbf { x } ^ { ( k ) } ) , \quad k = 0 , 1 , \cdots , K - 1 . +$$ + +If we set $\begin{array} { r } { \mathbf { W } _ { 1 } ^ { ( k ) } \equiv \frac { 1 } { L } \mathbf { D } ^ { T } } \end{array}$ , $\mathbf { W } _ { 2 } ^ { ( k ) } \equiv \mathbf { I } - \mathbf { \Pi } _ { L } ^ { 1 } \mathbf { D } ^ { T } \mathbf { D }$ , $\theta ^ { ( k ) } \equiv \frac { \ d _ { 1 } } { \ d { } _ { L } } \ d \lambda$ , then LISTA recovers ISTA. Given each pair of sparse vector and its noisy measurements $( \mathbf { x } ^ { * } , \mathbf { b } )$ , applying (4) from some initial point $\mathbf { x } ^ { ( 0 ) }$ and using $\mathbf { b }$ as the input yields $\mathbf { x } ^ { ( k ) }$ . Our goal is to choose the parameters $\begin{array} { r } { \Theta = \big \{ \mathbf { W } _ { 1 } ^ { ( k ) } , \mathbf { W } _ { w } ^ { ( k ) } , \theta ^ { ( k ) } \big \} _ { k = 0 , 1 , \dots , K - 1 } } \end{array}$ such that $\mathbf { x } ^ { ( k ) }$ is close to $\mathbf { x } ^ { * }$ for all sparse $\mathbf { x } ^ { * }$ following some distribution $\mathcal { P }$ . Therefore, given the distribution $\mathcal { P }$ , all parameters in $\Theta$ are subject to learning: + +$$ +\underset { \Theta } { \operatorname* { m i n i m i z e } } \mathbb { E } _ { \mathbf { x } ^ { * } , \mathbf { b } \sim \mathcal { P } } \left\| \mathbf { x } ^ { ( K ) } \Big ( \Theta , \mathbf { b } , \mathbf { x } ^ { ( 0 ) } \Big ) - \mathbf { x } ^ { * } \right\| _ { 2 } ^ { 2 } . +$$ + +This problem is approximately solved over a training dataset $\{ ( \mathbf { x } _ { i } ^ { * } , \mathbf { b } _ { i } ) \} _ { i = 1 } ^ { N }$ sampled from $\mathcal { P }$ + +Many empirical results, e.g., (Gregor & LeCun, 2010; Sprechmann et al., 2015; Wang et al., 2016b), show that a trained $K$ -layer LISTA (with $K$ usually set to $1 0 \sim 2 0 $ ) or its variants can generalize more than well to unseen samples $( \mathbf { x } ^ { \prime } , \mathbf { b } ^ { \prime } )$ from the same distribution and recover $\mathbf { x } ^ { \prime }$ from $\mathbf { b } ^ { \prime }$ to the same accuracy within one or two order-of-magnitude fewer iterations than the original ISTA. Additionally, the accuracies of the outputs $\big \{ \mathbf { x } ^ { ( k ) } \big \}$ of the layers $k = 1 , . . , K$ gradually improve. However, such networks will generalize worse when the input deviates from the training distribution (e.g., when D varies), in contrast to the classical iterative algorithms such as ISTA that are trainingfree and thus agnostic to the input distribution. The Analysis-Synthesis model (Rubinstein & Elad, 2014; Yang et al., 2016) could also be viewed as a special LISTA model with only one layer $K = 1$ ). + +More recently, the convolutional sparse coding (CSC), an extension of the sparse coding (1), gains increasingly attention in the machine learning area. (Sreter & Giryes, 2018) showed that the CSC could be similarly approximated and accelerated by a LISTA-type feed-forward network. (Tolooshams et al., 2018) designed a structure of sparse auto-encoder inspired by multi-layer CSC. (Papyan et al., 2016; Sulam et al., 2017) also revealed CSC as a potentially useful tool for understanding general convolutional neural networks (CNNs). + +# 1.1 RELATED WORK + +Despite the empirical success (Sprechmann et al., 2015; Wang et al., 2016a;b;c;d; Zhang & Ghanem, 2018; Zhou et al., 2018; Ito et al., 2018) in constructing fast trainable regressors for approximating iterative sparse solvers, the theoretical understanding of such approximations remains limited. + +A handful of recent works have been investigating the theory of LISTA. (Moreau & Bruna, 2017) re-factorized the Gram matrix of dictionary, by trying to nearly diagonalize the Gram matrix with a basis, subject to a small $\ell _ { 1 }$ perturbation. They thus re-parameterized LISTA a new factorized architecture that achieved similar acceleration gain to LISTA, hence ending up with an “indirect” proof. They concluded that LISTA can converge faster than ISTA, but still sublinearly. (Giryes et al., 2018) interpreted LISTA as a projected gradient descent descent (PGD) where the projection step was inaccurate, which enables a trade-off between approximation error and convergence speed. The latest work (Chen et al., 2018) presented the more related results to ours: they introduced necessary conditions for the LISTA weight structure in order to achieve asymptotic linear convergence of LISTA, which also proved to be a theoretical convergence rate upper bound. They also introduced a thresholding scheme for practically improving the convergence speed. Note that, none of the above works extended their discussions to CSC and its similar LISTA-type architectures. + +Several other works examined the theoretical properties of some sibling architectures to LISTA. (Xin et al., 2016) studied the model proposed by (Wang et al., 2016b), which unfolded/truncated the iterative hard thresholding (IHT) algorithm instead of ISTA, for approximating the solution to $\ell _ { 0 }$ - minimization. They showed that the learnable fast regressor can be obtained by using a transformed dictionary with improved restricted isometry property (RIP). However, their discussions are not applicable to LISTA directly, although IHT is linearly convergent (Blumensath & Davies, 2009) under rather strong assumptions. Their discussions were also limited to linear sparse coding and resulting fully-connected networks only. (Borgerding et al., 2017; Metzler et al., 2017) studied a similar learning-based model inspired from another LASSO solver, called approximated message passing (AMP). (Borgerding et al., 2017) showed the MMSE-optimality of an AMP-inspired model, but not accompanied with any convergence rate result. Also, the popular assumption in analyzing AMP algorithms (called “state evolution”) does not hold when analyzing ISTA. + +# 1.2 MOTIVATION AND CONTRIBUTIONS + +This paper presents multi-fold contributions in advancing the theoretical understanding of LISTA, beyond state-of-the-art results. Firstly, we show that the layer-wise weights in LISTA need not being learned from data. That is based on decoupling LISTA training into a data-free analytic optimization stage followed by a lighter-weight data-driven learning stage without compromising the optimal linear convergence rate proved in (Chen et al., 2018). We establish a minimum-coherence criterion between the desired LISTA weights and the dictionary D, which leads to an efficient algorithm that can analytically solve the former from the latter, independent of the distribution of x. The data-driven training is then reduced to learning layer-wise step sizes and thresholds only, which will fit the distribution of x. The new scheme, called Analytic LISTA (ALISTA), provides important insights into the working mechanism of LISTA. Experiments shows ALISTA to perform comparably with previous LISTA models (Gregor & LeCun, 2010; Chen et al., 2018) with much lighter-weight training. Then, we extend the above discussions and conclusions to CSC, and introduce an efficient algorithm to solve the convolutional version of coherence minimization. Further, we introduce a new robust LISTA learning scheme benefiting from the decoupled structure, by adding perturbations to $\mathbf { D }$ during training. The resulting model is shown to possess much stronger robustness when the input distribution varies, even when $\mathbf { D }$ changes to some extent, compared to classical LISTA models that learn to (over-)fit one specific $\mathbf { D }$ . + +# 2 ANALYTIC LISTA: CALCULATING WEIGHTS WITHOUT TRAINING + +We theoretically analyze the LISTA-CPSS model defined in (Chen et al., 2018): + +$$ +\mathbf { x } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \Big ( \mathbf { x } ^ { ( k ) } - ( \mathbf { W } ^ { ( k ) } ) ^ { T } \big ( \mathbf { D } \mathbf { x } ^ { ( k ) } - \mathbf { b } \big ) \Big ) , +$$ + +where $\mathbf { W } ^ { ( k ) } = [ \mathbf { w } _ { 1 } ^ { ( k ) } , \cdot \cdot \cdot , \mathbf { w } _ { M } ^ { ( k ) } ] \in \mathbb { R } ^ { N \times M }$ is a linear operator with the same dimensionality with D, $\mathbf { x } ^ { ( k ) } = \left[ x _ { 1 } ^ { ( k ) } , \cdot \cdot \cdot , x _ { M } ^ { ( k ) } \right]$ is the $k ^ { \mathrm { { t h } } }$ layer node. In (6), $\boldsymbol { \Theta } = \{ \mathbf { W } ^ { ( k ) } , \boldsymbol { \theta } ^ { ( k ) } \} _ { k }$ are parameters to train. Model (6) can be derived from (4) with $\mathbf { W } _ { 1 } ^ { ( k ) } = ( \mathbf { W } ^ { ( k ) } ) ^ { T }$ , $\mathbf { W } _ { 2 } ^ { ( k ) } = \mathbf { I } - \mathbf { W } _ { 1 } ^ { ( k ) } \mathbf { D }$ . (Chen et al., 2018) showed that (6) has the same representation capability with (4) on the sparse recovery problem, with a specifically light weight structure. + +Our theoretical analysis will further define and establish properties of “good” parameters $\Theta$ in (6), and then discuss how to analytically compute those good parameters rather than relying solely on black-box training. In this way, the LISTA model could be further significantly simplified, with little performance loss. The proofs of all the theorems in this paper are provided in the appendix. + +# 2.1 RECOVERY ERROR UPPER BOUND + +We start with an assumption on the “ground truth” signal $\mathbf { x } ^ { * }$ and the noise $\varepsilon$ . + +Assumption 1 (Basic assumptions). Signal $\mathbf { x } ^ { * }$ is sampled from the following set: + +$$ +\mathbf { x } ^ { * } \in \mathcal { X } ( B , s ) \triangleq \Big \{ \mathbf { x } ^ { * } \Big | | x _ { i } ^ { * } | \leq B , \forall i , \| \mathbf { x } ^ { * } \| _ { 0 } \leq s \Big \} . +$$ + +In other words, $\mathbf { x } ^ { * }$ is bounded and $s$ -sparse2 $s \geq 2 ,$ ). Furthermore, we assume $\varepsilon = 0$ . + +The zero-noise assumption is for simplicity of the proofs. Our experiments will show that our models are robust to noisy cases. + +The mutual coherence of the dictionary $\mathbf { D }$ is a significant concept in compressive sensing (Donoho & Elad, 2003; Elad, 2007; Lu et al., 2018). A dictionary with small coherence possesses better sparse recovery performance. Motivated by this point, we introduce the following definition. + +Definition 1. Given $\mathbf { D } \in \mathbb { R } ^ { N \times M }$ with each of its column normalized, we define the generalized mutual coherence: + +$$ +\widetilde { \mu } ( \mathbf { D } ) = \operatorname* { i n f } _ { \mathbf { W } \in \mathbb { R } ^ { N \times M } } \bigg \{ \operatorname* { m a x } _ { \substack { 1 \leq i \neq j \leq M } } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } \bigg \} . +$$ + +Additionally, We define ${ \mathcal { W } } ( \mathbf { D } ) = \left\{ \mathbf { W } \in \mathbb { R } ^ { N \times M } : \mathbf { W } \right.$ attains the infimum given $( \delta ) \}$ . A weight matrix W is “good” $f \mathbf { W } \in \mathcal { W } ( \mathbf { D } )$ . + +In the above definition, problem (8) is feasible and attainable, i.e., $\mathcal { W } ( \mathbf { D } ) \neq \mathcal { O }$ , which was proven in Lemma 1 of (Chen et al., 2018). + +Theorem 1 (Recovery error upper bound). Take any $\mathbf { x } ^ { * } \in \mathcal { X } ( B , s )$ , any $\mathbf { W } \in \mathcal { W } ( \mathbf { D } )$ , and any sequence $\begin{array} { r } { \gamma ^ { ( k ) } \in ( 0 , \frac { 2 } { 2 \tilde { \mu } s - \tilde { \mu } + 1 } ) } \end{array}$ . Using them, define the parameters $\{ \mathbf { W } ^ { ( k ) } , \theta ^ { ( k ) } \}$ : + +$$ +\mathbf { W } ^ { ( k ) } = \gamma ^ { ( k ) } \mathbf { W } , \quad \theta ^ { ( k ) } = \gamma ^ { ( k ) } \widetilde { \mu } ( \mathbf { D } ) \operatorname* { s u p } _ { \mathbf { x } ^ { * } \in \mathcal { X } ( B , s ) } \big \{ \| \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } ^ { * } \| _ { 1 } \big \} , +$$ + +while the sequence $\{ \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) \} _ { k = 1 } ^ { \infty }$ is generated by (6) using the above parameters and $\mathbf { x } ^ { ( 0 ) } = \mathbf { 0 }$ (Note that each $\mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } )$ depends only on $\theta ^ { ( k - 1 ) } , \theta ^ { ( k - 2 ) } , \dots .$ and defines $\theta ^ { ( k ) }$ ). Let Assumption $^ { l }$ hold with any $B > 0$ and $s < ( 1 + 1 / \tilde { \mu } ) / 2$ . Then, we have + +$$ +\operatorname { s u p p o r t } ( \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) ) \subset \mathbb { S } , \quad \| \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } ^ { * } \| _ { 2 } \leq s B \exp \Big ( - \sum _ { \tau = 0 } ^ { k - 1 } c ^ { ( \tau ) } \Big ) , \quad k = 1 , 2 , \dots +$$ + +where $\mathbb { S }$ is the support of $\mathbf { x } ^ { * }$ and $c ^ { ( k ) } = - \log \left( ( 2 \tilde { \mu } s - \tilde { \mu } ) \gamma ^ { ( k ) } + | 1 - \gamma ^ { ( k ) } | \right)$ is a positive constant. + +In Theorem 1, Eqn. (9) defines the properties of “good” parameters: + +• The weights $\mathbf { W } ^ { ( k ) }$ can be separated as the product of a scalar $\gamma ^ { ( k ) }$ and a matrix $\mathbf { W }$ independent of layer index $k$ , where $\mathbf { W }$ has small coherence with $\mathbf { D }$ . +• $\gamma ^ { ( k ) }$ is bounded in an interval. +• ${ \theta ^ { ( k ) } } / { \gamma ^ { ( k ) } }$ is proportional to the $\ell _ { 1 }$ error of the output of the $k ^ { \mathrm { { t h } } }$ layer. + +The factor $c ^ { ( k ) }$ takes the maximum at $\gamma ^ { ( k ) } = 1$ . If $\gamma ^ { ( k ) } \equiv 1$ , the recovery error converges to zero in a linear rate (Chen et al., 2018): + +$$ +\| \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } ^ { * } \| _ { 2 } \leq s B \exp \big ( - c k \big ) , +$$ + +where $c = - \log ( 2 \tilde { \mu } s - \tilde { \mu } ) \geq c ^ { ( k ) }$ . Although $\gamma ^ { ( k ) } \equiv 1$ gives the optimal theoretical upper bound if there are infinitely many layers $k = 0 , 1 , 2 , \cdots$ , it is not the optimal choice for finite $k$ . Practically, there are finitely many layers and $\gamma ^ { ( k ) }$ obtained by learning is bounded in an interval. + +# 2.2 RECOVERY ERROR LOWER BOUND + +In this subsection, we introduce a lower bound of the recovery error of LISTA, which illustrates that the parameters analytically given by (9) are optimal in the convergence order (linear). + +Assumption 2. The signal $\mathbf { x } ^ { * }$ is a random variable following the distribution $P _ { X }$ . Let $\mathbb { S } ~ =$ suppor $\mathbf { \Psi } ^ { * ( \mathbf { x } ^ { * } ) }$ . $P _ { X }$ satisfies: $2 \leq | \mathbb { S } | \leq s$ ; $\mathbb { S }$ uniformly distributes on the whole index set; nonzero part ${ \mathbf { x } } _ { \mathbb { S } } ^ { * }$ satisfies the uniform distribution with bound $B$ : $| x _ { i } ^ { * } | \leq B , \forall i \in \mathbb { S }$ . Moreover, the observation noise $\varepsilon = 0$ . + +Theorem 1 tells that an ideal weight $\mathbf { W } \in \mathcal { W } ( \mathbf { D } )$ satisfies $\mathbf { I } - \mathbf { W } ^ { T } \mathbf { D } \approx \mathbf { 0 }$ . But this cannot be met exactly in the overcomplete $\mathbf { D }$ case, i.e., $N < M$ . Definition 2 defines the set of matrices W such that $\dot { \mathbf { W } } ^ { T } \mathbf { D }$ is bounded away from the identity I. In Appendix D, we discuss the feasibility of (11). + +Definition 2. Given $\mathbf { D } \in \mathbb { R } ^ { N \times M } , s \geq 2 , { \bar { \sigma } } _ { \operatorname* { m i n } } > 0 ,$ , we define a set that $\mathbf { W } ^ { ( k ) }$ are chosen from: + +$$ +\bar { \mathcal { W } } ( \mathbf { D } , s , \bar { \sigma } _ { \operatorname* { m i n } } ) = \Big \{ \mathbf { W } \in \mathbb { R } ^ { N \times M } \Big | \sigma _ { \operatorname* { m i n } } \Big ( \mathbf { I } - ( \mathbf { W } _ { : , \mathrm { S } } ) ^ { T } \mathbf { D } _ { : , \mathrm { S } } \Big ) \geq \bar { \sigma } _ { \operatorname* { m i n } } , \forall \mathrm { S ~ w i t h } 2 \leq | \mathrm { S } | \leq s \Big \} . +$$ + +Based on Definition 2, we define a set that $\boldsymbol \Theta = \{ \mathbf W ^ { ( k ) } , \boldsymbol \theta ^ { ( k ) } \} _ { k = 0 } ^ { \infty }$ are chosen from: + +Definition 3. Let $\{ \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) \} _ { k = 1 } ^ { \infty }$ be generated by (6) with $\{ \mathbf { W } ^ { ( k ) } , \theta ^ { ( k ) } \} _ { k = 0 } ^ { \infty }$ and $\mathbf { x } ^ { ( 0 ) } = \mathbf { 0 }$ . Then we define $\tau$ as the set of parameters that guarantee there is no false positive in $\mathbf { x } ^ { ( k ) }$ : + +$$ +{ \mathcal { T } } = \Big \{ \{ \mathbf { W } ^ { ( k ) } \in \bar { \mathcal { W } } ( \mathbf { D } , s , \bar { \sigma } _ { m i n } ) , \theta ^ { ( k ) } \} _ { k = 0 } ^ { \infty } \Big | \mathrm { s u p p o r t } ( \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) ) \subset \mathbb { S } , \forall \mathbf { x } ^ { * } \in \mathcal { X } ( B , s ) , \forall k \Big \} +$$ + +The conclusion (10) demonstrates that $\tau$ is nonempty because “support $( \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) ) \subset \mathbb { S } ^ { \prime }$ is satisfied as long as $\theta ^ { ( k - 1 ) }$ large enough. Actually, $\tau$ contains almost all “good” parameters because considerable false positives lead to large recovery errors. With $\tau$ defined, we have: + +Theorem 2 (Recovery error lower bound). Let the sequence $\{ \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) \} _ { k = 1 } ^ { \infty }$ be generated by (6) with $\{ \mathbf { W } ^ { ( k ) } , \theta ^ { ( k ) } \} _ { k = 0 } ^ { \infty }$ aall $\mathbf { x } ^ { ( 0 ) } = \mathbf { 0 }$ . Under Assumption 2, for all parameters ave $\{ \mathbf { \bar { W } } ^ { ( k ) } , \theta ^ { ( k ) } \} _ { k = 0 } ^ { \infty } \in \mathcal { T }$ )}∞k=0 ∈ T and $\epsilon > 0$ + +$$ +\| \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } ^ { * } \| _ { 2 } \geq \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \exp ( - \bar { c } k ) , +$$ + +with probability at least $( 1 - \epsilon s ^ { 3 / 2 } - \epsilon ^ { 2 } )$ , where $\bar { c } = s \log ( 3 ) - \log ( \bar { \sigma } _ { m i n } ) .$ + +This theorem illustrates that, with high probability, the convergence rate of LISTA cannot be faster than a linear rate. Thus, the parameters given in (9), that leads to the linear convergence if $\gamma ^ { k }$ is bounded within an interval near 1, are optimal with respect to the order of convergence of LISTA. + +# 2.3 ANALYTIC LISTA: LESS PARAMETERS TO LEARN + +Following Theorems 1 and 2, we set $\mathbf { W } ^ { ( k ) } = \gamma ^ { ( k ) } \mathbf { W }$ , where $\gamma ^ { ( k ) }$ is a scalar, and propose Tied + +$$ +\mathbf { x } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \Big ( \mathbf { x } ^ { ( k ) } - \gamma ^ { ( k ) } \mathbf { W } ^ { T } ( \mathbf { D } \mathbf { x } ^ { ( k ) } - \mathbf { b } ) \Big ) , +$$ + +where $\boldsymbol { \Theta } = \left\{ \{ \gamma ^ { ( k ) } \} _ { k } , \{ \theta ^ { ( k ) } \} _ { k } , \mathbf { W } \right\}$ are parameters to train. The matrix $\mathbf { W }$ is tied over all the layers. Further, we notice that the selection of W from $\mathcal { W } ( \mathbf { D } )$ depends on $\mathbf { D }$ only. Hence we propose the analytic LISTA (ALISTA) that decomposes tied-LISTA into two stages: + +$$ +\mathbf { x } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \Big ( \mathbf { x } ^ { ( k ) } - \gamma ^ { ( k ) } \tilde { \mathbf { W } } ^ { T } ( \mathbf { D } \mathbf { x } ^ { ( k ) } - \mathbf { b } ) \Big ) , +$$ + +where $\tilde { \mathbf { W } }$ is pre-computed by solving the following problem (Stage 1)3: + +$$ +\begin{array} { r } { \tilde { { { \mathbf { W } } } } \in \underset { { \mathbf { W } } \in \mathbb { R } ^ { N \times M } } { \arg \operatorname* { m i n } } \left\| \mathbf { W } ^ { T } { \mathbf { D } } \right\| _ { F } ^ { 2 } , \quad \mathrm { s . t . } \left( { \mathbf { W } } _ { : , m } \right) ^ { T } { \mathbf { D } } _ { : , m } = 1 , \forall m = 1 , 2 , \cdots , M , } \end{array} +$$ + +Then with $\tilde { \mathbf { W } }$ fixed, $\{ \gamma ^ { ( k ) } , \theta ^ { ( k ) } \} _ { k }$ in (15) are learned from end to end (Stage 2). (16) reformulates (8) to minimizing the Frobenius norm of ${ \bf W } ^ { T } { \bf D }$ (a quadratic objective), over linear constraints. This is a standard convex quadratic program, which is easier to solve than to solve (8) directly. + +Table 1: Summary: variants of LISTA and the number of parameters to learn. + +
Vanilla LISTA (4)LISTA-CPSS (6)TiLISTA (14)ALISTA (15)
O(KM²+K+MN)O(KNM+K)O(NM+ K)O(K)
+ +# 3 CONVOLUTIONAL ANALYTIC LISTA + +We extend the analytic LISTA to the convolutional case in this section, starting from discussing the convolutional sparse coding (CSC). Many works studied CSC and proposed efficient algorithms for that (Bristow et al., 2013; Heide et al., 2015; Wohlberg, 2014; 2016; Papyan et al., 2017; GarciaCardona & Wohlberg, 2018; Wang et al., 2018; Liu et al., 2017; 2018). In CSC, the general linear transform is replaced by convolutions in order to learn spatially invariant features: + +$$ +{ \bf b } = \sum _ { m = 1 } ^ { M } { \bf d } _ { m } * { \bf x } _ { m } ^ { * } + \varepsilon , +$$ + +where each $\mathbf { d } _ { m }$ is a dictionary kernel (or filter). $\lbrace \mathbf { d } _ { m } \rbrace _ { m = 1 } ^ { M }$ is the dictionary of filters, $M$ denotes the number of filters. $\{ \mathbf { x } _ { m } ^ { * } \} _ { m = 1 } ^ { M }$ m=1 is the set of coefficient maps that are assumed to have sparse structure, + +and $^ *$ is the convolution operator. Now we consider 2D convolution and take4 $\mathbf b \in \mathbb R ^ { N ^ { 2 } } , \mathbf d _ { m } \in$ $\mathbb { R } ^ { D ^ { 2 } } , \mathbf { x } _ { m } \in \mathbb { R } ^ { ( N + D - 1 ) ^ { 2 } }$ . Equation (17) is pointwisely defined $\mathrm { a s } ^ { 5 }$ : + +$$ +\mathbf { b } ( i , j ) = \sum _ { k = 0 } ^ { D - 1 } \sum _ { l = 0 } ^ { D - 1 } \sum _ { m = 1 } ^ { M } \mathbf { d } _ { m } ( k , l ) \mathbf { x } _ { m } ( i + k , j + l ) + \varepsilon ( i , j ) , \quad 0 \leq i , j \leq N - 1 . +$$ + +We concatenate $\mathbf { d } _ { m } \mathbf { s }$ and $\mathbf { x } _ { m } \mathbf { s }$ : $\mathbf { d } = [ \mathbf { d } _ { 1 } , \cdots , \mathbf { d } _ { M } ] ^ { T }$ , $\mathbf { x } = [ \mathbf { x } _ { 1 } , \cdots , \mathbf { x } _ { M } ] ^ { T }$ , and rewrite (18) as: + +$$ +\boldsymbol { \mathbf { b } } = \sum _ { m = 1 } ^ { M } \mathbf { D } _ { \mathrm { c o n v } , m } ^ { N } ( \mathbf { d } _ { m } ) \mathbf { x } _ { m } + \varepsilon = \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } ) \mathbf { x } + \varepsilon , +$$ + +where the matrix $\mathbf D _ { \mathrm { c o n v } } ^ { N } ( \mathbf d ) = [ \mathbf D _ { \mathrm { c o n v } , 1 } ^ { N } ( \mathbf d _ { 1 } ) , \cdots , \mathbf D _ { \mathrm { c o n v } , M } ^ { N } ( \mathbf d _ { M } ) ] \in \mathbb R ^ { N ^ { 2 } \times ( N + D - 1 ) ^ { 2 } M }$ , depending on the signal size $N$ and the dictionary $\mathbf { d }$ , is defined in detail in (48) in Appendix C.2. + +From (17), the convolutional LISTA becomes a natural extension of the fully-connected LISTA (6): + +$$ +{ \bf x } _ { m } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \left( { \bf x } _ { m } ^ { ( k ) } - \left( { \bf w } _ { m } ^ { ( k ) } \right) ^ { \prime } \ast \Big ( \sum _ { \bar { m } = 1 } ^ { M } { \bf d } _ { \bar { m } } \ast { \bf x } _ { \bar { m } } ^ { ( k ) } - { \bf b } \Big ) \right) , \quad m = 1 , 2 , \cdots , M , +$$ + +where {w(k)m }Mm=1 share the same sizes with $\lbrace \mathbf { d } _ { m } \rbrace _ { m = 1 } ^ { M }$ and $( \cdot ) ^ { \prime }$ means a 180 rotation of the filter (Chalasani et al., 2013). We concatenate the filters together: $\dot { \mathbf { w } } ^ { ( k ) } = [ \mathbf { w } _ { 1 } ^ { ( k ) } , \cdot \cdot \cdot , \mathbf { w } _ { M } ^ { ( k ) } ] ^ { T } \in \mathbb { R } ^ { D ^ { 2 } M }$ . Parameters to train are $\boldsymbol { \Theta } = \{ \mathbf { w } ^ { ( k ) } , \boldsymbol { \theta } ^ { ( k ) } \} _ { k }$ . + +Let $\mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ^ { ( k ) } )$ be the matrix induced by dictionary $\mathbf { w } ^ { ( k ) }$ with the same dimensionality as $\mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } )$ nv. Since convolution can be written as a matrix form (19), (20) is equivalent to + +$$ +\begin{array} { r } { \mathbf { x } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \Big ( \mathbf { x } ^ { ( k ) } - ( \mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ^ { ( k ) } ) ) ^ { T } \big ( \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } ) \mathbf { x } ^ { ( k ) } - \mathbf { b } \big ) \Big ) . } \end{array} +$$ + +Then by just substituting can be applied to the conv $\mathbf { D } , \mathbf { W } ^ { ( k ) }$ with LIST $\mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } ) , \mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ^ { ( k ) } )$ respectively, Theorems 1 and 2 + +Proposition 1. Let $\mathbf { D } = \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } )$ and $\mathbf { W } ^ { ( k ) } = \mathbf { W } _ { \mathrm { c o n v } } ^ { N } \big ( \mathbf { w } ^ { ( k ) } \big )$ . With Assumption 1 and other settings the same with those in Theorem $^ { l }$ , $( I O )$ holds. With Assumption 2 and other settings the same with those in Theorem 2, (13) holds. + +Similar to the fully connected case (15), based on the results in Proposition 1, we should set $\mathbf { w } _ { m } ^ { ( k ) } =$ $\gamma _ { m } ^ { ( k ) } \tilde { \mathbf { w } } _ { m }$ , $m = 1 , 2 , \cdots , M$ , where $\tilde { \mathbf { w } } = [ \tilde { \mathbf { w } } _ { 1 } , \cdots , \tilde { \mathbf { w } } _ { M } ] ^ { T }$ is chosen from + +$$ +\tilde { \mathbf { w } } \in \mathcal { W } _ { \mathrm { c o n v } } ^ { N } = \underset { \mathbf { w } _ { m } \cdot \mathbf { d } _ { m } = 1 , 1 \leq m \leq M } { \arg \operatorname* { m i n } } \left\| \left( \mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) \right) ^ { T } \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } ) \right\| _ { F } ^ { 2 } . +$$ + +However, (22) is not as efficient to solve as (16). To see that, matrices $\mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } )$ and $\mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } )$ are both of size $N ^ { 2 } \times ( N + D - 1 ) ^ { 2 } M$ , the coherence matrix $\left( \mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) \right) ^ { T } \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } )$ is thus of size $( N + D - 1 ) ^ { 2 } M \times ( N + D - 1 ) ^ { 2 } M .$ In the typical application setting of CSC, b is usually an image rather than a small patch. For example, if the image size is $1 0 0 \times 1 0 0$ , dictionary size is $7 \times 7 \times 6 4$ , $N = 1 0 0 , D = 7 , M = 6 4$ , then $( \dot { N } + D - 1 ) ^ { 2 } \bar { M } \times ( N + D - 1 ) ^ { 2 } M \approx 5 \times \dot { 1 } 0 ^ { 1 1 }$ . + +# 3.1 CALCULATING CONVOLUTIONAL WEIGHTS ANALYTICALLY AND EFFICIENTLY + +To overcome the computational challenge of solving (22), we exploit the following circular convolution as an efficient approximation: + +$$ +\mathbf { b } ( i , j ) = \sum _ { k = 0 } ^ { D - 1 } \sum _ { l = 0 } ^ { D - 1 } \sum _ { m = 1 } ^ { M } \mathbf { d } _ { m } ( k , l ) \mathbf { x } _ { m } \big ( ( i + k ) _ { \mathrm { m o d } N } , ( j + l ) _ { \mathrm { m o d } N } \big ) + \varepsilon ( i , j ) , \quad 0 \leq i , j \leq N - 1 , +$$ + +where $ { \mathbf { b } } \in \mathbb { R } ^ { N ^ { 2 } } , \mathbf { d } _ { m } \in \mathbb { R } ^ { D ^ { 2 } } , { \mathbf { x } } _ { m } \in \mathbb { R } ^ { N ^ { 2 } }$ . Similar to (18), we rewrite (23) in a compact way: + +$$ +\mathbf { b } = \sum _ { m = 1 } ^ { M } \mathbf { D } _ { \mathrm { c i r } , m } ^ { N } ( \mathbf { d } _ { m } ) \mathbf { x } _ { m } + \varepsilon = \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( \mathbf { d } ) \mathbf { x } + \varepsilon , +$$ + +where ${ \bf D } _ { \mathrm { c i r } } ^ { N } ( { \bf d } ) : \mathbb { R } ^ { N ^ { 2 } M } \mathbb { R } ^ { N ^ { 2 } }$ is a matrix depending on the signal size $N$ and the dictionary $\mathbf { d }$ Then the coherence minimization with the circular convolution is given by + +$$ +\mathcal { W } _ { \mathrm { c i r } } ^ { N } = \underset { \mathbf { w } _ { m } \cdot \mathbf { d } _ { m } = 1 , \ 1 \leq m \leq M } { \arg \operatorname* { m i n } } \left\| \left( \mathbf { W } _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } ) \right) ^ { T } \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( \mathbf { d } ) \right\| _ { F } ^ { 2 } . +$$ + +The following theorem motivates us to use the solution to (24) to approximate that of (22) + +Theorem 3. The solution sets of (22) and (24) satisfy the following properties: + +$I . \mathcal { W } _ { \mathrm { c i r } } ^ { N } = \mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } , \forall N \ge 2 D - 1 .$ + +2. If at least one of the matrices $\{ \mathbf { D } _ { \mathrm { c i r } , 1 } ^ { 2 D - 1 } , \cdot \cdot \cdot , \mathbf { D } _ { \mathrm { c i r } , M } ^ { 2 D - 1 } \}$ is non-singular, $\mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ involves only a unique element. Furthermore, + +$$ +\operatorname * { l i m } _ { N \infty } \mathcal { W } _ { \mathrm { c o n v } } ^ { N } = \mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } . +$$ + +The solution set $\mathcal { W } _ { \mathrm { c i r } } ^ { N }$ is not related with the image size $N$ as long as $N \geq 2 D - 1$ , thus one can deal with a much smaller-size problem (let $N = 2 D - 1 \mathrm { \ Y }$ ). Further, (25) indicates that as $N$ gets (much) larger than $D$ , the boundary condition becomes less important. Thus, one can use $\mathcal { W } _ { \mathrm { c i r } } ^ { 2 \breve { D } - 1 }$ to approximate $\mathcal { W } _ { \mathrm { c o n v } } ^ { N }$ . In Appendix E.2, we introduce the algorithm details of solving (24). + +Based on Proposition 1 and Theorem 3, we obtain the convolutional ALISTA: + +$$ +\mathbf { x } _ { m } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \left( \mathbf { x } _ { m } ^ { ( k ) } - \gamma _ { m } ^ { ( k ) } \left( \tilde { \mathbf { w } } _ { m } \right) ^ { \prime } \ast \bigg ( \sum _ { \bar { m } = 1 } ^ { M } \mathbf { d } _ { \bar { m } } \ast \mathbf { x } _ { \bar { m } } ^ { ( k ) } - \mathbf { b } \bigg ) \right) , \quad m = 1 , 2 , \cdots , M , +$$ + +where $\tilde { \mathbf { w } } = [ \tilde { \mathbf { w } } _ { 1 } , \cdots , \tilde { \mathbf { w } } _ { M } ] ^ { T } \in \mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ and $\Theta = \{ \{ \gamma _ { m } ^ { ( k ) } \} _ { m , k } , \{ \theta ^ { ( k ) } \} _ { k } \}$ are the parameters to train. (26) is a simplified form, compared to the empirically unfolded CSC model recently proposed in (Sreter & Giryes, 2018) + +# 4 ROBUST ALISTA TO MODEL PERTURBATION + +Many applications, such as often found in surveillance video scenarios (Zhao et al., 2011; Han et al., 2013), can be formulated as sparse coding models whose dictionaries are subject to small dynamic perturbations (e.g, slowly varied over time). Specifically, the linear system model (1) may have uncertain $\mathbf { D }$ : $\tilde { \textbf { D } } = \textbf { D } + \varepsilon _ { D }$ , where $\varepsilon _ { D }$ is some small stochastic perturbation. Classical LISTA entangles the learning of all its parameters, and the trained model is tied to one static D. The important contribution of ALISTA is to decompose fitting W w.r.t. D, from adapting other parameters $\{ \gamma ^ { ( k ) } , \theta ^ { ( k ) } \} _ { k }$ to training data. + +In this section, we develop a robust variant of ALISTA that is a fast regressor not only for a given $\mathbf { D }$ , but all its randomly perturbations $\tilde { \bf D }$ to some extent. Up to our best knowledge, this approach is new. Robust ALISTA can be sketched as the following empirical routine (at each iteration): + +• Sample a perturbed dictionary $\tilde { \bf D }$ . Sample $\mathbf { x }$ and $\varepsilon$ to generate b w.r.t. $\tilde { \bf D }$ . • Apply Stage 1 of ALISTA w.r.t. $\tilde { \bf D }$ and obtain $\tilde { \mathbf { W } }$ ; however, instead of an iterative minimization algorithm, we use a neural network that unfolds that algorithm to produce $\tilde { \mathbf { W } }$ . • Apply Stage 2 of ALISTA w.r.t. $\tilde { \mathbf { W } }$ , D, $\mathbf { x }$ , and b to obtain $\{ \gamma ^ { ( k ) } , \theta ^ { ( k ) } \} _ { k }$ . + +In Robust ALISTA above, $\tilde { \bf D }$ becomes a part of the data for training the neural network that generates $\tilde { \mathbf { W } }$ . This neural network is faster to apply than the minimization algorithm. One might attempt to use $\tilde { \bf D }$ in the last step, rather than $\mathbf { D }$ , but $\tilde { \bf D }$ makes training less stable, potentially because of larger weight variations between training iterations due to the random perturbations in $\tilde { \bf D }$ . We observe that using $\mathbf { D }$ stabilizes training better and empirically achieves a good prediction. More details of training Robust ALISTA are given in Appendix G. + +# 5 NUMERICAL RESULTS + +In this section, we conduct extensive experiments on both synthesized and real data to demonstrate:6 • We experimentally validate Theorems 1 and 2, and show that ALISTA is as effective as classical LISTA (Gregor & LeCun, 2010; Chen et al., 2018)but is much easier to train. • Similar conclusions can be drawn for convolutional analytic LISTA. • The robust analytic LISTA further shows remarkable robustness in sparse code prediction, given that $\mathbf { D }$ is randomly perturbed within some extent. + +Notation For brevity, we let LISTA denote the vanilla LISTA model (4) in (Gregor & LeCun, 2010); LISTA-CPSS refers to the lately-proposed fast LISTA variant (Chen et al., 2018) with weight coupling and support selection; TiLISTA is the tied LISTA (14); and ALISTA is our proposed Analytic LISTA (15). If the model is for convolutional case, then we add “Conv” as the prefix for model name, such as “Conv ALISTA” that represents the convolutional analytic LISTA. + +# 5.1 VALIDATION OF THEOREMS 1 AND 2 (ANALYTIC LISTA) + +We follow the same $N = 2 5 0$ , $M = 5 0 0$ setting as (Chen et al., 2018) by default. We sample the entries of $\mathbf { D }$ i.i.d. from the standard Gaussian distribution, $\mathbf { D } _ { i j } \sim \mathcal { N } ( 0 , 1 / N )$ and then normalize its columns to have the unit $\ell _ { 2 }$ norm. We fix a dictionary $\mathbf { D }$ in this section. To generate sparse vectors $\mathbf { x } ^ { * }$ , we decide each of its entry to be non-zero following the Bernoulli distribution with $p _ { b } = 0 . 1$ . The values of the non-zero entries are sampled from the standard Gaussian distribution. A test set of 1000 samples generated in the above manner is fixed for all tests in our simulations. The analytic weight $W$ that we use in the ALISTA is obtained by solving (16). + +All networks used (vanilla LISTA, LISTA-CPSS, TiLISTA and ALISTA) have the same number of 16 layers. We also include two classical iterative solvers: ISTA and FISTA. We train the networks with four different levels of noises: SNR (Signal-to-Noise Rati $\mathbf { \delta } _ { 0 } ) = 2 0 , 3 0 , 4 0$ , and $\infty$ . While our theory mainly discussed the noise-free case $\mathrm { S N R } = \infty$ ), we hope to empirically study the algorithm performance under noise too. As shown in Figure 1, the $\mathbf { X }$ -axes denotes the indices of layers for the networks, or the number of iterations for the iterative algorithms. The y-axes represent the NMSE (Normalized Mean Squared Error) in the decibel (dB) unit: + +$$ +\begin{array} { r } { \mathrm { N M S E } _ { \mathrm { d B } } ( \hat { \mathbf { x } } , \mathbf { x } ^ { * } ) = 1 0 \log _ { 1 0 } \left( \mathbb { E } \| \hat { \mathbf { x } } - \mathbf { x } ^ { * } \| ^ { 2 } / \mathbb { E } \| \mathbf { x } ^ { * } \| ^ { 2 } \right) , } \end{array} +$$ + +where $\mathbf { x } ^ { * }$ is the ground truth and $\hat { \bf x }$ is the estimated one. + +![](images/971fc90a1cbe56d421357c61684b1c5712be35575d937c498da5ae4642154c58.jpg) +Figure 1: Justification of Theorems 1 and 2: comparision among LISTA variants. + +In Figure 1 (a) noise-less case, all four learned models apparently converge much faster than two iterative solvers (ISTA/FISTA curves almost overlap in this y-scale, at the small number of iterations). Among the four networks, classical-LISTA is inferior to the other three by an obvious margin. LISTA-CPSS, TiLISTA and ALISTA perform comparably: ALISTA is observed to eventually achieve the lowest NMSE. Figure 1(a) also supports Theorem 2, that all networks have at most linear convergence, regardless of how freely their parameters can be end-to-end learned. + +Figure 1 (b) - (d) further show that even in the presence of noise, ALISTA can empirically perform comparably with LISTA-CPSS and TiLISTA, and stay clearly better than LISTA and ISTA/FISTA. Always note that ALISTA the smallest amount of parameters to learn from the end-to-end training (Stage 2). The above results endorse that: i) the optimal LISTA layer-wise weights could be structured as $\mathbf { W } ^ { ( k ) } = \gamma ^ { ( k ) } \mathbf { W }$ ; and ii) W could be analytically solved rather than learned from data, without incurring performance loss. We also observe the significant reduction of training time for ALISTA: while LISTA-CPSS of the same depth took ${ \sim } 1 . 5$ hours to train, ALISTA was trained within only 6 minutes (0.1 hours) to achieve comparable performance, on the same hardware (one 1080 Ti on server). + +![](images/b4ad8ecd57838661b04ad8c14453ba10544c6080dccc8fdb5991b9989b099d41.jpg) +Figure 2: Justification of Theorem 1 (noiseless case): parameters obtained by training satisfy (9). + +We further supply Figures 2 and 3 to justify Theorem 1 from different perspectives. Figure 2 plots the learned parameters $\{ \bar { \gamma ^ { ( k ) } } , \bar { \theta ^ { ( k ) } } \}$ in ALISTA (Stage 2), showing that they satisfy the properties proposed in Theorem 1: $\gamma ^ { ( k ) }$ bounded; ${ \theta } ^ { \bar { ( k ) } }$ and $\gamma ^ { ( k ) }$ is proportional to $\begin{array} { r } { \operatorname* { s u p } _ { \mathbf { x } ^ { * } } \| \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } ^ { * } \| _ { 1 } ~ ( ^ { * } \mathrm { s u p } _ { \mathbf { x } ^ { * } } , } \end{array}$ is taken over the test set). Figure 3 reports the average magnitude7 of the false positives and the true positives in $\mathbf { \Delta x } ^ { k } ( \mathbf { x } ^ { * } )$ of ALISTA: the “true positives” curve draws the values of $\mathbb { E } \{ \| \mathbf { x } _ { \mathbb { S } } ^ { k } ( \mathbf { x } ^ { * } ) \| _ { 2 } ^ { 2 } / \| \mathbf { x } ^ { \hat { k } } ( \mathbf { x } ^ { * } ) \| _ { 2 } ^ { 2 } \}$ w.r.t. $k$ (the expectation is taken over the test set), while “false positives” for ${ \mathbb { E } } \{ \| \mathbf { x } _ { \mathbb { S } ^ { c } } ^ { k } ( \mathbf { x } ^ { * } ) \| _ { 2 } ^ { 2 } / \| \mathbf { x } ^ { k } ( \mathbf { x } ^ { * } ) \| _ { 2 } ^ { 2 } \}$ . False positives take up small proportion over the positives, which supports the Theorem 1 conclusion that support $( \mathbf { x } ^ { k } ( \mathbf { x } ^ { \ast } ) ) \subset \mathbb { S }$ . + +![](images/113a9556b7deff18282564eac7e07763703a959bbb96cb26d5149e31de0c5122.jpg) +Figure 3: Justification of Theorem 1 (noiseless case): Proportion of false positives vs true positives in $\mathbf { x } ^ { k } ( \mathbf { x } ^ { * } )$ . + +# 5.2 VALIDATION OF THEOREM 3 (CONVOLUTIONAL ANALYTIC LISTA) + +For convolutional cases, we use real image data to verify Theorem 3. We train a convolutional dictionary $\mathbf { d }$ with $D = 7 , M = 6 4$ on the BSD500 training set (400 images), using the Algorithm 1 in (Liu et al., 2018). We then use it for problems (22) and (24) and solve them with different $N s$ . + +In Table 2, we take $\mathbf { w } _ { \mathrm { c i r } } ^ { N } \in \mathcal { W } _ { \mathrm { c i r } } ^ { N }$ , $\mathbf { w } ^ { * } \in \mathcal { W } _ { \mathrm { c i r } } ^ { 5 0 }$ (consider 50 as large enough) For this example, $\mathcal { W } _ { \mathrm { c i r } } ^ { N }$ has only one element. Table 2 shows that $\mathbf { w } _ { \mathrm { c i r } } ^ { N } = \mathbf { w } ^ { * }$ for $N \geq 1 3$ , i.e., the solution of the problem (24) is independent of if $N \geq 2 D - 1$ , justifying the first conclusion in Theorem 3. In Table shows valida 3, we take $\mathbf { w } _ { \mathrm { c o n v } } ^ { N } \to \mathbf { w } ^ { * }$ $\mathbf { \bar { w } } _ { \mathrm { c o n v } } ^ { N } \in \mathcal { W } _ { \mathrm { c o n v } } ^ { N }$ conv cir co, i.e., the solution of the problem (22) d conclusion of Theorem 3. Visualized and $\mathbf { w ^ { * } } \in \mathbf { w } _ { \mathrm { c i r } } ^ { 1 3 }$ , where $\mathcal { W } _ { \mathrm { c o n v } } ^ { N }$ also has only one element. Table 3 o that of (24) as is displayed in increases,pendix F. $\mathbf { w } ^ { * } \in \mathbf { w } _ { \mathrm { c i r } } ^ { 1 3 }$ + +Besides validating Theorem 3, we also present a real image denoising experiment to verify the effectiveness of Conv ALISTA. The detailed settings and results are presented in Appendix $_ \mathrm { H }$ . + +Table 2: Validation of Conclusion 1 in Theorem 3. $D = 7$ . $\mathbf { w } _ { \mathrm { c i r } } ^ { N } \in \mathcal { W } _ { \mathrm { c i r } } ^ { N }$ and $\mathbf { w } ^ { * } \in \mathcal { W } _ { \mathrm { c i r } } ^ { 5 0 }$ + +
lwir- w*||2/1lw*||2
N=10N= 11N = 12N=13N=15N = 20
2.0×10-29.3×10-33.9 ×10-31.4 ×10-128.8×10-135.9×10-13
+ +Table 3: Validation of Conclusion 2 in Theorem 3. $D = 7$ . $\mathbf { w } _ { \mathrm { c o n v } } ^ { N } \in \mathcal { W } _ { \mathrm { c o n v } } ^ { N }$ and $\mathbf { w } ^ { * } \in \mathbf { w } _ { \mathrm { c i r } } ^ { 1 3 }$ + +
|/wconv - w*||2/lw*|2
N=3N=5N= 10N=15N = 20
0.18920.08500.02840.01610.0113
+ +![](images/d4471e4ca71aa2a0f014d99d6cc87823b8becfcf52e21856f57b86aa7e1731b3.jpg) +Figure 4: Validation of Robust ALISTA. + +# 5.3 VALIDATION OF ROBUST ALISTA + +We empirically verify the effectiveness of Robust ALISTA, by sampling the dictionary perturbation $\varepsilon _ { D }$ entry-wise i.i.d. from another Gaussian distribution $\mathcal { N } ( \bar { 0 _ { } } , \sigma _ { m a x } ^ { 2 } )$ . We choose $\sigma _ { m a x } = 0 . 0 2$ and 0.03. Other simulation settings are by default the same as in Section 5.1. We then build the Robust ALISTA model, following the strategy in Section 4 and using a 4-layer encoder for approximating its second step (see Appendix G for details). Correspondingly, we compare Robust ALISTA with TiLISTA and ALISTA with specific data augmentation: we straightforwardly augment their training sets, by including all data generated with randomly perturbed $\breve { \tilde { \mathbf { D } } } \mathbf { s }$ when training Robust ALISTA. We also include the data-free FISTA algorithm into the comparison. + +Figure 4 plots the results when the trained models are applied on the testing data, generated with the same dictionary and perturbed by $\mathcal { N } ( 0 , \sigma _ { t } )$ . We vary $\sigma _ { t }$ from zero to slightly above $\sigma _ { m a x }$ . Not surprisingly, FISTA is unaffected, while the other three data-driven models all slight degrade as $\sigma _ { t }$ increases. Compared to the augmented TiLISTA and ALISTA whose performance are both inferior to FISTA, the proposed Robust ALISTA appears to be much more favorable in improving robustness to model perturbations. In both $\sigma _ { m a x }$ cases, it consistently achieves much lower NMSE than FISTA, even when $\sigma _ { t }$ has slightly surpassed $\sigma _ { m a x }$ . Although the NMSE of ALISTA may decrease faster if $\sigma _ { t }$ continues growing larger, such decrease could be alleviated by improving $\sigma _ { m a x }$ in training, e.g., by comparing $\sigma _ { m a x } = 0 . 0 2$ and 0.03. Robust ALISTA demonstrates remarkable robustness and maintains the best NMSE performance, within at least the $[ 0 , \sigma _ { m a x } ]$ range. + +# 6 CONCLUSIONS AND FUTURE WORK + +Based on the recent theoretical advances of LISTA, we have made further steps to reduce the training complexity and improve the robustness of LISTA. Specifically, we no longer train any matrix for LISTA but directly use the solution to an analytic minimization problem to solve for its layer-wise weights. Therefore, only two scalar sequences (stepsizes and thresholds) still need to be trained. Excluding the matrix from training is backed by our theoretical upper and lower bounds. The resulting method, Analytic LISTA or ALISTA, is not only faster to train but performs as well as the state-of-the-art variant of LISTA by (Chen et al., 2018). This discovery motivates us to further replace the minimization algorithm by its unfolding neural network, and train this neural network to more quickly produce the weight matrix. The resulting algorithm is used to handle perturbations in the model dictionary — we only train once for a dictionary with all its small perturbations. Our future work will investigate the theoretical sensitivity of ALISTA (and its convolutional version) to noisy measurements. + +# REFERENCES + +Thomas Blumensath and Mike E. Davies. Iterative hard thresholding for compressed sensing. Applied and Computational Harmonic Analysis, 27(3):265 – 274, 2009. + +Mark Borgerding, Philip Schniter, and Sundeep Rangan. AMP-inspired deep networks for sparse linear inverse problems. 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Restricted strong convexity and its applications to convergence analysis of gradient-type methods in convex optimization. Optimization Letters, 9(5):961–979, 2015. + +Jian Zhang and Bernard Ghanem. ISTA-Net: Interpretable optimization-inspired deep network for image compressive sensing. In IEEE CVPR, 2018. + +Bin Zhao, Li Fei-Fei, and Eric P Xing. Online detection of unusual events in videos via dynamic sparse coding. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pp. 3313–3320. IEEE, 2011. + +Joey Tianyi Zhou, Kai Di, Jiawei Du, Xi Peng, Hao Yang, Sinno Jialin Pan, Ivor W Tsang, Yong Liu, Zheng Qin, and Rick Siow Mong Goh. SC2Net: Sparse LSTMs for sparse coding. In AAAI Conference on Artificial Intelligence, 2018. + +# A PROOF OF THEOREM 1 + +In this proof, we use the notion $\mathbf { x } ^ { ( k ) }$ to replace $\mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } )$ for simplicity. We fix $\mathbf { D }$ in the proof, $\tilde { \mu } ( \mathbf { D } )$ can be simply written as $\tilde { \mu }$ . + +Before proving Theorem 1, we present and prove a lemma. + +Lemma 1. With all the settings the same with those in Theorem $^ { l }$ , we have + +$$ +\mathrm { s u p p o r t } ( \mathbf { x } ^ { ( k ) } ) \subset \mathbb { S } , \quad \forall k . +$$ + +In another word, there are no false positives in $\mathbf { x } ^ { ( k ) }$ : $x _ { i } ^ { ( k ) } = 0 , \forall i \notin \mathbb { S } , \forall k$ + +Proof. Take arbitrary $\mathbf { x } ^ { * } \in \mathcal { X } ( B , s )$ . We prove Lemma 1 by induction. As $k = 0$ , (27) is satisfied since $\mathbf { x } ^ { ( 0 ) } = \mathbf { 0 }$ . Fixing $k$ , and assuming support $( \mathbf { x } ^ { ( k ) } ) \subset \mathbb { S }$ , we have + +$$ +\begin{array} { r l } & { x _ { i } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \Big ( x _ { i } ^ { ( k ) } - \gamma ^ { ( k ) } ( \mathbf { W } _ { : , i } ) ^ { T } \big ( \mathbf { D x } ^ { ( k ) } - \mathbf { b } \big ) \Big ) } \\ & { \quad \quad \quad = \eta _ { \theta ^ { ( k ) } } \Big ( - \gamma ^ { ( k ) } \displaystyle \sum _ { j \in \mathbb { S } } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } \big ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \big ) \Big ) , \quad \forall i \notin \mathbb { S } . } \end{array} +$$ + +By (9), the thresholds are taken as $\theta ^ { ( k ) } = \tilde { \mu } \gamma ^ { ( k ) } \operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k ) } - \mathbf { x } ^ { * } \| _ { 1 } \}$ . Also, since $\mathbf { W } \in \mathcal { W } ( \mathbf { D } )$ , we have $| ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } | \leq \tilde { \mu }$ for all $j \neq i$ . Thus, for all $i \not \in { \mathbb S }$ , + +$$ +\begin{array} { l } { { \displaystyle \theta ^ { ( k ) } \geq \widetilde { \mu } \gamma ^ { ( k ) } \| \mathbf { x } ^ { ( k ) } - \mathbf { x } ^ { * } \| _ { 1 } = \displaystyle \sum _ { j \in \mathrm { s u p p o r t } ( \mathbf { x } ^ { ( k ) } ) } \widetilde { \mu } \gamma ^ { ( k ) } \big | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \big | = \displaystyle \sum _ { j \in \mathbb { S } } \widetilde { \mu } \gamma ^ { ( k ) } \big | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \big | } } \\ { { \displaystyle \geq \Big | - \gamma ^ { ( k ) } \sum _ { j \in \mathbb { S } } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) \Big | , } } \end{array} +$$ + +which implies x(ki +1) = 0, ∀i /∈ S by the definition of ηθ(k) , i.e., + +$$ +\mathrm { s u p p o r t } ( \mathbf { x } ^ { ( k + 1 ) } ) \subset \mathbb { S } +$$ + +By induction, (27) is proved. + +With Lemma 1, we are able to prove Theorem 1 now. + +Proof of Theorem $^ { l }$ . Take arbitrary $\mathbf { x } ^ { * } \in \mathcal { X } ( B , s )$ . For all $i \in \mathbb { S }$ , by (27), we obtain + +$$ +\begin{array} { r l } & { x _ { i } ^ { ( k + 1 ) } = \eta _ { \theta ^ { ( k ) } } \left( x _ { i } ^ { ( k ) } - \gamma ^ { ( k ) } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } ) \right) } \\ & { \quad \quad \quad \in x _ { i } ^ { ( k ) } - \gamma ^ { ( k ) } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } ) - \theta ^ { ( k ) } \partial \ell _ { 1 } ( x _ { i } ^ { ( k + 1 ) } ) , } \end{array} +$$ + +where $\partial \ell _ { 1 } ( x )$ is the sub-gradient of $| x | , x \in \mathbb { R }$ : + +$$ +\partial { \boldsymbol { \ell } } _ { 1 } ( x ) = { \left\{ \begin{array} { l l } { \{ \operatorname { s i g n } ( x ) \} } & { { \mathrm { i f ~ } } x \neq 0 , } \\ { [ - 1 , 1 ] } & { { \mathrm { i f ~ } } x = 0 . } \end{array} \right. } +$$ + +The choice of $\mathbf { W } \in \mathcal { W } ( \mathbf { D } )$ gives $( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , i } = 1$ . Thus, + +$$ +\begin{array} { r l } & { \quad x _ { i } ^ { ( k ) } - \gamma ^ { ( k ) } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ( { \mathbf x } _ { \mathbb { S } } ^ { ( k ) } - { \mathbf x } _ { \mathbb { S } } ^ { * } ) } \\ & { = x _ { i } ^ { ( k ) } - \gamma ^ { ( k ) } \underset { j \in \mathbb { S } , j \neq i } { \sum } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) - \gamma ^ { ( k ) } ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) } \\ & { = x _ { i } ^ { * } - \gamma ^ { ( k ) } \underset { j \in \mathbb { S } , j \neq i } { \sum } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) + ( 1 - \gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) . } \end{array} +$$ + +Then the following inclusion formula holds for all $i \in \mathbb S$ , + +$$ +\mathbf { \Phi } _ { i } ^ { ( k + 1 ) } - \mathbf { \Phi } _ { x _ { i } ^ { * } } ^ { * } \in - \gamma ^ { ( k ) } \sum _ { \substack { j \in \mathbb { S } , j \neq i } } ( \mathbf { W } _ { : , i } ) ^ { T } \mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) - \theta ^ { ( k ) } \partial \ell _ { 1 } ( x _ { i } ^ { ( k + 1 ) } ) + ( 1 - \gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) . +$$ + +By the definition of $\partial \ell _ { 1 }$ , every element in $\partial \ell _ { 1 } ( x ) , \forall x \in \mathbb { R }$ has a magnitude less than or equal to 1. Thus, for all $i \in \mathbb S$ , + +$$ +\begin{array} { r l } & { | x _ { i } ^ { ( k + 1 ) } - x _ { i } ^ { * } | \le \displaystyle \sum _ { j \in \mathbb S , j \ne i } \gamma ^ { ( k ) } \Big | ( \mathbf W _ { : , i } ) ^ { T } \mathbf D _ { : , j } \Big | | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } | + \theta ^ { ( k ) } + | 1 - \gamma ^ { ( k ) } | | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } | } \\ & { \qquad \le \widetilde { \mu } \gamma ^ { ( k ) } \displaystyle \sum _ { j \in \mathbb S , j \ne i } | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } | + \theta ^ { ( k ) } + | 1 - \gamma ^ { ( k ) } | | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } | . } \end{array} +$$ + +Equation (27) implies $\| \mathbf { x } ^ { ( k ) } - \mathbf { x } ^ { * } \| _ { 1 } = \| \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } \| _ { 1 }$ for all $k$ . Then + +$$ +\begin{array} { r l } { { \| { \mathbf { x } } ^ { ( k + 1 ) } - { \mathbf { x } } ^ { * } \| _ { 1 } = \sum _ { i \in \mathbb { S } } \big | x _ { i } ^ { ( k + 1 ) } - x _ { i } ^ { * } \big | } } \\ & { \leq \sum _ { i \in \mathbb { S } } \Big ( \tilde { \mu } \gamma ^ { ( k ) } \sum _ { j \in \mathbb { S } , j \neq i } \big | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \big | + \theta ^ { ( k ) } + \big | 1 - \gamma ^ { ( k ) } \big | \big | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } \big | \Big ) } \\ & { = \tilde { \mu } \gamma ^ { ( k ) } ( \big | \mathbb { S } \big | - 1 ) \sum _ { i \in \mathbb { S } } \big | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } \big | + \theta ^ { ( k ) } \big | \mathbb { S } \big | + \big | 1 - \gamma ^ { ( k ) } \big | \big | { \mathbf { x } } ^ { ( k ) } - { \mathbf { x } } ^ { * } \big | \big | _ { 1 } } \\ & { = \tilde { \mu } \gamma ^ { ( k ) } ( \big | \mathbb { S } \big | - 1 ) \big | \big | { \mathbf { x } } ^ { ( k ) } - { \mathbf { x } } ^ { * } \big | \big | _ { 1 } + \theta ^ { ( k ) } \big | \mathbb { S } \big | + \big | 1 - \gamma ^ { ( k ) } \big | \big | \big | { \mathbf { x } } ^ { ( k ) } - { \mathbf { x } } ^ { * } \big | \big | . } \end{array} +$$ + +Taking supremum of the above inequality over $\mathbf { x } ^ { * } \in \mathcal { X } ( B , s )$ , by $| \mathbb { S } | \le s$ , + +$$ +\operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k + 1 ) } - \mathbf { x } ^ { * } \| _ { 1 } \} \le \Big ( \tilde { \mu } \gamma ^ { ( k ) } ( s - 1 ) + | 1 - \gamma ^ { ( k ) } | \Big ) \operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k ) } - \mathbf { x } ^ { * } \| _ { 1 } \} + \theta ^ { ( k ) } s . +$$ + +By the value of $\theta ^ { ( k ) }$ given in (9), we have + +$$ +\operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k + 1 ) } - \mathbf { x } ^ { * } \| _ { 1 } \} \leq \Big ( \gamma ^ { ( k ) } ( 2 \tilde { \mu } s - \tilde { \mu } ) + | 1 - \gamma ^ { ( k ) } | \Big ) \operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k ) } - \mathbf { x } ^ { * } \| _ { 1 } \} . +$$ + +Let $c ^ { ( \tau ) } = - \log \left( ( 2 \tilde { \mu } s - \tilde { \mu } ) \gamma ^ { ( \tau ) } + | 1 - \gamma ^ { ( \tau ) } | \right)$ . Then, by induction, + +$$ +\operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k + 1 ) } - \mathbf { x } ^ { * } \| _ { 1 } \} \leq \exp \Big ( - \sum _ { \tau = 0 } ^ { k } c ^ { ( \tau ) } \Big ) \operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( 0 ) } - \mathbf { x } ^ { * } \| _ { 1 } \} \leq \exp \Big ( - \sum _ { \tau = 0 } ^ { k } c ^ { ( \tau ) } \Big ) s B . +$$ + +Since $\| \mathbf { x } \| _ { 2 } \leq \| \mathbf { x } \| _ { 1 }$ for any $\mathbf { x } \in \mathbb { R } ^ { n }$ , we can get the upper bound for $\ell _ { 2 }$ norm: + +$$ +\operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k + 1 ) } - \mathbf { x } ^ { * } \| _ { 2 } \} \leq \operatorname* { s u p } _ { \mathbf { x } ^ { * } } \{ \| \mathbf { x } ^ { ( k + 1 ) } - \mathbf { x } ^ { * } \| _ { 1 } \} \leq s B \exp \Big ( - \sum _ { \tau = 0 } ^ { k } c ^ { ( \tau ) } \Big ) . +$$ + +The assumption $s < ( 1 + 1 / \tilde { \mu } ) / 2$ gives $2 \tilde { \mu } s - \tilde { \mu } < 1$ . If $0 < \gamma ^ { ( k ) } \leq 1$ , we have $c ^ { ( k ) } > 0$ . If $1 < \gamma ^ { ( k ) } < 2 / ( 1 + 2 \tilde { \mu } s - \tilde { \mu } )$ , we have + +$$ +( 2 \tilde { \mu } s - \tilde { \mu } ) \gamma ^ { ( k ) } + | 1 - \gamma ^ { ( k ) } | = ( 2 \tilde { \mu } s - \tilde { \mu } ) \gamma ^ { ( k ) } + \gamma ^ { ( k ) } - 1 < 1 , +$$ + +which implies $c ^ { ( k ) } > 0$ . Theorem 1 is proved. + +# B PROOF OF THEOREM 2 + +Proof of Theorem 2. We fix $\mathbf { D }$ and sample a $\mathbf { x } ^ { * } \sim P _ { X }$ . + +If we can prove + +$$ +P \Big ( ( 1 3 ) \mathrm { d o e s ~ n o t ~ h o l d } \Big | \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) \leq \epsilon | \mathbb { S } | + \epsilon ^ { | \mathbb { S } | } , +$$ + +then the lower bound (13) in Theorem 2 is proved by + +$$ +\begin{array} { r l } & { P \Big ( ( 1 3 ) \mathrm { h o l d s } \Big ) = \displaystyle \sum _ { \mathbb { S } , 2 \leq | \mathbb { S } | \leq s } P \Big ( ( 1 3 ) \mathrm { h o l d s } \Big | \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) P \Big ( \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) } \\ & { \qquad \geq ( 1 - \epsilon s ^ { 3 / 2 } - \epsilon ^ { 2 } ) \displaystyle \sum _ { 2 \leq | \mathbb { S } | \leq s } P \Big ( \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) } \\ & { \qquad = 1 - \epsilon s ^ { 3 / 2 } - \epsilon ^ { 2 } . } \end{array} +$$ + +Now we fix $k$ and prove inequality (28) by three steps: + +# Step 1: If (13) does not hold, then what condition $\mathbf { x } ^ { * }$ should satisfy? + +Fixing $k$ , we define a set ${ \mathcal X } ^ { ( k ) } ( \epsilon )$ , which involves all the $\mathbf { x } ^ { * }$ that does not satisfy (13): + +$$ +\mathcal { X } ^ { ( k ) } ( \epsilon ) = \{ ( 1 3 ) \mathrm { d o e s ~ n o t ~ h o l d } \} = \Big \{ \mathbf { x } ^ { * } \Big | \| \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } ^ { * } \| _ { 2 } < \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \Big ( \frac { \bar { \sigma } _ { \operatorname* { m i n } } } { 3 ^ { s } } \Big ) ^ { k } \Big \} . +$$ + +Let $\mathbb { S } = \operatorname { s u p p o r t } ( \mathbf { x } ^ { * } )$ . For $\mathbf { x } ^ { * } \in \mathcal { X } ^ { ( k ) } ( \epsilon )$ , we consider two cases: + +1. $| x _ { i } ^ { * } | > \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } ( \bar { \sigma } _ { \operatorname* { m i n } } / 3 ^ { s } ) ^ { k } , \forall i \in \mathbb { S } .$ +2. $| x _ { i } ^ { * } | \leq \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \big ( \bar { \sigma } _ { \operatorname* { m i n } } / 3 ^ { s } \big ) ^ { k }$ , for some $i \in \mathbb S$ . + +If case 1 holds, we obtain that the support of $\mathbf { x } ^ { ( k ) }$ is exactly the same with that of $\mathbf { x } ^ { * }$ : + +$$ +\operatorname { s u p p o r t } ( \mathbf { x } ^ { ( k ) } ( \mathbf { x } ^ { * } ) ) = \mathbb { S } . +$$ + +Then the relationship between $\mathbf { x } ^ { ( k ) }$ and $\mathbf { x } ^ { ( k - 1 ) }$ can be reduced to an affine transform: + +$$ +\begin{array} { r l } & { \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } = \eta _ { \theta ^ { ( k ) } } \left( \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } ( \mathbf { D } \mathbf { x } ^ { ( k - 1 ) } - \mathbf { b } ) \right) } \\ & { \quad \quad = \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } ) - \theta ^ { ( k - 1 ) } \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } ) . } \end{array} +$$ + +Subtracting $\mathbf { x } ^ { * }$ from the two sides of (29), we obtain + +$$ +\Big \| \big ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } \big ) ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } ) - \theta ^ { ( k - 1 ) } \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } ) \Big \| _ { 2 } = \| \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } \| _ { 2 } = \| \mathbf { x } ^ { ( k ) } - \mathbf { x } ^ { * } \| _ { 2 } , +$$ + +where the last equality is due to Definition 3. Thus, for all $\mathbf { x } ^ { * } \in \mathcal { X } ^ { ( k ) } ( \epsilon )$ , if case 1 holds, we have + +$$ +\| \big ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } \big ) ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } ) - \theta ^ { ( k - 1 ) } \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } ) \Big \| _ { 2 } \leq \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \big ( \bar { \sigma } _ { \operatorname* { m i n } } / 3 ^ { s } \big ) ^ { k } . +$$ + +Multiplying both sides of (30) by $( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ) ^ { - 1 }$ , we have + +$$ +\begin{array} { r l } & { \quad \| \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } - \mathbf { x } _ { \mathbb { S } } ^ { * } - \theta ^ { ( k - 1 ) } ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ) ^ { - 1 } \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } ) \| _ { 2 } } \\ & { { \leq } \| ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ) ^ { - 1 } \| _ { 2 } \cdot \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } ( \bar { \sigma } _ { \operatorname* { m i n } } / 3 ^ { s } ) ^ { k } \leq \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } ( \bar { \sigma } _ { \operatorname* { m i n } } ) ^ { k - 1 } 3 ^ { - k s } , } \end{array} +$$ + +where the last inequality is due to (11). Let $\tilde { \mathbf { x } } ^ { ( k - 1 ) }$ denote the bias of $\mathbf { x } ^ { ( k - 1 ) }$ : + +$$ +\begin{array} { r } { \tilde { \mathbf { x } } ^ { ( k - 1 ) } \triangleq \boldsymbol { \theta } ^ { ( k - 1 ) } ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } ) ^ { - 1 } \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } ) , } \end{array} +$$ + +then we get a condition that $\mathbf { x } ^ { * }$ satisfies if case 1 holds: + +$$ +\begin{array} { r } { \boldsymbol { \chi } ^ { ( k - 1 ) } ( \epsilon ) = \Bigl \{ \mathbf { x } ^ { * } \Big | \bigl \| \mathbf { x } _ { \mathbb { S } } ^ { ( k - 1 ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } _ { \mathbb { S } } ^ { * } - \tilde { \mathbf { x } } ^ { ( k - 1 ) } ( \mathbf { x } ^ { * } ) \bigr \| _ { 2 } \le \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } ( \bar { \sigma } _ { \operatorname* { m i n } } ) ^ { k - 1 } \mathbb { 3 } ^ { - k s } \Bigr \} . } \end{array} +$$ + +If case 2 holds, $\mathbf { x } ^ { * }$ belongs to the following set: + +$$ +\begin{array} { r } { \tilde { \boldsymbol { \chi } } ^ { ( k ) } ( \epsilon ) = \Bigl \{ \mathbf { x } ^ { * } \Big | | \boldsymbol { x } _ { i } ^ { * } | \le \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \big ( \bar { \sigma } _ { \operatorname* { m i n } } / 3 ^ { s } \big ) ^ { k } , \mathrm { ~ f o r ~ s o m e ~ } i \in \mathbb { S } \Bigr \} . } \end{array} +$$ + +Then for any $\mathbf { x } ^ { * } \in \mathcal { X } ^ { ( k ) } ( \epsilon )$ , either $\mathbf { x } ^ { * } \in \mathcal { X } ^ { ( k - 1 ) } ( \epsilon )$ or $\mathbf { x } ^ { * } \in \tilde { \mathcal { X } } ^ { ( k ) } ( \epsilon )$ holds. In another word, + +$$ +\mathcal { X } ^ { ( k ) } ( \epsilon ) \subset \tilde { \mathcal { X } } ^ { ( k ) } ( \epsilon ) \cup \mathcal { X } ^ { ( k - 1 ) } ( \epsilon ) . +$$ + +Step 2: By imitating the construction of ${ \mathcal { X } } ^ { ( k ) } ( \epsilon )$ , we construct + +$$ +\mathcal { X } ^ { ( k - 2 ) } ( \epsilon ) , \mathcal { X } ^ { ( k - 3 ) } ( \epsilon ) , \cdot \cdot \cdot . +$$ + +Similar to Step 1, we divide $\chi ^ { ( k - 1 ) } ( \epsilon )$ into two sets: $\tilde { \boldsymbol { \chi } } ^ { ( k - 1 ) } ( \epsilon )$ and $\chi ^ { ( k - 2 ) } ( \epsilon )$ , then we divide $\chi ^ { ( k - 2 ) } ( \epsilon )$ into $\tilde { \chi } ^ { ( k - 2 ) } ( \epsilon )$ and $\chi ^ { ( k - 3 ) } ( \epsilon )$ . Repeating the process, until dividing $\mathcal { X } ^ { ( 1 ) } ( \epsilon )$ into $\tilde { \chi } ^ { ( 1 ) } ( \epsilon )$ and ${ \mathcal X } ^ { ( 0 ) } ( \epsilon )$ . + +By induction, we have + +$$ +\mathcal { X } ^ { ( k ) } ( \epsilon ) \subset \tilde { \mathcal { X } } ^ { ( k ) } ( \epsilon ) \cup \tilde { \mathcal { X } } ^ { ( k - 1 ) } ( \epsilon ) \cup \tilde { \mathcal { X } } ^ { ( k - 2 ) } ( \epsilon ) \cup \cdots \cup \tilde { \mathcal { X } } ^ { ( 1 ) } ( \epsilon ) \cup \mathcal { X } ^ { ( 0 ) } ( \epsilon ) , +$$ + +where the sets are defined as follows for all $j = 0 , 1 , 2 , \cdots , k$ : + +$$ +\begin{array} { r l } & { \tilde { { \boldsymbol { \chi } } } ^ { ( k - j ) } ( \epsilon ) = \Bigl \{ \mathbf { x } ^ { * } \Big | | x _ { i } ^ { * } + \tilde { x } _ { i } ^ { ( k - j ) } ( \mathbf { x } ^ { * } ) | < \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \big ( \bar { \sigma } _ { \operatorname* { m i n } } \big ) ^ { k - j } 3 ^ { - k s } , \mathrm { ~ f o r ~ s o m e ~ } i \in \mathbb { S } . \Bigr \} , } \\ & { { \boldsymbol { \chi } } ^ { ( k - j ) } ( \epsilon ) = \Bigl \{ \mathbf { x } ^ { * } \Big | \| \mathbf { x } _ { \mathbb { S } } ^ { ( k - j ) } ( \mathbf { x } ^ { * } ) - \mathbf { x } _ { \mathbb { S } } ^ { * } - \tilde { \mathbf { x } } ^ { ( k - j ) } ( \mathbf { x } ^ { * } ) \| _ { 2 } \le \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } \big ( \bar { \sigma } _ { \operatorname* { m i n } } \big ) ^ { k - j } 3 ^ { - k s } \Bigr \} } \end{array} +$$ + +and the bias is defined as following for all $j = 0 , 1 , 2 , \cdots , k$ : + +$$ +\tilde { \mathbf { x } } ^ { ( k - j ) } ( \mathbf { x } ^ { * } ) = \sum _ { t = 1 } ^ { j } \left( \mathbf { I } - \left( \mathbf { W } _ { : , \mathbb { S } } ^ { ( k - j + t - 1 ) } \right) ^ { T } \mathbf { D } _ { : , \mathbb { S } } \right) ^ { - t } \boldsymbol { \theta } ^ { ( k - j + t - 1 ) } \mathrm { s i g n } \big ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - j + t ) } ( \mathbf { x } ^ { * } ) \big ) . +$$ + +# Step 3: Estimating the probabilities of all the sets in (31). + +By (31), we have + +$$ +\begin{array} { r l } { { P \Big ( \mathbf { x } ^ { * } \in \mathcal { X } ^ { ( k ) } ( \epsilon ) \Big | \operatorname { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) } \quad } & { } \\ & { \leq \displaystyle \sum _ { j = 1 } ^ { k - 1 } P \Big ( \mathbf { x } ^ { * } \in \tilde { \mathcal { X } } ^ { ( k - j ) } ( \epsilon ) \Big | \operatorname { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) + P \Big ( \mathbf { x } ^ { * } \in \mathcal { X } ^ { ( 0 ) } ( \epsilon ) \Big | \operatorname { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) . } \end{array} +$$ + +Now we have to prove that each of the above terms is small, then $P ( \mathbf { x } ^ { * } \in \mathcal { X } ^ { ( k ) } ( \epsilon ) | \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } )$ is small and (28) will be proved. + +Define a set of $n$ -dimensional sign numbers + +$$ +\operatorname { S i } ( n ) = { \Big \{ } ( s _ { 1 } , s _ { 2 } , \cdots , s _ { n } ) { \Big | } s _ { i } \in \{ 0 , - 1 , 1 \} , \forall i = 1 , \cdots , n { \Big \} } . +$$ + +Since $\mathrm { s i g n } \big ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - j + t ) } \big ) \ \in \ \mathrm { S i } ( | \mathbb { S } | )$ for all $t = 1 , 2 , \cdots , j$ , $\{ \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - j + t ) } ) \} _ { t = 1 } ^ { j }$ has finitely possible values. Let $\mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k - j + t ) } ) = \mathbf { s } ^ { ( t ) }$ for $t = 1 , 2 , \cdots , j$ . Then $\tilde { x } _ { i } ^ { ( k - j ) } ( \mathbf { x } ^ { * } )$ is independent of $\mathbf { x } ^ { * }$ and can be written as $\tilde { x } _ { i } ^ { ( k - j ) } ( \mathbf { s } ^ { ( 1 ) } , \mathbf { s } ^ { ( 2 ) } , \cdot \cdot \cdot , \mathbf { s } ^ { ( j ) } )$ . Thus, we have + +$$ +\begin{array} { r l } & { \quad P ( x ^ { * } \in \tilde { \mathcal { X } } ^ { ( k - \delta ) } ( e ) \operatorname* { l a n p p o r } ( x ^ { * } ) = 8 ) } \\ & { = \displaystyle \sum _ { i \in \mathbb { S } } \displaystyle \sum _ { s \in \{ x \} \in \{ 0 , 1 \} \leq n \leq i \leq | \mathbb { S } } \sum _ { s \in \{ 1 \} | s | \in \mathcal { S } } \sum _ { s \in \{ 1 \} | s | \in \mathcal { S } } \sum _ { s \in \{ 1 \} | s | } } \\ & { \quad P \Big ( | x _ { s } ^ { * } + \tilde { x } _ { s } ^ { ( k ) } ( s \cdot s ) ( x ) | \times | e | \mathcal { X } | | \Big ) \cdot \mathrm { ~ e ~ i s ' ~ s h ( \tilde { x } | \tilde { x } ) } } \\ & \leq \displaystyle \sum _ { i \in \mathbb { S } } \sum _ { s \in \{ x \} \in \{ 1 \} \leq n \} \sum _ { s \in \{ 1 \} \atop | s \in \{ 1 \} } \sum _ { s \in \{ 1 \} \in \{ 1 \} } \sum _ { | s | \in \mathcal { S } } \sum _ { s \in \{ 1 \} | s | } \Big ( x _ { s } ^ { ( k ) } - \tilde { x } _ { s } ^ { ( k ) } - s \cdot k \cdot s _ { s } \cdot \operatorname* { l i g n } ( \mathbf { x } _ { s } ^ { ( k ) } ) = \mathbf { s } ^ { ( 1 ) } , \cdots , \ s \operatorname* { l i p } ( \mathbf { x } _ { s } ^ { ( k - \delta + 1 ) } ) = \mathbf { s } ^ { ( \delta ) } \Big | s \Big ) } \\ & { \quad \times \displaystyle \sum _ { i \in \mathbb { S } } \Big ( | x _ { s } ^ { * } + \tilde { x } _ { s } ^ { ( k ) } ( s \cdot s ) ( | \mathbf { x } | ) \cdot \ s | ^ { 2 } } \\ & { \quad \times \displaystyle \sum _ { s \in \{ x \} \leq n \leq i \leq | \mathbb { S } } \sum _ { s \in \{ 1 \} } \sum _ { s \in \{ 1 \} } \sum _ { s \in \{ 1 \} \leq n \leq i \leq | \mathbb { S } } \sum _ { s \in \{ 1 \} } E \Big ( | s | B ( \sigma _ { \operatorname* { m i n } } ) ^ { k - \delta } - s \cdot | s | \operatorname* { s u p p o r } ( \mathbf { x } ^ { * } ) = \mathbf { s } \Big ) } \\ & \leq \displaystyle \sum _ { s \in \{ 1 \} \leq n \leq i \leq | \mathbb { S } | } \sum _ s \in \{ 1 \end{array} +$$ + +where the second inequality comes from the uniform distribution of ${ \mathbf { x } } _ { \mathbb { S } } ^ { * }$ (Assumption 2), the last inequality comes from $| \mathbb { S } | \le s$ . + +The last term, due to the uniform distribution of ${ \mathbf { x } } _ { \mathbb { S } } ^ { * }$ and $\mathbf { x } ^ { ( 0 ) } = \mathbf { 0 }$ , can be bounded by + +$$ +\begin{array} { r l } & { \quad P ( \mathbf { x } ^ { * } \in \mathcal { X } ^ { ( 0 ) } ( \epsilon ) | \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } ) } \\ & { = P \Big ( \| \mathbf { x } ^ { * } + \tilde { \mathbf { x } } ^ { ( 0 ) } ( \mathbf { x } ^ { * } ) \| _ { 2 } \leq \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } 3 ^ { - k s } \Big | \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } \Big ) } \\ & { = \displaystyle \sum _ { \mathbf { s } ^ { ( 1 ) } \in \mathbb { S } ( | \mathbb { S } | ) } \sum _ { \mathbf { s } ^ { ( 2 ) } \in \mathrm { S i } ( | \mathbb { S } | ) } \cdot \sum _ { \mathbf { s } ^ { ( k ) } \in \mathrm { S i } ( | \mathbb { S } | ) } } \\ & { \quad P \Big ( \| \mathbf { x } ^ { * } + \tilde { \mathbf { x } } ^ { ( 0 ) } ( \mathbf { x } ^ { * } ) \| _ { 2 } \leq \epsilon \| \mathbf { x } ^ { * } \| _ { 2 } 3 ^ { - k s } , \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( 1 ) } ) = \mathbf { s } ^ { ( 1 ) } , \cdots , \mathrm { s i g n } ( \mathbf { x } _ { \mathbb { S } } ^ { ( k ) } ) = \mathbf { s } ^ { ( k ) } \Big | \mathrm { s u p p o r t } ( \mathbf { x } ^ { * } ) = \mathbb { S } } \\ & { \leq 3 ^ { k | \mathbb { S } | } \Big ( ( \epsilon 3 ^ { - k s } ) ^ { | \mathbb { S } | } \Big ) \leq \epsilon ^ { | \mathbb { S } | } . } \end{array} +$$ + +Then we obtain + +$$ +\begin{array} { l } { { \displaystyle P ( { \bf x } ^ { * } \in \mathcal { X } ^ { ( k ) } ( \epsilon ) \vert \mathrm { s u p p o r t } ( { \bf x } ^ { * } ) = \mathbb { S } ) } \ ~ } \\ { \displaystyle \leq \sum _ { j = 0 } ^ { k - 1 } \epsilon \vert \mathbb { S } \vert ^ { 3 / 2 } ( \bar { \sigma } _ { \operatorname* { m i n } } ) ^ { k - j } \mathbb { 3 } ^ { ( j - k ) \vert \mathbb { S } \vert } + \epsilon ^ { \vert \mathbb { S } \vert } = \sum _ { j = 1 } ^ { k } \epsilon \vert \mathbb { S } \vert ^ { 3 / 2 } ( \bar { \sigma } _ { \operatorname* { m i n } } ) ^ { j } 3 ^ { - j \vert \mathbb { S } \vert } + \epsilon ^ { \vert \mathbb { S } \vert } } \ \\ { { \displaystyle = \epsilon \vert \mathbb { S } \vert ^ { 3 / 2 } \frac { \bar { \sigma } _ { \operatorname* { m i n } } 3 ^ { - \vert \mathbb { S } \vert } } { 1 - \bar { \sigma } _ { \operatorname* { m i n } } 3 ^ { - \vert \mathbb { S } \vert } } \Big ( 1 - ( \bar { \sigma } _ { \operatorname* { m i n } } 3 ^ { - \vert \mathbb { S } \vert } ) ^ { k } \Big ) + \epsilon ^ { \vert \mathbb { S } \vert } \leq \epsilon \vert \mathbb { S } \vert ^ { 3 / 2 } + \epsilon ^ { \vert \mathbb { S } \vert } } . } \end{array} +$$ + +Then (28) is proved. + +# C PROOF OF THEOREM 3 + +There are two conclusions in Theorem 3. We prove the two conclusions in the following two subsections respectively. + +# C.1 PROOF OF CONCLUSION 1. + +Before proving Conclusion 1, we analyze the operator $\mathbf { D } _ { \mathrm { c i r } } ^ { N }$ in detail. + +The circular convolution (23) is equivalent with: + +$$ +\mathbf { b } ( i , j ) = \sum _ { k = 0 } ^ { N - 1 } \sum _ { l = 0 } ^ { N - 1 } \sum _ { m = 1 } ^ { M } \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( i , j ; k , l , m ) \mathbf { x } _ { m } ( k , l ) , \quad 0 \leq i , j \leq N - 1 , +$$ + +where the circulant matrix is element-wise defined as: + +$$ +\begin{array} { r } { \mathrm { \mathfrak { I } } _ { \mathrm { c i r } } ^ { N } ( i , j ; k , l , m ) = \left\{ \mathbf { d } _ { m } \left( ( k - i ) _ { \mathrm { m o d } N } , ( l - j ) _ { \mathrm { m o d } N } \right) , \ : \ : \ : 0 \leq ( k - i ) _ { \mathrm { m o d } N } , ( l - j ) _ { \mathrm { m o d } N } \leq D - 1 \right. } \\ { 0 , \ : \ : \ : \mathrm { \ o t h e r s } } \end{array} +$$ + +Similarly, the corresponding circulant matrix $\mathbf { W } _ { \mathrm { c i r } } ^ { N } ( i , j ; k , l , m )$ of dictionary w is: + +$$ +N _ { \mathrm { c i r } } ^ { N } ( i , j ; k , l , m ) = \left\{ \begin{array} { l l } { { \displaystyle { \bf w } _ { m } \left( ( k - i ) _ { \mathrm { m o d } N } , ( l - j ) _ { \mathrm { m o d } N } \right) } , } & { { \displaystyle 0 \leq ( k - i ) _ { \mathrm { m o d } N } , ( l - j ) _ { \mathrm { m o d } N } \leq D - 1 } } \\ { { 0 , } } & { { \mathrm { o t h e r s } } } \end{array} \right. +$$ + +As we defined in Section 3, $\mathbf { b }$ is a vector. With $\mathbf { x } = [ \mathbf { x } _ { 1 } , \cdots , \mathbf { x } _ { M } ] ^ { T }$ , $\mathbf { x }$ is a vector. Then the operator $\mathbf { D } _ { \mathrm { c i r } } ^ { N }$ is a matrix, where $( i , j )$ is its row index and $( k , l , m )$ is its column index. + +Define a function measuring the difference between $i$ and $k$ : + +$$ +I ( i , k ) \triangleq ( k - i ) _ { \mathrm { m o d } N } , \quad 0 \leq i , k \leq N - 1 . +$$ + +The coherence between $\mathbf { D } _ { \mathrm { c i r } } ^ { N } ( i , j ; k , l , m )$ and $\mathbf { W } _ { \mathrm { c i r } } ^ { N } ( i , j ; k , l , m )$ : $\mathbf { B } _ { \mathrm { c o h } } = ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N }$ is elementwise defined by: + +$$ +\begin{array} { r l } & { \mathbf { B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) = \displaystyle \sum _ { i = 0 } ^ { N - 1 } \sum _ { j = 0 } ^ { N - 1 } \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( i , j ; k _ { 1 } , l _ { 1 } , m _ { 1 } ) \mathbf { W } _ { \mathrm { c i r } } ^ { N } ( i , j ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) } \\ & { \qquad = \displaystyle \sum _ { i \in \mathbb { Z } ( k _ { 1 } , k _ { 2 } ) } \sum _ { j \in \mathcal { I } ( l _ { 1 } , l _ { 2 } ) } \mathbf { d } _ { m _ { 1 } } \big ( I ( i , k _ { 1 } ) , I ( j , l _ { 1 } ) \big ) \mathbf { w } _ { m _ { 2 } } \big ( I ( i , k _ { 2 } ) , I ( j , l _ { 2 } ) \big ) . } \end{array} +$$ + +where + +$$ +\begin{array} { r l } & { \mathcal { T } ( k _ { 1 } , k _ { 2 } ) = \{ i | 0 \leq i \leq N - 1 , 0 \leq I ( i , k _ { 1 } ) \leq D - 1 , 0 \leq I ( i , k _ { 2 } ) \leq D - 1 \} , } \\ & { \mathcal { I } ( l _ { 1 } , l _ { 2 } ) = \{ j | 0 \leq j \leq N - 1 , 0 \leq I ( j , l _ { 1 } ) \leq D - 1 , 0 \leq I ( j , l _ { 2 } ) \leq D - 1 \} . } \end{array} +$$ + +Lemma 2. Given $N \geq 2 D - 1$ , it holds that: + +(a) $\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \neq \emptyset$ if and only if “ $0 \leq ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N } \leq D - 1 ^ { \prime \prime } o r ^ { \ast } 0 < ( k _ { 2 } - k _ { 1 } ) _ { \mathrm { m o d } N } \leq D - 1 ^ { \prime }$ holds. +(b) $\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \neq \emptyset$ if and only if “ $0 \leq ( l _ { 1 } - l _ { 2 } ) _ { \mathrm { m o d } N } \leq D - 1 ^ { \prime \prime } o r ^ { \textit { \infty } } 0 < ( l _ { 2 } - l _ { 1 } ) _ { \mathrm { m o d } N } \leq D - 1 ^ { \prime }$ holds. + +Proof. Now we prove Conclusion (a). Firstly, we prove “if.” If $0 \leq ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N } \leq D - 1$ and $N \geq 2 D - 1$ , we have + +$$ +\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \big \{ ( k _ { 1 } - \delta ) _ { \mathrm { m o d } N } \big | \delta \in \mathbb { Z } , ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N } \leq \delta \leq D - 1 \big \} \neq \emptyset . +$$ + +If $0 < ( k _ { 2 } - k _ { 1 } ) _ { \mathrm { m o d } N } \leq D - 1$ and $N \geq 2 D - 1$ , we have + +$$ +\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \big \{ ( k _ { 2 } - \delta ) _ { \mathrm { m o d } N } \big | \delta \in \mathbb { Z } , ( k _ { 2 } - k _ { 1 } ) _ { \mathrm { m o d } N } \leq \delta \leq D - 1 \big \} \neq \emptyset . +$$ + +Secondly, we prove “only if.” If $\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \neq \emptyset$ , we can select an $i \in \mathcal { T } ( k _ { 1 } , k _ { 2 } )$ . Let $r _ { 1 } = ( k _ { 1 } -$ $i ) _ { \mathrm { m o d } N }$ and $r _ { 2 } = ( k _ { 2 } - i ) _ { \mathrm { m o d } N }$ . By the definition of $\mathcal { T } ( k _ { 1 } , k _ { 2 } )$ , we have $0 \leq r _ { 1 } , r _ { 2 } \leq D - 1$ . Two cases should be considered here. Case 1: $r _ { 1 } \geq r _ { 2 }$ . Since $0 \le r _ { 1 } - r _ { 2 } \le D - 1 \le N - 1$ , it holds that $r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \mathrm { m o d } N }$ . Thus, + +$$ +\begin{array} { r l } & { r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \mathrm { m o d } N } = \left( ( k _ { 1 } - i ) _ { \mathrm { m o d } N } - ( k _ { 2 } - i ) _ { \mathrm { m o d } N } \right) _ { \mathrm { m o d } N } } \\ & { ~ = \left( ( k _ { 1 } - i ) - ( k _ { 2 } - i ) \right) _ { \mathrm { m o d } N } } \\ & { ~ = ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N } . } \end{array} +$$ + +The equality $\begin{array} { r } { \mathbf { \dot { \mathbf { \varphi } } } 0 \leq r _ { 1 } - r _ { 2 } \leq D - { \mathbf { \varphi } } 1 ^ { , } } \end{array}$ leads to the conclusion $0 \leq ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N } \leq D - 1 ^ { \mathfrak { M } }$ . In case 2 where $r _ { 1 } < r _ { 2 }$ , we can obtain $0 < ( k _ { 2 } - k _ { 1 } ) _ { \mathrm { m o d } N } \leq D - 1$ with the similar arguments. + +Conclusion (b) can be proved by the same argument with the proof of (a). Lemma 2 is proved. + +Now we fix $k _ { 1 } , l _ { 1 }$ and consider what values of $k _ { 2 } , l _ { 2 }$ give $\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \neq \emptyset$ and $\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \neq \emptyset$ . Define four index sets given $0 \leq k _ { 1 } , l _ { 1 } \leq N - 1$ : + +$$ +\begin{array} { l c l } { { } } & { { } } & { { \displaystyle \mathcal { K } ( k _ { 1 } ) = \{ k | 0 \leq ( k _ { 1 } - k ) _ { \mathrm { m o d } N } \leq D - 1 \} } } \\ { { } } & { { } } & { { \displaystyle \bar { \mathcal { K } } ( k _ { 1 } ) = \{ k | 0 < ( k - k _ { 1 } ) _ { \mathrm { m o d } N } \leq D - 1 \} } } \end{array} +$$ + +$$ +\mathcal { L } ( l _ { 1 } ) = \{ l | 0 \leq ( l _ { 1 } - l ) _ { \mathrm { m o d } N } \leq D - 1 \} +$$ + +$$ +\bar { \mathcal { L } } ( l _ { 1 } ) = \{ l | 0 < ( l - l _ { 1 } ) _ { \mathrm { m o d } N } \leq D - 1 \} +$$ + +Lemma 3. If $N \geq 2 D - 1$ , we have: + +(a) The cardinality of $\begin{array} { r } { \mathcal { K } ( k _ { 1 } ) , \bar { \mathcal { K } } ( k _ { 1 } ) \colon | \mathcal { K } ( k _ { 1 } ) | = D , | \bar { \mathcal { K } } ( k _ { 1 } ) | = D - 1 . } \end{array}$ + +(b) $\mathcal { K } ( k _ { 1 } ) \cap \bar { \mathcal { K } } ( k _ { 1 } ) = \emptyset .$ . + +(c) The cardinality of $\mathcal { L } ( l _ { 1 } ) , \bar { \mathcal { L } } ( l _ { 1 } ) \colon | \mathcal { L } ( l _ { 1 } ) | = D , | \bar { \mathcal { L } } ( l _ { 1 } ) | = D - 1 .$ + +(d) $\mathcal { L } ( l _ { 1 } ) \cap \bar { \mathcal { L } } ( l _ { 1 } ) = \emptyset$ + +Proof. Now we prove Conclusion (a). The set $\kappa ( k _ { 1 } )$ can be equivalently written as + +$$ +\mathcal { K } ( k _ { 1 } ) = \{ ( k _ { 1 } - r _ { k } ) _ { \mathrm { m o d } N } | r _ { k } = 0 , 1 , \cdots , D - 1 \} +$$ + +Let $k ( r _ { k } ) = ( k _ { 1 } - r _ { k } ) _ { \mathrm { m o d } N }$ . We want to show that $k ( r _ { k } ^ { 1 } ) \neq k ( r _ { k } ^ { 2 } )$ as long as $r _ { k } ^ { 1 } \neq r _ { k } ^ { 2 }$ . Without loss of generality, we assume $0 \leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \leq D - 1$ . By the definition of modulo operation, There exist two integers $q , q ^ { \prime }$ such that + +$$ +k ( r _ { k } ^ { 1 } ) = q N + k _ { 1 } - r _ { k } ^ { 1 } , \quad k ( r _ { k } ^ { 2 } ) = q ^ { \prime } N + k _ { 1 } - r _ { k } ^ { 2 } . +$$ + +Suppose $k ( r _ { k } ^ { 1 } ) = k ( r _ { k } ^ { 2 } )$ . Taking the difference between the above two equations, we obtain $r _ { k } ^ { 2 } \mathrm { ~ - ~ }$ $r _ { k } ^ { 1 } = ( q ^ { \prime } - \overset { \cdot } { q } ) N$ , i.e, $N$ divides $\overline { { r _ { k } ^ { 2 } } } - r _ { k } ^ { 1 }$ . However, $0 \leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \leq D - \mathrm { i }$ implies $1 \leq r _ { k } ^ { 2 } - r _ { k } ^ { 1 } \leq$ $D - 1 \leq N - 1$ k k, which contradicts with $^ { \circ } N$ dividing $r _ { k } ^ { 2 } - r _ { k } ^ { 1 }$ k .” Thus, it holds that $k ( r _ { k } ^ { 1 } ) \stackrel { \sim } { = } k ( \stackrel { \sim } { r } _ { k } ^ { 2 } )$ . Then we have $| \kappa ( k _ { 1 } ) | = D$ . + +In the same way, we have + +$$ +\bar { \mathcal { K } } ( k _ { 1 } ) = \{ ( k _ { 1 } + r _ { k } ) _ { \mathrm { m o d } N } | r _ { k } = 1 , 2 , \cdot \cdot \cdot , D - 1 \} +$$ + +and $| \bar { \kappa } ( k _ { 1 } ) | = D - 1$ . Conclusion (a) is proved. + +Now we prove Conclusion (b). Suppose ${ \mathcal { K } } ( k _ { 1 } ) \cap { \bar { \mathcal { K } } } ( k _ { 1 } ) \neq \emptyset$ . Pick a $k _ { 2 } \in \mathcal { K } ( k _ { 1 } ) \cap \bar { \mathcal { K } } ( k _ { 1 } )$ . Let $r _ { 3 } = ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N }$ and $r _ { 4 } = ( k _ { 2 } - k _ { 1 } ) _ { \mathrm { m o d } { N } }$ . Then we have $0 \leq r _ { 3 } \leq D - 1$ and $0 < r _ { 4 } \le D - 1$ . By the definition of modulo operation, There exist two integers $q , q ^ { \prime }$ such that + +$$ +k _ { 1 } - k _ { 2 } = q N + r _ { 3 } , \quad k _ { 2 } - k _ { 1 } = q ^ { \prime } N + r _ { 4 } +$$ + +which imply + +$$ +r _ { 3 } + r _ { 4 } + ( q + q ^ { \prime } ) N = 0 . +$$ + +However, $0 < r _ { 3 } + r _ { 4 } \le 2 D - 2$ contradicts with $\dot { \boldsymbol { q } } \in \mathbb { Z } , \boldsymbol { q } ^ { \prime } \in \mathbb { Z } , N \in \mathbb { Z } , N \geq 2 D - 1$ .” Conclusion (b) is proved. + +Conclusions (c) and (d) are actually the same with Conclusions (a) and (b) respectively. Thus, it holds that + +$$ +\begin{array} { r l } & { \mathcal { L } ( l _ { 1 } ) = \{ ( l _ { 1 } - r _ { l } ) _ { \mathrm { m o d } N } | r _ { l } = 0 , 1 , \cdots , D - 1 \} } \\ & { \bar { \mathcal { L } } ( l _ { 1 } ) = \{ ( l _ { 1 } + r _ { l } ) _ { \mathrm { m o d } N } | r _ { l } = 1 , 2 , \cdots , D - 1 \} } \end{array} +$$ + +and $| \mathcal { L } ( l _ { 1 } ) | = D , | \bar { \mathcal { L } } ( l _ { 1 } ) | = D - 1$ . Lemma 3 is proved. + +With the preparations, we can prove Conclusion 1 of Theorem 3 now. + +Proof of Theorem 3, Conclusion $^ { l }$ . Firstly we fix $k _ { 1 } \in \{ 0 , 1 , \cdots , N - 1 \}$ and consider $k _ { 2 } \in \mathcal { K } ( k _ { 1 } )$ . Let $r _ { k } \stackrel { \cdot } { = } ( k _ { 1 } - k _ { 2 } ) _ { \mathrm { m o d } N }$ . Then equation (37) implies that, for any $i \in \mathcal { T } ( k _ { 1 } , k _ { 2 } )$ , there exists a $\delta$ $( r _ { k } \le \delta \le D - 1 )$ such that + +$$ +\begin{array} { r l } & { I ( i , k _ { 1 } ) = \bigr ( k _ { 1 } - ( k _ { 1 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta ) _ { \mathrm { m o d } N } = \delta , } \\ & { I ( i , k _ { 2 } ) = \bigr ( k _ { 2 } - ( k _ { 1 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta - r _ { k } ) _ { \mathrm { m o d } N } = \delta - r _ { k } . } \end{array} +$$ + +Now we consider another case for $k _ { 2 }$ : $k _ { 2 } \in \bar { \mathcal { K } } ( k _ { 1 } )$ , $r _ { k } = ( k _ { 2 } - k _ { 1 } ) _ { \mathrm { m o d } N }$ . Equation (38) implies that, for any $i \in \mathcal { T } ( k _ { 1 } , k _ { 2 } )$ , there exists a $\delta$ $r _ { k } \le \delta \le D - 1 \}$ ) such that + +$$ +\begin{array} { r l } & { I ( i , k _ { 1 } ) = \bigr ( k _ { 1 } - ( k _ { 2 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta - r _ { k } ) _ { \mathrm { m o d } N } = \delta - r _ { k } , } \\ & { I ( i , k _ { 2 } ) = \bigr ( k _ { 2 } - ( k _ { 2 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta ) _ { \mathrm { m o d } N } = \delta . } \end{array} +$$ + +Similarly, for any $l _ { 1 } \in \{ 0 , 1 , \cdots , N - 1 \}$ and $l _ { 2 } \in \mathcal { L } ( l _ { 1 } )$ , we denote $r _ { l } = ( l _ { 1 } - l _ { 2 } ) _ { \mathrm { m o d } N }$ . For any $j \in \mathcal { I } ( l _ { 1 } , l _ { 2 } )$ , there exists a $\delta$ $( r _ { l } \le \delta \le D - 1 )$ such that + +$$ +\begin{array} { r l } & { I ( j , l _ { 1 } ) = \bigr ( l _ { 1 } - ( l _ { 1 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta ) _ { \mathrm { m o d } N } = \delta , } \\ & { I ( j , l _ { 2 } ) = \bigr ( l _ { 2 } - ( l _ { 1 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta - r _ { l } ) _ { \mathrm { m o d } N } = \delta - r _ { l } . } \end{array} +$$ + +Another case for $l _ { 2 }$ : $l _ { 2 } ~ \in ~ \bar { \mathcal { L } } ( l _ { 1 } )$ , $r _ { l } = ( l _ { 2 } - l _ { 1 } ) _ { \mathrm { m o d } N }$ . For any $j \in \mathcal { I } ( l _ { 1 } , l _ { 2 } )$ , there exists a $\delta$ $( r _ { l } \le \delta \le D - 1 )$ such that + +$$ +\begin{array} { r l } & { I ( j , l _ { 1 } ) = \bigr ( l _ { 1 } - ( l _ { 2 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta - r _ { l } ) _ { \mathrm { m o d } N } = \delta - r _ { l } , } \\ & { I ( j , l _ { 2 } ) = \bigr ( l _ { 2 } - ( l _ { 2 } - \delta ) _ { \mathrm { m o d } N } \bigr ) _ { \mathrm { m o d } N } = ( \delta ) _ { \mathrm { m o d } N } = \delta . } \end{array} +$$ + +Now let us consider the following function. By results in Lemmas 2 and 3, we have + +$$ +\begin{array} { r l r } { { f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } ) = \sum _ { k _ { 2 } = 0 } ^ { N - 1 } \sum _ { l _ { 2 } = 0 } ^ { N - 1 } ( { \bf B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) ) ^ { 2 } } } \\ & { } & { = f _ { 1 } + f _ { 2 } + f _ { 3 } + f _ { 4 } , } \end{array} +$$ + +where + +$$ +\begin{array} { r l r } { { f _ { 1 } = \sum _ { k _ { 2 } \in \mathcal { K } ( k _ { 1 } ) } \sum _ { l _ { 2 } \in \mathcal { L } ( l _ { 1 } ) } \Big ( \mathbf { B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \Big ) ^ { 2 } } } \\ & { } & \\ & { } & { f _ { 2 } = \sum _ { { k _ { 2 } \in \widehat { \mathcal { K } } ( k _ { 1 } ) } } \sum _ { l _ { 2 } \in \mathcal { L } ( l _ { 1 } ) } \Big ( \mathbf { B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \Big ) ^ { 2 } } \\ & { } & \\ & { } & { f _ { 3 } = \sum _ { { k _ { 2 } \in \mathcal { K } ( k _ { 1 } ) } } \sum _ { l _ { 2 } \in \widehat { \mathcal { L } } ( l _ { 1 } ) } \Big ( \mathbf { B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \Big ) ^ { 2 } } \\ & { } & \\ & { } & { f _ { 4 } = \sum _ { { k _ { 2 } \in \widehat { \mathcal { K } } ( k _ { 1 } ) } } \sum _ { l _ { 2 } \in \widehat { \mathcal { L } } ( l _ { 1 } ) } \Big ( \mathbf { B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \Big ) ^ { 2 } . } \end{array} +$$ + +Combining equations (39), (41), (43) and (45), we obtain + +$$ +f _ { 1 } = \sum _ { r _ { k } = 0 } ^ { D - 1 } \sum _ { r _ { l } = 0 } ^ { D - 1 } \sum _ { \delta _ { k } = r _ { k } } ^ { D - 1 } \sum _ { \delta _ { l } = r _ { l } } ^ { D - 1 } \Big ( \mathbf { d } _ { m _ { 1 } } ( \delta _ { k } , \delta _ { l } ) \mathbf { w } _ { m _ { 2 } } ( \delta _ { k } - r _ { k } , \delta _ { l } - r _ { l } ) \Big ) ^ { 2 } . +$$ + +Combining (40), (41), (44) and (45), we obtain + +$$ +f _ { 2 } = \sum _ { r _ { k } = 1 } ^ { D - 1 } \sum _ { r _ { l } = 0 } ^ { D - 1 } \sum _ { \delta _ { k } = r _ { k } } ^ { D - 1 } \sum _ { \delta _ { l } = r _ { l } } ^ { D - 1 } \Big ( \mathbf { d } _ { m _ { 1 } } ( \delta _ { k } - r _ { k } , \delta _ { l } ) \mathbf { w } _ { m _ { 2 } } ( \delta _ { k } , \delta _ { l } - r _ { l } ) \Big ) ^ { 2 } . +$$ + +Combining (39), (42), (43) and (46), we obtain + +$$ +f _ { 3 } = \sum _ { r _ { k } = 0 } ^ { D - 1 } \sum _ { r _ { l } = 1 } ^ { D - 1 } \sum _ { \delta _ { k } = r _ { k } } ^ { D - 1 } \sum _ { \delta _ { l } = r _ { l } } ^ { D - 1 } \Big ( \mathbf { d } _ { m _ { 1 } } ( \delta _ { k } , \delta _ { l } - r _ { l } ) \mathbf { w } _ { m _ { 2 } } ( \delta _ { k } - r _ { k } , \delta _ { l } ) \Big ) ^ { 2 } . +$$ + +Combining (40), (42), (44) and (46), we obtain + +$$ +f _ { 4 } = \sum _ { r _ { k } = 1 } ^ { D - 1 } \sum _ { r _ { l } = 1 } ^ { D - 1 } \sum _ { \delta _ { k } = r _ { k } } ^ { D - 1 } \sum _ { \delta _ { l } = r _ { l } } ^ { D - 1 } \Big ( \mathbf { d } _ { m _ { 1 } } ( \delta _ { k } - r _ { k } , \delta _ { l } - r _ { l } ) \mathbf { w } _ { m _ { 2 } } ( \delta _ { k } , \delta _ { l } ) \Big ) ^ { 2 } . +$$ + +By the above explicit formulas of $f _ { i } , 1 \le i \le 4$ , we have $f _ { 1 } , f _ { 2 } , f _ { 3 } , f _ { 4 }$ are all independent of $k _ { 1 } , l _ { 1 }$ and $N$ . They are only related with $m _ { 1 } , m _ { 2 }$ for fixed $\mathbf { d }$ and $\mathbf { m }$ . Thus, we are able to denote $f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } )$ as $f ( m _ { 1 } , m _ { 2 } )$ for simplicity. Consequently, + +$$ +\begin{array} { r l } { \displaystyle \frac { 1 } { N ^ { 2 } } \| ( \mathbf { D } _ { \mathrm { c r } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c r i } } ^ { N } \| _ { F } ^ { 2 } = \frac { 1 } { N ^ { 2 } } \sum _ { { \boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \sum _ { i = 0 } ^ { N - 1 } \sum _ { \iota _ { 2 } = 0 } ^ { M } \sum _ { m = 1 } ^ { M } \sum _ { m = 1 } ^ { M } \Big ( \mathbf { B } _ { \mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \Big ) ^ { 2 } } & { } \\ { = \frac { 1 } { N ^ { 2 } } \sum _ { { \boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \sum _ { \iota _ { 2 } = 0 } ^ { M } \sum _ { m = 1 } ^ { M } \sum _ { \iota _ { 2 } = 1 } ^ { M } f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } ) } & { } \\ { = \frac { 1 } { N ^ { 2 } } \sum _ { { \boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \sum _ { \iota _ { 2 } = 0 } ^ { N - 1 } \sum _ { m = 1 } ^ { M } \sum _ { m = 2 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } & { } \\ { = \frac { 1 } { N ^ { 2 } } \sum _ { { \boldsymbol k } _ { 1 } = 0 } ^ { M } \sum _ { \iota _ { 1 } = 0 } ^ { M } \sum _ { m = 1 } ^ { M } \sum _ { \iota _ { 2 } = 1 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } & { } \\ { = \frac { 1 } { N ^ { 2 } } \cdot N ^ { 2 } \cdot \displaystyle \sum _ { m = 1 } ^ { M } \sum _ { \iota _ { 2 } = 0 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) = \sum _ { m _ { 1 } = 1 } ^ { M } \sum _ { \iota _ { 2 } = 1 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } \end{array} +$$ + +Thus, $\begin{array} { r } { \frac { 1 } { N ^ { 2 } } \| ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N } \| _ { F } ^ { 2 } } \end{array}$ is dependent of $N$ : + +$$ +\frac { 1 } { N ^ { 2 } } \| ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N } \| _ { F } ^ { 2 } = \frac { 1 } { ( 2 D - 1 ) ^ { 2 } } \| ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } \| _ { F } ^ { 2 } , \quad \forall N \geq 2 D - 1 , +$$ + +which implies $\mathcal { W } _ { \mathrm { c i r } } ^ { N } = \mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ , $\forall N \geq 2 D - 1$ + +# C.2 PROOF OF CONCLUSION 2. + +Before proving Conclusion 2, let us analyze the relationship between $\mathbf { D } _ { \mathrm { { c o n v } } } ^ { N }$ and DN+D−1. + +Similar to $\mathbf { D } _ { \mathrm { c i r } }$ , we use $( i , j )$ as the row index and $( k , l , m )$ as the column index of $\mathbf { D } _ { \mathrm { c o n v } }$ . For $0 \leq i , j \leq N - 1 , 1 \leq m \leq M$ , + +$$ +\mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) = \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( i , j ; k , l , m ) = { \left\{ \begin{array} { l l } { \mathbf { d } _ { m } ( k - i , l - j ) , } & { 0 \leq k - i , l - j \leq D - 1 } \\ { 0 , } & { k , l { \mathrm { ~ t a k e n ~ a s ~ o t h e r s } } } \end{array} \right. } +$$ + +Matrixmatrix $ { \mathbf { D } } _ { \mathrm { c i r } } ^ { N + D - 1 }$ is of dimensf dimension $( N + D - 1 ) ^ { 2 } \times ( N + D - 1 ) ^ { 2 } M$ $0 \leq i , j \leq N + D - 2$ DNcon v i $( N ) ^ { 2 } \times ( N + D - 1 ) ^ { 2 } M$ $0 \leq i , j \leq N - 1 .$ $\mathbf { D } _ { \mathrm { { c o n v } } } ^ { N }$ block in DN+D−1cir , i.e., + +$$ +\begin{array} { r } { \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } = \left[ \begin{array} { c } { \mathbf { D } _ { \mathrm { c o n v } } ^ { N } } \\ { \Delta _ { \mathbf { D } } ^ { N } } \end{array} \right] . } \end{array} +$$ + +The matrix $\Delta _ { \mathbf { D } } ^ { N }$ is of dimension $( ( N + D - 1 ) ^ { 2 } - N ^ { 2 } ) \times ( N + D - 1 ) ^ { 2 } M \colon$ + +$$ +\Delta _ { \mathbf { D } } ^ { N } = \left[ \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \cdot , \cdot ) \right] , \quad ( i , j ) \in \mathcal { T } _ { \Delta } +$$ + +where + +$$ +\begin{array} { r l } & { \mathcal { Z } _ { \Delta } = \mathcal { Z } _ { 1 } \cup \mathcal { Z } _ { 2 } \cup \mathcal { Z } _ { 3 } } \\ & { \mathcal { T } _ { 1 } = \{ ( i , j ) | N \le i \le N + D - 2 , 0 \le j \le N - 1 \} } \\ & { \mathcal { T } _ { 2 } = \{ ( i , j ) | 0 \le i \le N - 1 , N \le j \le N + D - 2 \} } \\ & { \mathcal { Z } _ { 3 } = \{ ( i , j ) | N \le i \le N + D - 2 , N \le j \le N + D - 2 \} . } \end{array} +$$ + +Similarly, + +$$ +\mathbf { W } _ { \mathrm { c i r } } ^ { N + D - 1 } = \left[ \mathbf { W } _ { \mathrm { c o n v } } ^ { N } \right] , \quad \Delta _ { \mathbf { W } } ^ { N } = \left[ \mathbf { W } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot ; \cdot ) \right] , \quad ( i , j ) \in \mathcal { T } _ { \Delta } . +$$ + +Then, + +$$ +( \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N + D - 1 } = ( \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c o n v } } ^ { N } + ( \boldsymbol { \Delta } _ { \mathbf { D } } ^ { N } ) ^ { T } \boldsymbol { \Delta } _ { \mathbf { W } } ^ { N } . +$$ + +Lemma 4. For any $( i , j ) \in \mathcal { I } _ { \Delta }$ , one has + +$$ +\begin{array} { r } { \| \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \cdot , : ) \| _ { 2 } ^ { 2 } = \| \mathbf { d } \| _ { 2 } ^ { 2 } , } \\ { \| \mathbf { W } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \cdot , : ) \| _ { 2 } ^ { 2 } = \| \mathbf { w } \| _ { 2 } ^ { 2 } . } \end{array} +$$ + +Proof. Equation (35) implies that, for $( i , j ) \in \mathcal { T } _ { 1 } , 1 \le m \le M$ , + +$$ +\begin{array} { r } { \mathsf { \Pi } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) = \left\{ \begin{array} { l l } { \mathbf { d } _ { m } ( k - i , l - j ) , } & { i \le k \le N + D - 2 , j \le l \le j + D - 1 } \\ { \mathbf { d } _ { m } ( k - i + N + D - 1 , l - j ) , } & { 0 \le k \le i - N , j \le l \le j + D - 1 } \\ { 0 , } & { k , l \mathrm { ~ t a k e n ~ a s ~ o t h e r s } } \end{array} \right. } \end{array} +$$ + +Thus, for any $( i , j ) \in \mathcal { T } _ { 1 }$ , + +$$ +\| \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \cdot , \cdot ) \| _ { 2 } ^ { 2 } = \sum _ { k = 0 } ^ { N + D - 2 } \sum _ { l = 0 } ^ { N + D - 2 } \sum _ { m = 1 } ^ { M } \left| \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) \right| ^ { 2 } = \| \mathbf { d } \| _ { 2 } ^ { 2 } +$$ + +Similarly, + +$$ +\begin{array} { r } { \| \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \cdot , \cdot ) \| _ { 2 } ^ { 2 } = \| \mathbf { d } \| _ { 2 } ^ { 2 } , \quad ( i , j ) \in \mathcal { T } _ { 2 } \cup \mathcal { T } _ { 3 } . } \end{array} +$$ + +Equation (50) is proved. With the same argument, equation (51) is also proved. + +Lemma 5. If $N \geq 2 D - 1$ , we have + +$$ +\begin{array} { r } { \| ( \Delta _ { \mathbf { D } } ^ { N } ) ^ { T } \Delta _ { \mathbf { W } } ^ { N } \| _ { F } ^ { 2 } \leq \big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \big ) ( 2 D - 1 ) ^ { 2 } \| \mathbf { d } \| _ { 2 } ^ { 2 } \| \mathbf { w } \| _ { 2 } ^ { 2 } . } \end{array} +$$ + +Proof. For simplicity, we denote two row vectors: + +$$ +\begin{array} { r l } & { \mathbf { d } _ { i , j } \triangleq { \mathbf { D } } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \colon , \colon ) \in \mathbb { R } ^ { 1 \times ( N + D - 1 ) ^ { 2 } M } } \\ & { \mathbf { w } _ { i , j } \triangleq \mathbf { W } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \cdot , \colon , \colon ) \in \mathbb { R } ^ { 1 \times ( N + D - 1 ) ^ { 2 } M } } \end{array} +$$ + +Then, + +$$ +\Vert ( \Delta _ { \mathbf { D } } ^ { N } ) ^ { T } \Delta _ { \mathbf { W } } ^ { N } \Vert _ { F } ^ { 2 } = \bigg \Vert \sum _ { ( i , j ) \in \mathcal { Z } _ { \Delta } } \mathbf { d } _ { i , j } ^ { T } \mathbf { w } _ { i , j } \bigg \Vert _ { F } ^ { 2 } = \sum _ { ( i _ { 1 } , j _ { 1 } ) \in \mathcal { Z } _ { \Delta } } \sum _ { ( i _ { 2 } , j _ { 2 } ) \in \mathcal { Z } _ { \Delta } } \left. \mathbf { d } _ { i _ { 1 } , j _ { 1 } } ^ { T } \mathbf { w } _ { i _ { 1 } , j _ { 1 } } , \mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } \mathbf { w } _ { i _ { 2 } , j _ { 2 } } \right. _ { F } , +$$ + +where + +$$ +\mathbf { \mathop { ' } d } _ { i _ { 1 } , j _ { 1 } } ^ { T } \mathbf { \boldsymbol { w } } _ { i _ { 1 } , j _ { 1 } } , \mathbf { \boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \mathbf { \boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } \bigg \rangle _ { F } = \operatorname { t r a c e } \Bigl ( \mathbf { w } _ { i _ { 1 } , j _ { 1 } } ^ { T } \mathbf { \boldsymbol { d } } _ { i _ { 1 } , j _ { 1 } } \mathbf { \boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \mathbf { \boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } \Bigr ) = \bigl ( \mathbf { \boldsymbol { d } } _ { i _ { 1 } , j _ { 1 } } \mathbf { \boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \bigr ) \cdot \bigl ( \mathbf { \boldsymbol { w } } _ { i _ { 1 } , j _ { 1 } } \mathbf { \boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \bigr ) . +$$ + +Since + +$$ +\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } = \sum _ { k = 0 } ^ { N - 1 } \sum _ { l = 0 } ^ { N - 1 } \sum _ { m = 1 } ^ { M } = \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i _ { 1 } , j _ { 1 } ; k , l , m ) \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ( i _ { 2 } , j _ { 2 } ; k , l , m ) , +$$ + +with the same argument in Lemma 2, we have: $\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } \neq 0$ implies + +$$ +\begin{array} { r l } & { i _ { 2 } \in \mathcal { I } _ { \Delta } ^ { \prime } \triangleq \{ i | 0 \le ( i _ { 1 } - i ) _ { \mathrm { m o d } ( N + D - 1 ) } \le D - 1 \mathrm { o r } 0 \le ( i - i _ { 1 } ) _ { \mathrm { m o d } ( N + D - 1 ) } \le D - 1 \} } \\ & { j _ { 2 } \in \mathcal { I } _ { \Delta } ^ { \prime } \triangleq \{ j | 0 \le ( j _ { 1 } - j ) _ { \mathrm { m o d } ( N + D - 1 ) } \le D - 1 \mathrm { o r } 0 \le ( j - j _ { 1 } ) _ { \mathrm { m o d } ( N + D - 1 ) } \le D - 1 \} } \end{array} +$$ + +Then + +$$ +\begin{array} { r l } { \| ( \Delta _ { \mathbf { D } } ^ { N } ) ^ { T } \Delta _ { \mathbf { W } } ^ { N } \| _ { F } ^ { 2 } = } & { \displaystyle \sum _ { ( i _ { 1 } , j _ { 1 } ) \in \mathbb { Z } } \displaystyle \sum _ { \Delta } \sum _ { i _ { 2 } \in \mathbb { Z } _ { \Delta } ^ { \prime } } \displaystyle \sum _ { j _ { 2 } \in \mathcal { I } _ { \Delta } ^ { \prime } } ( \mathbf { d } _ { i _ { 1 } , j _ { 1 } } \mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } ) \cdot ( \mathbf { w } _ { i _ { 1 } , j _ { 1 } } \mathbf { w } _ { i _ { 2 } , j _ { 2 } } ^ { T } ) } \\ { \leq } & { \displaystyle \sum _ { ( i _ { 1 } , j _ { 1 } ) \in \mathbb { Z } } \displaystyle \sum _ { \Delta } \sum _ { i _ { 2 } \in \mathbb { Z } _ { \Delta } ^ { \prime } } \displaystyle \sum _ { j _ { 2 } \in \mathcal { I } _ { \Delta } ^ { \prime } } \| \mathbf { d } \| _ { 2 } ^ { 2 } \| \mathbf { w } \| _ { 2 } ^ { 2 } } \\ { } & { = | \mathcal { Z } _ { \Delta } | \cdot | \mathcal { I } _ { \Delta } ^ { \prime } | \cdot | \mathcal { I } _ { \Delta } ^ { \prime } | \cdot \| \mathbf { d } \| _ { 2 } ^ { 2 } \| \mathbf { w } \| _ { 2 } ^ { 2 } } \\ { } & { = \big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \big ) ( 2 D - 1 ) ^ { 2 } \| \mathbf { d } \| _ { 2 } ^ { 2 } \| \mathbf { w } \| _ { 2 } ^ { 2 } , } \end{array} +$$ + +where the inequality in the second line follows from (50) and (51). Inequality (52) is proved. + +With these preparations, we can prove Theorem 3, Conclusion 2 now. + +Proof of Theorem 3, Conclusion 2. Define set + +$$ +\mathcal { W } _ { \mathrm { n o r m a l } } = \Big \{ \mathbf { w } \in \mathbb { R } ^ { D ^ { 2 } M } \Big | \mathbf { w } _ { m } \cdot \mathbf { d } _ { m } = 1 , \forall m = 1 , \cdots , M \Big \} . +$$ + +Since $\mathbf { d } \in \mathcal { W } _ { \mathrm { n o r m a l } }$ , the set is nonempty: + +$$ +\mathcal { W } _ { \mathrm { n o r m a l } } \neq \emptyset . +$$ + +Define functions $F _ { \mathrm { c o n v } } ^ { N } : \mathbb { R } ^ { D ^ { 2 } M } \to \mathbb { R } , F _ { \mathrm { c i r } } ^ { N } : \mathbb { R } ^ { D ^ { 2 } M } \to \mathbb { R } .$ + +$$ +\begin{array} { r l } & { F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) = \displaystyle \frac { 1 } { N + D - 1 } \Big \| \big ( \mathbf { D } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { d } ) \big ) ^ { T } \mathbf { W } _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) \Big \| _ { F } + \imath \gamma _ { \mathrm { n o r m a l } } ( \mathbf { w } ) } \\ & { ~ F _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } ) = \displaystyle \frac { 1 } { N } \Big \| ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( \mathbf { d } ) ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } ) \Big \| _ { F } + \imath \gamma _ { \mathrm { n o r m a l } } ( \mathbf { w } ) } \end{array} +$$ + +By the definitions of $\mathcal { W } _ { \mathrm { c o n v } } ^ { N } , \mathcal { W } _ { \mathrm { c i r } } ^ { N }$ , we have + +$$ +\mathcal { W } _ { \mathrm { c o n v } } ^ { N } = \underset { \mathbf { w } } { \arg \operatorname* { m i n } } F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) , \quad \mathcal { W } _ { \mathrm { c i r } } ^ { N } = \underset { \mathbf { w } } { \arg \operatorname* { m i n } } F _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } ) +$$ + +Step 1: Proving $F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } )$ uniformly converges to $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } )$ on $X \cap \mathcal { W } _ { \mathrm { n o r m a l } }$ for any compact set X ⊂ RD2M. + +We arbitrarily choose such a compact set $X$ . Based on (47), (49) and (52), one has, for all w $\in$ X ∩ Wnormal, + +$$ +\begin{array} { r l } & { \| F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) \| = | F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) - F _ { \mathrm { c i r } } ^ { N + D - 1 } ( \mathbf { w } ) | } \\ & { \qquad = \displaystyle \frac { 1 } { N + D - 1 } \Big | \Big \| \big ( \mathbf { D } _ { \mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N + D - 1 } \Big \| _ { F } - \Big \| \big ( \mathbf { D } _ { \mathrm { c o n v } } ^ { N } \big ) ^ { T } \mathbf { W } _ { \mathrm { c o n v } } ^ { N } \Big \| _ { F } } \\ & { \qquad \leq \displaystyle \frac { 1 } { N + D - 1 } \Big \| ( \Delta _ { \mathbf { D } } ^ { N } ) ^ { T } \Delta _ { \mathbf { W } } ^ { N } \Big \| _ { F } } \\ & { \qquad \leq \displaystyle \frac { \sqrt { \big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \big ) } ( 2 D - 1 ) } { N + D - 1 } \| \mathbf { d } \| _ { 2 } \| \mathbf { w } \| _ { 2 } } \\ & { \qquad \leq \displaystyle \frac { ( 2 D - 1 ) \sqrt { 2 ( D - 1 ) } } { \sqrt { N + D - 1 } } \| \mathbf { d } \| _ { 2 } \| \mathbf { w } \| _ { 2 } . } \end{array} +$$ + +Thus, there exists a constant $B > 0$ , which is independent of $N$ , such that + +$$ +| F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) | \leq \frac { B } { \sqrt { N } } \operatorname* { s u p } _ { \mathbf { w } \in X \cap \mathcal { W } _ { \mathrm { n o r m a l } } } \| \mathbf { w } \| , \quad \forall \mathbf { w } \in X \cap \mathcal { W } _ { \mathrm { n o r m a l } } . +$$ + +Step 2: Proving $F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } )$ epigraphically converges8 to $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } )$ + +We want to show, at each point w it holds that + +$$ +\begin{array} { r l } & { \underset { N \infty } { \operatorname* { l i m } \operatorname* { i n f } } F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ^ { N } ) \geq F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) \quad \mathrm { f o r ~ e v e r y ~ s e q u e n c e ~ } \mathbf { w } ^ { N } \mathbf { w } } \\ & { \underset { N \infty } { \operatorname* { l i m } \operatorname* { s u p } } F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ^ { N } ) \leq F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) \quad \mathrm { f o r ~ s o m e ~ s e q u e n c e ~ } \mathbf { w } ^ { N } \mathbf { w } } \end{array} +$$ + +Firstly, we prove (56). We arbitrarily pick a sequence $\{ \mathbf { w } ^ { N } \} _ { N = 0 } ^ { \infty }$ such that $\mathbf { w } ^ { N } \to \mathbf { w }$ + +$\notin \mathcal { W } _ { \mathrm { n o r m a l } }$ , o $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) = + \infty$ . Since s, one h $\mathcal { W } _ { \mathrm { n o r m a l } }$ et, therfor all , $N ^ { + }$ such that, $\mathbf { w } ^ { N } \notin \mathcal { W } _ { \mathrm { n o r m a l } }$ $N \geq N ^ { + }$ $F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ^ { N } ) = + \infty$ $N \geq N ^ { + }$ + +$$ +\operatorname* { l i m } _ { N \to \infty } \operatorname* { i n f } _ { C \mathrm { c o n v } } ( \mathbf { w } ^ { N } ) = F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) = + \infty . +$$ + +If $\mathbf { v } \in \mathcal { W } _ { \mathrm { n o r m a l } }$ , two cases should be considered. The first case is that any subsequences of $\{ \mathbf { w } ^ { N } \} _ { N = 0 } ^ { \infty }$ re not kept within . Then we have $\mathcal { W } _ { \mathrm { n o r m a l } }$ , i.e., there exists a $N ^ { + }$ such that $\mathbf { w } ^ { \mathbf { \bar { \boldsymbol { N } } } } \notin \mathcal { W } _ { \mathrm { n o r m a l } }$ for $N \geq N ^ { + }$ + +$$ +\operatorname* { l i m } _ { N \to \infty } \operatorname* { i n f } _ { C \mathrm { c o n v } } ( \mathbf { w } ^ { N } ) = + \infty > F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) . +$$ + +The second case is that there exists a subsequence $\{ \mathbf { w } ^ { N _ { k } } \} _ { k = 0 } ^ { \infty } \subset \{ \mathbf { w } ^ { N } \} _ { N = 0 } ^ { \infty }$ such that + +$$ +\mathbf { w } ^ { N _ { k } } \in { \mathcal { W } } _ { \mathrm { n o r m a l } } , \quad \forall k = 0 , 1 , 2 , \cdots . +$$ + +Since $\mathbf { w } ^ { N }$ converges to $\mathbf { w }$ , any subsequences should be Cauchy. Given any Cauchy sequence $\{ \mathbf { w } ^ { N _ { k } } \} _ { k = 0 } ^ { \infty }$ in finite dimensional Euclidean space, there exists a compact set $X$ such that + +$$ +\mathbf { w } ^ { N _ { k } } \in X , \quad \forall k = 0 , 1 , 2 , \cdot \cdot \cdot +$$ + +Let $B ^ { \prime } = \operatorname* { s u p } _ { \mathbf { w } \in X \cap \mathcal { W } _ { \mathrm { n o r m a l } } } \left\| \mathbf { w } \right\|$ . By (55), we obtain + +$$ +\begin{array} { r l } { \big | F _ { \mathrm { c o n v } } ^ { N _ { k } } \big ( \mathbf { w } ^ { N _ { k } } \big ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) \big | \le \big | F _ { \mathrm { c o n v } } ^ { N _ { k } } ( \mathbf { w } ^ { N _ { k } } ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } \big ( \mathbf { w } ^ { N _ { k } } \big ) \big | + \big | F _ { \mathrm { c i r } } ^ { 2 D - 1 } \big ( \mathbf { w } ^ { N _ { k } } \big ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } \big ( \mathbf { w } \big ) \big | } & { } \\ { \le \displaystyle \frac { B B ^ { \prime } } { \sqrt { N _ { k } } } + \big | F _ { \mathrm { c i r } } ^ { 2 D - 1 } \big ( \mathbf { w } ^ { N _ { k } } \big ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } \big ( \mathbf { w } \big ) \big | . } & { } \end{array} +$$ + +For any  > 0, by tinuity o F 2D−1cir the con for all f . ,a are able tosuch that $K > 0$ hat . T $| F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ^ { N _ { k } } ) -$ $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf w ) | \ < \ \epsilon$ $\operatorname* { m a x } ( K , K ^ { \prime } )$ , we have $| F _ { \mathrm { c o n v } } ^ { N _ { k } } ( \mathbf { w } ^ { N _ { k } } ) - F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) | < 2 \epsilon$ $k \geq K$ $K ^ { \prime }$ , i.e., $N _ { K ^ { \prime } } \geq ( B B ^ { \prime } / \epsilon ) ^ { 2 }$ $k \geq$ + +$$ +\operatorname* { l i m } _ { k \infty } F _ { \mathrm { c o n v } } ^ { N _ { k } } ( \mathbf { w } ^ { N _ { k } } ) = F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) . +$$ + +mulation point of The above conclusion holds for all subsequences $+ \infty$ because $F _ { \mathrm { c o n v } } ^ { N } ( \mathbf { w } ) = F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) = + \infty$ k=0 normal. All the other accumulation points of for all $\{ \mathbf { w } ^ { N _ { k } } \} _ { k = 0 } ^ { \infty } \subset \mathcal { W } _ { \mathrm { n o r m a l } }$ $\mathbf { w } \notin \mathcal { W } _ { \mathrm { n o r m a l } }$ . Thus, . $\{ \bar F _ { \mathrm { c o n v } } ^ { \bar { N } } ( \mathbf { w } ^ { N } ) \} _ { N = 0 } ^ { \infty }$ $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } )$ is an accu- must + +$$ +\operatorname* { l i m } _ { N \to \infty } \operatorname* { i n f } _ { C \mathrm { c o n v } } ( \mathbf { w } ^ { N } ) = F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) < + \infty . +$$ + +Secondly, we prove (57). We set $\mathbf { w } ^ { N } = \mathbf { w }$ for all $N = 0 , 1 , 2 , \cdots$ . Then (57) is a direct result of (55). + +Step 3: proving (25). Define + +$$ +G ( \mathbf { w } ) = \left. ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } \right. _ { F } ^ { 2 } . +$$ + +We want to show that $G ( \mathbf { w } )$ is strongly convex. + +$\tilde { \mathbf { w } } _ { i } \in \mathbb { R } ^ { ( 2 D - 1 ) ^ { 2 } }$ be the $i ^ { \mathrm { { t h } } }$ column of $\mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ + +$$ +\mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } = \left[ \tilde { \mathbf { w } } _ { 1 } , \tilde { \mathbf { w } } _ { 2 } , \cdots , \tilde { \mathbf { w } } _ { ( 2 D - 1 ) ^ { 2 } M } \right] +$$ + +Then + +$$ +G ( \mathbf { w } ) = \sum _ { i = 1 } ^ { ( 2 D - 1 ) ^ { 2 } M } ( \tilde { \mathbf { w } } _ { i } ) ^ { T } \Big ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \Big ) \tilde { \mathbf { w } } _ { i } . +$$ + +$\tilde { \mathbf { w } } \in \mathbb { R } ^ { ( 2 D - 1 ) ^ { 4 } M }$ vectorize $\mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ + +$$ +\begin{array} { r } { \tilde { { \bf w } } = \left[ ( \tilde { \bf w } _ { 1 } ) ^ { T } , ( \tilde { \bf w } _ { 2 } ) ^ { T } , \cdots , ( \tilde { \bf w } _ { ( 2 D - 1 ) ^ { 2 } M } ) ^ { T } \right] ^ { T } . } \end{array} +$$ + +Then $G ( \mathbf { w } )$ can be written as a quadratic form of w˜ : + +$$ +G ( \mathbf { w } ) = \tilde { \mathbf { w } } ^ { T } Q \tilde { \mathbf { w } } , +$$ + +where + +$$ +\begin{array} { r } { Q = \left[ \begin{array} { l l l } { \left( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \right) } & & \\ & { \cdots } & \\ & & { \left( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \right) } \end{array} \right] . } \end{array} +$$ + +As long as at least one of the matrices {D2D−1cir,0 , $\{ \mathbf { D } _ { \mathrm { c i r } , 0 } ^ { 2 D - 1 } , \cdot \cdot \cdot , \mathbf { D } _ { \mathrm { c i r } , M - 1 } ^ { 2 D - 1 } \}$ in on-singular, is positive d $\mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ is full row rank, which implies that D2D−1cir ( $\mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { D } _ { \mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T }$ $Q$ + +The transform between w and $\tilde { \mathbf { w } }$ is linear. We denote the transform as $T$ , i.e., + +$$ +\tilde { \mathbf { w } } = T \mathbf { w } . +$$ + +It’s trivial that $\| \tilde { \mathbf { w } } \| _ { 2 } ^ { 2 } = 0$ implies $\| \mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } \| _ { F } ^ { 2 } = 0 .$ . By the definition of $\mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ , $\| \mathbf { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } \| _ { F } ^ { 2 } = 0$ implies $\| \mathbf { w } \| _ { 2 } ^ { 2 } = 0$ . Thus, linear operator $T$ is full column rank. Thus, $T ^ { T } Q T$ is positive definite, and + +$$ +G ( \mathbf { w } ) = \mathbf { w } ^ { T } ( T ^ { T } Q T ) \mathbf { w } +$$ + +is strongly convex. Then $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( { \bf w } ) = \sqrt { G ( { \bf w } ) } + \iota _ { \mathcal { W } _ { \mathrm { n o r m a l } } } ( { \bf w } )$ has only one minimizer, i.e., $\mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ involves only a unique element. + +Now we check the conditions of Propositions 7.32(c) and 7.33 in (Rockafellar & Wets, 2009) to apply them. + +1. F N $F _ { \mathrm { c o n v } } ^ { N } \xrightarrow [ ] { \mathrm { e } } F _ { \mathrm { c i r } } ^ { 2 D - 1 }$ . This is proved in Step 2. + +2. $F _ { \mathrm { c i r } } ^ { 2 D - 1 }$ vel bounded. Since must be level bounde $G ( \mathbf { w } )$ is strongly convex, $F _ { \mathrm { c i r } } ^ { 2 D - 1 } ( \mathbf { w } ) ~ = ~ \sqrt { G ( \mathbf { w } ) } ~ +$ $\iota _ { \mathcal { W } _ { \mathrm { n o r m a l } } } ( \mathbf { w } )$ + +3. $F _ { \mathrm { c i r } } ^ { 2 D - 1 } \not \equiv + \infty$ . Since $\mathcal { W } _ { \mathrm { n o r m a l } }$ is nonempty (54), dom $F _ { \mathrm { c i r } } ^ { 2 D - 1 } \neq \emptyset$ , $F _ { \mathrm { c i r } } ^ { 2 D - 1 }$ s not con- i $+ \infty$ + +$F _ { \mathrm { c o n v } } ^ { N }$ $F _ { \mathrm { c o n v } } ^ { N }$ + +5 . F 2D−1 and F Ncon are all lower semi-continuous and proper. This condition follows from the fact that the functions F 2D−1 and F N are all continuous functions defined on a nonempty closed convex domain $\mathcal { W } _ { \mathrm { n o r m a l } }$ . + +Applying Proposition 7.32(c), we have $\{ F _ { \mathrm { c o n v } } ^ { N } \}$ is eventually level bounded. If we arbitrarily pick a $\mathbf { w } ^ { N } \in \mathcal { W } _ { \mathrm { c o n v } } ^ { N }$ and let $\mathbf { w } _ { \mathrm { c i r } }$ convbe the unique point in $\mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 }$ . Applying Proposition 7.33, we have $\mathbf { w } ^ { N } \to \mathbf { w } _ { \mathrm { c i r } }$ . Bsets o: ts, 2009), we obtain the convergence of the. $\{ \mathcal { W } _ { \mathrm { c o n v } } ^ { N } \}$ $\begin{array} { r } { \operatorname* { l i m } _ { N \to \infty } \mathcal { W } _ { \mathrm { c o n v } } ^ { N } = \mathcal { W } _ { \mathrm { c i r } } ^ { 2 D - 1 } } \end{array}$ + +# D DISCUSSION OF DEFINITION 2 (11) + +In this section, we want to numerically show that, given typical $\mathbf { D }$ and $s$ , there is a $\bar { \sigma } _ { \operatorname* { m i n } } > 0$ such that a random generated matrix $\mathbf { W } \in \mathbf { \bar { \mathcal { W } } } ( \mathbf { D } , s , \bar { \sigma } _ { \operatorname* { m i n } } )$ . However, given $\mathbf { D }$ and $\mathbf { W }$ , it’s intractable to completely check (11): + +$$ +\sigma _ { \operatorname* { m i n } } \Big ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } \Big ) \geq \bar { \sigma } _ { \operatorname* { m i n } } , \forall \mathbb { S } \mathrm { ~ w i t h ~ } 2 \leq | \mathbb { S } | \leq s . +$$ + +The reason is that there are extremely large amount of possible $\mathbb { S }$ s. For example, we take $M =$ $2 5 0 , N = 5 0 0 , s = 5 0$ . There are totally + +$$ +\binom { 5 0 0 } { 5 0 } + \binom { 5 0 0 } { 4 9 } + \cdot \cdot \cdot + \binom { 5 0 0 } { 2 } +$$ + +possible $\mathbb { S } s$ satisfying $2 \leq | \mathbb { S } | \leq s$ . It’s impossible to check (11) on all possible $\mathbb { S } s$ + +Instead of checking all possible $\mathbb { S } s$ , we sample $5 0 0 0 \mathbb { S } \mathrm { s }$ from the whole set: + +$$ +\mathcal S ^ { \prime } \subset S = \{ \mathbb S : \mathbb S \subset \{ 1 , 2 , \cdots , 5 0 0 \} | 2 \leq | S | \leq s \} , +$$ + +where $S ^ { \prime }$ is the set of all the samples. Then we estimate $\bar { \sigma } _ { \mathrm { m i n } }$ with the following quantity: + +$$ +\bar { \sigma } ^ { \prime } ( \mathbf { D } , \mathbf { W } ) = \operatorname* { m i n } _ { \mathbb { S } \in \mathcal { S } ^ { \prime } } \left\{ \sigma _ { \operatorname* { m i n } } \Big ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } \Big ) \right\} +$$ + +Furthermore, we use the same $\mathbf { D }$ as that in Section 5 and generate $1 0 0 0 ~ \mathbf { W } \mathbf { s }$ with each entry i.i.d sampled from the normal distribution. Then we normalize each column of the generated Ws. This technique is commonly used in sparse coding. Finally, we report the distribution of $\bar { \sigma } ^ { \prime } ( \mathbf { D } , \mathbf { W } )$ with the fixed $\mathbf { D }$ and the 1000 sampled Ws in Figure 5. + +Figure 5 demonstrates that, with the fixed $\mathbf { D }$ , most of the random generated Ws have a $\bar { \sigma } ^ { \prime } ( \mathbf { D } , \mathbf { W } )$ within the interval [0.25, 0.35]. Thus, the numerical results support our claim: with high probability, a random generated W satisfies + +$$ +\operatorname* { m i n } _ { \mathbb { S } \in \mathcal { S } } \left\{ \sigma _ { \operatorname* { m i n } } \Big ( \mathbf { I } - ( \mathbf { W } _ { : , \mathbb { S } } ) ^ { T } \mathbf { D } _ { : , \mathbb { S } } \Big ) \right\} \geq \bar { \sigma } _ { \operatorname* { m i n } } > 0 , +$$ + +that is, $\mathbf { W } \in \bar { \mathcal { W } } ( \mathbf { D } , s , \bar { \sigma } _ { \operatorname* { m i n } } )$ + +# E EFFICIENT ALGORITHM TO CALCULATE ANALYTIC WEIGHTS + +# E.1 AN EFFICIENT ALGORITHM TO SOLVE (16) + +In this section, we introduce an algorithm to solve (16) (we copy (16) below to facilitate reading): + +$$ +\operatorname* { m i n } _ { \mathbf { W } \in \mathbb { R } ^ { N \times M } } \left\| \mathbf { W } ^ { T } \mathbf { D } \right\| _ { F } ^ { 2 } , \quad \mathrm { s . t . } \left( \mathbf { W } _ { : , m } \right) ^ { T } \mathbf { D } _ { : , m } = 1 , \forall m = 1 , 2 , \cdots , M , +$$ + +![](images/212353f30a28fcf8c4d3a4fb864f91d09298479d20268f8523dd485628469f57.jpg) +Figure 5: Discussion of Definition 2: distribution of $\bar { \sigma } ^ { \prime } ( \mathbf { D } , \mathbf { W } )$ on random generated Ws. + +By the definition of the Frobenius norm, it holds that + +$$ +\| \mathbf { W } ^ { T } \mathbf { D } \| _ { F } ^ { 2 } = \| ( \mathbf { W } ^ { T } \mathbf { D } ) ^ { T } \| _ { F } ^ { 2 } = \| \mathbf { D } ^ { T } \mathbf { W } \| _ { F } ^ { 2 } . +$$ + +Thus, the above problem is equivalent with + +$$ +\operatorname* { m i n } _ { \mathbf { W } \in \mathbb { R } ^ { N \times M } } \left\| \mathbf { D } ^ { T } \mathbf { W } \right\| _ { F } ^ { 2 } , \quad \mathrm { s . t . } \left( \mathbf { D } _ { : , m } \right) ^ { T } \mathbf { W } _ { : , m } = 1 , \forall m = 1 , 2 , \cdots , M . +$$ + +We apply projected gradient descent (PGD) to solve the above problem. The gradient of $\| \mathbf { D } ^ { T } \mathbf { W } \| _ { F } ^ { 2 }$ is $\nabla \| \mathbf { D } ^ { T } \mathbf { W } \| _ { F } ^ { 2 } = \mathbf { D } \mathbf { D } ^ { T } \mathbf { W }$ . Denote the set by + +$$ +\mathcal { W } = \{ \mathbf { W } \in \mathbb { R } ^ { N \times M } | ( \mathbf { D } _ { : , m } ) ^ { T } \mathbf { W } _ { : , m } = 1 , \forall m = 1 , 2 , \cdots , M . \} +$$ + +Then the projection onto $\mathcal { W }$ can be calculated by + +$\mathrm { \mathrm { \mathrm { ~ \ p ~ } } } _ { \mathrm { r o j } _ { \mathcal { W } } } ( { \bf W } ) = { \bf W } + \Delta { \bf W } , \Delta { \bf W } = \left[ ( 1 - ( { \bf D } _ { : , 1 } ) ^ { T } { \bf W } _ { : , 1 } ) { \bf W } _ { : , 1 } , \ \cdots , \ ( 1 - ( { \bf D } _ { : , M } ) ^ { T } { \bf W } _ { : , M } ) { \bf W } _ { : , M } \right]$ With these formulas, we are able to write down the PGD, which is listed in Algorithm 1. + +# Algorithm 1: Projected gradient descent for solving (16) + +Input: Dictionary D ∈ RN×M . +Initialize: Let $\dot { \mathbf { W } } ^ { 0 } = \mathbf { D }$ . + +1 for $j = 0 , 1 , 2 , \ldots$ until convergence do + +# 3 end + +Output: $\mathbf { W } ^ { J }$ , where $J$ is the last iterate. + +In each step, calculating the gradient has the complexity of $O ( N ^ { 2 } M )$ because $\mathbf { D } \mathbf { D } ^ { T }$ can be precomputed. Calculating the projection takes $O ( N M )$ time consumptions. Due to the objective function to minimize in (16) is restricted strongly convex, Algorithm 1 is linear convergent (Zhang & Cheng, 2015). To get an $\epsilon$ -accurate solution, PGD takes ${ \cal O } ( \log ( 1 / \epsilon ) )$ steps. Thus, the complexity of Algorithm 1 is ${ \cal O } ( \log ( 1 / \epsilon ) N ^ { 2 } M )$ . We should note that the bounds given in Table 1 are the number of parameters to train, not the training complexity. The training complexity can be estimated by “Number of iterations $\times$ complexity of back-propagation”, i.e., $\bar { O } ( I B \bar { K } N \bar { M } )$ ,where $I$ is the number of iterations for training, $B$ is the batch size , and $K$ is the number of layers. Actually, Algorithm 1 (Stage 1) only takes a few seconds on an example of $\mathbf { D } : 2 5 0 \times 5 0 0$ , while the training process (Stage 2) of, for example, ALISTA, takes around 0.1 hours. + +# E.2 AN EFFICIENT ALGORITHM TO SOLVE (24) + +In this section, we introduce an algorithm to solve (24) (we copy (24) below to facilitate reading): + +$$ +\operatorname* { m i n } _ { \mathbf { w } \in \mathbb { R } ^ { D ^ { 2 } M } } \Big \| \big ( \mathbf { W } _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } ) \big ) ^ { T } \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( \mathbf { d } ) \Big \| _ { F } ^ { 2 } . +$$ + +Similarly, by (58), the above problem is equivalent with + +$$ +\operatorname* { m i n } _ { \mathbf { w } \in \mathbb { R } ^ { D ^ { 2 } M } } \Big \| \big ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ( \mathbf { d } ) \big ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } ) \Big \| _ { F } ^ { 2 } . +$$ + +Since the circular convolution is very efficient to calculate in the frequency domain, we consider solving (59) utilizing the fast Fourier transform (FFT). + +Firstly, we introduce the operators $\mathbf { D } _ { \mathrm { c i r } } ^ { N } ( \mathbf { d } ) , \mathbf { W } _ { \mathrm { c i r } } ^ { N } ( \mathbf { w } )$ in the frequency domain. To simplify the notation, we denote the operators as $\mathbf { D } _ { \mathrm { c i r } } ^ { N }$ and $\mathbf { W } _ { \mathrm { c i r } } ^ { N }$ respectively. Let $\mathcal { F }$ be the FFT operator. Thus, ${ \bf b } = { \bf D } _ { \mathrm { c i r } } ^ { N } { \bf x }$ is equivalent with + +$$ +\mathcal { F } \mathbf { b } = \mathcal { F } \mathbf { D } _ { \mathrm { c i r } } ^ { N } \mathcal { F } ^ { H } \mathcal { F } \mathbf { x } . +$$ + +Let $\hat { \mathbf { b } } = \mathcal { F } \mathbf { b } , \hat { \mathbf { x } } = \mathcal { F } \mathbf { x }$ be the frequency domain signals, let $\hat { \mathbf { D } } _ { \mathrm { c i r } } ^ { N } = \mathcal { F } \mathbf { D } _ { \mathrm { c i r } } ^ { N } \mathcal { F } ^ { H }$ be the frequency domain operator. The above equation is: + +$$ +\hat { \mathbf { b } } = \hat { \mathbf { D } } _ { \mathrm { c i r } } ^ { N } \hat { \mathbf { x } } . +$$ + +The frequency domain operator $\hat { \mathbf { D } } _ { \mathrm { c i r } } ^ { N }$ is much cheaper to calculate than the operator $\mathbf { D } _ { \mathrm { c i r } } ^ { N }$ in the spacial domain because it is block diagonal (Wohlberg, 2016). Specifically, we zero pad d to $N \times N$ and do FFT: $\hat { \mathbf { d } } _ { m } = \mathrm { F F T } \left( \mathrm { z e r o p a d } ( \mathbf { d } _ { m } , N - D ) \right)$ , then the above operator can be explicitly written as: + +$$ +\hat { \mathbf { b } } = \sum _ { m = 1 } ^ { M } \overline { { \hat { \mathbf { d } } _ { m } } } \odot \hat { \mathbf { x } } _ { m } , +$$ + +where ¯· means complex conjugate. This is due to $\mathbf { D } _ { \mathrm { c i r } } ^ { N }$ is actually cross-correlation, not convolution (see (18)). Cross-correlation is equal to the transpose of convolution. Thus, there should be complex conjugate in the frequency domain. + +Further, since + +$$ +\begin{array} { r l } & { \| ( \hat { \mathbf { D } } _ { \mathrm { c i r } } ^ { N } ) ^ { H } \hat { \mathbf { W } } _ { \mathrm { c i r } } ^ { N } \| _ { F } ^ { 2 } = \| ( \mathcal { F } \mathbf { D } _ { \mathrm { c i r } } ^ { N } \mathcal { F } ^ { H } ) ^ { H } \mathcal { F } \mathbf { W } _ { \mathrm { c i r } } ^ { N } \mathcal { F } ^ { H } \| _ { F } ^ { 2 } = \| \mathcal { F } ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N } \mathcal { F } ^ { H } \| _ { F } ^ { 2 } } \\ & { \qquad = \| ( \mathbf { D } _ { \mathrm { c i r } } ^ { N } ) ^ { T } \mathbf { W } _ { \mathrm { c i r } } ^ { N } \| _ { F } ^ { 2 } , } \end{array} +$$ + +problem (59) is equivalent with + +$$ +\operatorname* { m i n } _ { \mathbf { w } \in \mathbb { R } ^ { D ^ { 2 } M } } \Big \| \big ( \hat { \mathbf { D } } _ { \mathrm { c i r } } ^ { N } \big ) ^ { H } \hat { \mathbf { W } } _ { \mathrm { c i r } } ^ { N } \Big \| _ { F } ^ { 2 } , +$$ + +which can be efficiently solved by the frequency domain ISTA in (Liu et al., 2017). The details are outlined in Algorithm 2. + +# F VISUALIZATION OF THE ANALYTIC CONVOLUTIONAL WEIGHTS + +Fig. 6 visualizes the dictionary d A-LISTA simulation of Section 5 $( 7 \times 7 \times 6 4 )$ and the weights ned by Algorith $\tilde { \mathbf { w } } \in \mathcal { W } _ { \mathrm { c i r } } ^ { 1 3 }$ , used in the convolutionalendix E.2. + +# G ALGORITHM DETAILS OF TRAINING ROBUST ALISTA + +# G.1 MODEL ARCHITECTURE + +Inspired by the a similar unrolling and truncating fashion in LISTA, we can approximately solve the coherence minimization problem (16) using a similar finite-layer neural network that is unfolded from iterative algorithms. Because the linear constraints in (16) are hard to enforce in deep neural networks, we first relax it to the following form: + +$$ +\underset { \mathbf { W } \in \mathbb { R } ^ { N \times M } } { \arg \operatorname* { m i n } } \left\| \mathbf { Q } \odot ( \mathbf { D } ^ { T } \mathbf { W } - \pmb { I } _ { M } ) \right\| _ { F } ^ { 2 } , +$$ + +# Algorithm 2: Frequency-domain ISTA for solving (24) + +Input: Dictionary $\mathbf { d } = [ \mathbf { d } _ { 1 } , \boldsymbol { \cdot \cdot \cdot } , \mathbf { d } _ { M } ] ^ { T } , \mathbf { d } _ { m } \in \mathbb { R } ^ { D ^ { 2 } } , m = 1 , 2 , \cdots , M .$ Initialize: Let $\mathbf { w } ^ { \bar { 0 } } = \mathbf { d }$ . +for $j = 0 , 1 , 2 , \ldots$ until convergence do + +2 + +Zeropad and FFT: + +$$ +\hat { \mathbf { w } } _ { m } ^ { j } = \mathrm { F F T } \Big ( \mathrm { z e r o p a d } \big ( \mathbf { w } _ { m } ^ { j } , N - D \big ) \Big ) , \quad m = 1 , \cdots , M . +$$ + +3 Compute frequency domain gradient: + +$$ +( \nabla f ) _ { m } = \Big ( \sum _ { m = 1 } ^ { M } \hat { \mathbf { d } } _ { m } \odot \bar { \hat { \mathbf { d } } } _ { m } \Big ) \odot \hat { \mathbf { w } } _ { m } ^ { j } , \quad m = 1 , \cdots , M , +$$ + +where ¯· represents the conjugate of a complex number. + +4 + +Compute the next iterate: + +$$ +\mathbf { w } _ { m } ^ { j + 1 } = \mathrm { P r o j } _ { \mathcal { W } _ { \mathrm { n o r m a l } } } \Big ( \mathrm { I F F T } \big ( \hat { \mathbf { w } } _ { m } ^ { j } - \boldsymbol { \eta } ( \nabla f ) _ { m } \big ) \Big ) , \quad m = 1 , \cdots , M , +$$ + +where the set $\mathcal { W } _ { \mathrm { n o r m a l } }$ is defined in (53). + +# 5 end + +Output: $\mathbf { w } ^ { J }$ , where $J$ is the last iterate. + +![](images/e098e2cc3cfc2da2fdab50ba85dada72320489488e18bbff5cca05da392227dc.jpg) +Figure 6: A visualization of convolutional kernels d and $\tilde { \mathbf { w } }$ , which is obtained by Algorithm 2 and used in the convolutional A-LISTA. w˜ keeps the high-frequency texture in d. The support of w is small, most of the pixels in w are zeros. Then the coherence between shifted $\mathbf { d }$ and w is nearly 0. + +where $\odot$ is the Hadamard product and $\mathbf { Q }$ is a weight matrix that put more penalty on errors on diagonals, because entries on the diagonal will be far smaller than off-diagonal. The above relaxed coherence minimization can be solved using the gradient descent algorithm: + +$$ +\mathbf { W } ^ { ( k + 1 ) } = \mathbf { W } ^ { ( k ) } - \gamma ^ { ( k ) } \mathbf { D } ( \mathbf { Q } ^ { 2 } \odot ( \mathbf { D } ^ { T } \mathbf { W } ^ { ( k ) } - \pmb { I } _ { M } ) ) . +$$ + +By unfolding (61) and truncate to $K$ steps, and considering the $\gamma ^ { ( k ) } \mathbf { D } ^ { T }$ outside the residual as learnable parameters $\mathbf { B }$ , we will have a deep neural network $\mathbf { W } = E ( \mathbf { D } )$ as a coherence minimizer. We call it a Stage 1 encoder as it encodes a dictionary $\mathbf { D }$ into a weight matrix, that can be used in the Stage 2 of ALISTA, refered as a decoder. One layer of this model is shown in Fig. 7(a). + +The illustration of the whole feed-forward robust model is shown in Fig. 7(b). The two parts, the encoder and the decoder, can be jointly trained to gain the most from data-driven learning. We further adopt pre-training and curriculum learning to stabilize training, as to be discussed below. + +![](images/6776e69f5cec6be82f55156bb9af67711806163f340570dab157b388fb069086.jpg) +Figure 7: Feed-Forward Analytic LISTA. + +# G.2 MODEL TRAINING + +To stabilize the training process, we train the model in two stages: the pre-training stage and curriculum (joint) training stage. + +Pre-Training Stage. We first pre-train the encoder and the decoder individually. The pre-training of the decoder, e.g., ALISTA, follows the standard training procedure in Section 5.1, without bothering the perturbations of $_ { D }$ . On the other hand, the encoder will always see perturbed dictionaries $\tilde { \cal D } = { \cal D } + \varepsilon _ { \cal D }$ , where $\varepsilon _ { D }$ ’s entries are sampled from i.i.d. normal distribution with zero mean and $\sigma _ { p r e } ^ { 2 }$ variance, and update its weight to minimize loss function defined by (60). The $\sigma _ { p r e }$ is a hyperparameter that we manually select for the pre-training stage, with a default value of 0.01. We use an exponentially decaying learning rate for encoder pre-training with an initial value $\alpha _ { p r e } = 1 0 ^ { - 4 }$ . + +Curriculum (Joint) Training Stage. After the pre-training stage, we concatenate these two parts and do joint training. However, a direct end-to-end tuning was observed to cause much instability, due to the randomness in weights. Inspired by the curriculum learning technique, we first perturb the dictionaries with smaller standard deviations and gradually increase the perturbation level during training. Specifically, starting from a small standard deviation $\sigma ^ { t } = \sigma ^ { 0 }$ , the curriculum joint training procedure repeats the routine below: + +• First uniformly sample a batch of standard deviations $\{ \sigma _ { i } \} _ { i = 1 } ^ { B _ { D } }$ from $[ 0 , \sigma ^ { t } ]$ , where $B _ { D }$ is the batch size for perturbations of the original dictionary $\mathbf { D }$ . Use the sampled standard deviations to sample $B _ { D }$ perturbations and apply to $\mathbf { D }$ and then normalized them to get $\{ \tilde { \bf D } \} _ { i = 1 } ^ { B _ { D } }$ . +• Sample a batch of sparse codes $\{ \mathbf { x } _ { j } \} _ { j = 1 } ^ { B _ { x } }$ from a pre-defined Gaussian-Bernoulli distribution; the supports of the sparse codes are decided i.i.d. by a Bernoulli distribution to have around $1 0 \%$ non-zero entries; and the magnitudes are sampled from i.i.d. standard Gaussian. $B _ { x }$ is the batch size for sparse codes in training. +• Measure $\mathbf { y } _ { i , j } = \tilde { \mathbf { D } } _ { i } \mathbf { x } _ { j }$ . Then $( \mathbf { y } _ { i , j } , \mathbf { x } _ { j } , \tilde { \mathbf { D } } _ { i } )$ forms a tripelet of training sample. Note that only Robust ALISTA needs $\tilde { \bf D } _ { i }$ as part of the training samples. +• Feed in the data and update the encoder and decoder with learning rates $\alpha _ { e }$ and $\alpha _ { d }$ , using the Adam Optimizer, respectively. +• Increase $\sigma ^ { t }$ to the next larger value, after $C$ training batches. +• Repeat the above steps, until the value of $\sigma ^ { t }$ exceeds the pre-defined $\sigma _ { m a x }$ , that represents the maximal standard deviation to sample the dictionary perturbation. + +In the experiment, we have $B _ { D } = 4$ , $B _ { x } = 1 6$ , $C = 5 0 0 0 0$ , $\alpha _ { e } = 1 0 ^ { - 6 }$ , $\alpha _ { d } = 1 0 ^ { - 4 }$ and $\sigma _ { m a x } \in$ $\{ 0 . 0 2 , 0 . { \bar { 0 } } 3 \}$ and $\sigma ^ { t }$ is obtained by linearly interpolating between $[ 0 , \sigma _ { m a x } ]$ for $L - 1$ times. We choose $L = 5$ ; hence $\sigma ^ { t }$ takes $\scriptstyle { \frac { i } { 5 } } \sigma _ { m a x } , i = 1 , 2 , \ldots , 5$ in order. + +# H RESULTS OF NATURAL IMAGE DENOISING USING CONV ALISTA + +The natural image denoising experiment is conducted on the same BSD 500 dataset using the 400- image training set, 50-image validation set and 50-image test set. We convert them all to grayscale, and then add $\sigma = 2 0$ Gaussian i.i.d. noise. We train both Conv LISTA (i.e., model (20)) and Conv ALISTA (i.e., model (26)). Both networks have 5 layers, with the same dictionary D obtained from the training set by solving (24). We reconstruct the denoised images using by convolving the learned feature maps with the original dictionary D. The mean-square-error (MSE) between denoised and clean images are adopted as the network training loss, as inspired by (Zhou et al., 2018). + +Six popular benchmark images (adding $\sigma = 2 0$ noise) are tested and reported in Table 4. The APSNR denotes the average PSNR over all images and the A-Times represents the average inference time (in seconds) for denoising one image. We compare Conv LISTA and Conv ALISTA, as well as the classical KSVD denoising algorithm (Elad & Aharon, 2006) and the recent CSC denoising algorithm with gradient regularization (CSC-GR) (Wohlberg, 2018). The results show that Conv LISTA and Conv ALISTA (without heavy tuning done for their optimal performance) can perform comparably with KSVD and outperforms CSC-GR, but with tremendously faster inference speeds than KSVD/CSC-GR. More importantly, Conv LISTA and Conv ALISTA only have marginal performance differences, validating again the analytic weights in convolutional cases. + +Table 4: Peak Signal to Noise Ratio (PSNR) Comparision between Conv LISTA and Conv ALISTA. + +
ModelImage PSNR (dB)A-PSNRA-Time
LennaHousePepperCoupleBoatsBarbara
KSVD31.0333.2430.9731.7131.0030.4731.4024.70
CSC-GR28.4129.1127.3929.3128.3527.1928.297.56
Conv LISTA31.2632.7731.0031.8930.7829.5331.210.012
Conv ALISTA31.0132.4630.8131.8530.5829.7231.070.014
+ +# I RESULTS OF ABLATION STUDIES IN ROBUSTNESS EXPERIMENTS + +As one anonymous reviewer kindly pointed out, Robust ALISTA has larger parameter space over TiLISTA and ALISTA trained. Therefore, they suggested that we increased the number of layers in TiLISTA, ALISTA and the baseline model LISTA-CPSS in (Chen et al., 2018) to see if their performance in this above evaluation setting can be improved in that way. Note that LISTA-CPSS has tens of layers, hence actually containing more parameters than robust ALISTA. In addition, the reviewers also suggested a set of ablation studies to investigate whether more layers in the encoder can endorse the model better adaptivity to higher level perturbations. We conduct the suggested experiments and present the results in this section. + +# I.1 NUMBER OF LAYERS IN TILISTA, ALISTA AND LISTA-CPSS + +As the reviewers pointed out, the comparison we present in Section. 5.3 might be unfair because robust ALISTA contains much more parameters (because it contains a 4-layer encoder) comparing to ALISTA, which only learns two series of scalars, and TiLISTA which has just one more matrix weight than ALISTA. Therefore, we add the following experiments to consolidate our claim on the effectiveness of the robust ALISTA model: + +• we increase the number of layers of TiLISTA and ALISTA which are then trained in the same data augmentation setting as we do in Section. 5.3, to see if they could yield comparitive robustness against dictionary perturbations; we also compare the robust ALISTA with the baseline LISTA-CPSS model in Chen et al. (2018), which contains tens of layers of independent weight matrices, thus having even more parameters than robust ALISTA. This comparison could consolidate our claim that the outstanding adaptiveness to dictionary perturbations of robust ALISTA is brought by its encoder-decoder structure rather than its learning capacity alone. + +The results are shown in Table. 5, where the performances are measured with NMSE in dB, which is defined in Section 5.1. The “Augmented” prefix means the models are trained in the data augmentation setting. $\sigma$ is the standard deviation of the Gaussian distribution that is used to generate the dictioanry perturbations. $T$ stands for the number of layers (in the case of robust ALISTA it means the nubmer of layers of the ALISTA decoder, with a 4-layer encoder). We follow the training strategy and settings explained in Appendix G, with $\sigma _ { m a x } = 0 . 0 2$ during training. + +On one hand, the comparison of performances of ALISTA, TiLISTA and LISTA-CPSS shows results that are consistent to the intuition that larger parameter space yields larger learning capacity, and therefore, better adaptiveness (LISTA-CPSS $>$ TiLISTA $>$ ALISTA). On the other hand, we can also find that ALISTA with more layers has worse performance. We think this observation is also reasonable for two reasons: 1) adding more layers in ALISTA does not enlarge the parameter volume significantly because it has only two scalar parameters in each layer; noting that ALISTA uses a fixed, analytically solved weight matrix, if this weight matrix is not compatible with the perturbed dictionary, more layers can even hurt the performance instead of improving. Lastly, it’s clearly shown that robust ALISTA outperforms LISTA-CPSS, even if it contains less parameters. This proves that the encoding process that adaptively transforms the perturbed dictiories is necessary to achieve good robustness against perturbations in dictionaries. + +
σ of perturbationsduring testing0.00010.0010.010.0150.020.025
AugmentedALISTAT=16-26.58-25.87-15.49-11.71-8.84-6.74
T=20-24.43-24.46-15.39-11.77-8.94-6.82
T=24-24.12-24.00-15.45-11.68-8.81-6.70
AugmentedTiLISTAT=16-27.76-27.18-16.83-12.95-9.81-7.55
T=20-28.13-28.54-17.15-12.98-9.83-7.58
T=24-26.08-27.27-17.34-13.14-9.91-7.61
AugmentedLISTA-CPSST=16-27.93-27.18-16.96-12.99-9.93-7.70
T=20-28.17-27.33-16.95-13.00-9.94-7.71
T=24-30.30-29.24-16.86-12.97-9.94-7.70
Robust ALISTAT=16-62.47-62.41-62.02-61.50-60.67-45.00
+ +Table 5: The results (recovery NMSE in dB) of ablation study on the influence of model capacity towards the model robustness against dictionary perturbations. + +# I.2 ABLATION STUDY ON THE DEPTHS OF ENCODERS IN ROBUST ALISTA + +Another constructive suggestion from the reviewers is to design an ablation study to investigate the influence of the depth of encoders in robust ALISTA on its adaptivity to dictionary perturbations. A natural intuition is that, adding more layers to the encoder can increase its ability to sustain larger perturbation levels. But is this true? + +Therefore, we train another two robust ALISTA models, with a 5-layer and a 6-layer encoders respectively and 16-layer ALISTA decoders for both, and compare them with the originally reported robust ALISTA model with a 4-layer encoder and a 16-layer ALISTA decoders. All three models use one pretrained decoder, and pretrain their encoders using the same method (see Appendix G). We only use $\sigma _ { m a x } = 0 . 0 2$ during training. One thing to notice is that we observe unstable training process if we use default initial learning rates $\alpha _ { p r e } = 1 0 ^ { - 4 }$ in the pre-training stage and $\alpha _ { e } = 1 0 ^ { - } \overline { { 6 } }$ in the joint training stage when encoders have 5 or 6 layers. Therefore, we use $\alpha _ { p r e } ^ { \prime } = 1 0 ^ { - 5 }$ for the 6-layer encoder in the pre-training stage, and in the joint training stage use a decreased and uniform initial learning rate $\alpha _ { e } ^ { \prime } = 1 0 ^ { - 9 }$ for the three encoders while keeping the default initial learning rate $\alpha _ { d } = 1 0 ^ { - 4 }$ for decoders. The other settings remain the same. + +Results are shown in Table. 6. The performances are measured with NMSE in dB, defined in Section 5.1. From the table we can see that encoders do show better robustness when they have more layers, i.e. larger learning capacity. + +
# Encoder Layerso of perturbations during testing
0.00010.0010.010.0150.020.025
4-68.57-68.56-67.94-66.86-64.84-56.63
5-69.34-69.34-69.02-68.49-67.20-65.55
6-70.38-70.33-69.92-69.22-67.72-65.60
+ +Table 6: The results of ablation study on the influence of the depths of encoder towards the model robustness against dictionary perturbations. \ No newline at end of file diff --git a/parse/train/B1lnzn0ctQ/B1lnzn0ctQ_content_list.json b/parse/train/B1lnzn0ctQ/B1lnzn0ctQ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..28f89ab160fa03af4032b79ada446f2e7a9d4d4e --- /dev/null +++ b/parse/train/B1lnzn0ctQ/B1lnzn0ctQ_content_list.json @@ -0,0 +1,6544 @@ +[ + { + "type": "text", + "text": "ALISTA: ANALYTIC WEIGHTS ARE AS GOOD ASLEARNED WEIGHTS IN LISTA", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jialin Liu∗ Department of Mathematics University of California, Los Angeles liujl11@math.ucla.edu ", + "bbox": [ + 184, + 170, + 433, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xiaohan Chen∗ \nDepartment of Computer Science and Engineering \nTexas A&M University \nchernxh@tamu.edu ", + "bbox": [ + 482, + 170, + 813, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Wotao Yin ", + "text_level": 1, + "bbox": [ + 550, + 248, + 625, + 261 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zhangyang Wang \nDepartment of Computer Science and Engineering \nTexas A&M University \natlaswang@tamu.edu ", + "bbox": [ + 183, + 247, + 516, + 304 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Mathematics University of California, Los Angeles wotaoyin@math.ucla.edu ", + "bbox": [ + 549, + 262, + 797, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 340, + 544, + 354 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks based on unfolding an iterative algorithm, for example, LISTA (learned iterative shrinkage thresholding algorithm), have been an empirical success for sparse signal recovery. The weights of these neural networks are currently determined by data-driven “black-box” training. In this work, we propose Analytic LISTA (ALISTA), where the weight matrix in LISTA is computed as the solution to a data-free optimization problem, leaving only the stepsize and threshold parameters to data-driven learning. This significantly simplifies the training. Specifically, the data-free optimization problem is based on coherence minimization. We show our ALISTA retains the optimal linear convergence proved in (Chen et al., 2018) and has a performance comparable to LISTA. Furthermore, we extend ALISTA to convolutional linear operators, again determined in a data-free manner. We also propose a feed-forward framework that combines the data-free optimization and ALISTA networks from end to end, one that can be jointly trained to gain robustness to small perturbations in the encoding model. ", + "bbox": [ + 233, + 372, + 764, + 566 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 594, + 336, + 609 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sparse vector recovery, or sparse coding, is a classical problem in source coding, signal reconstruction, pattern recognition and feature selection. There is an unknown sparse vector $\\mathbf { x } ^ { * } =$ $[ x _ { 1 } ^ { * } , \\cdot \\cdot \\cdot , x _ { M } ^ { * } ] ^ { T } \\in \\mathbb { R } ^ { M }$ . We observe its noisy linear measurements: ", + "bbox": [ + 176, + 613, + 823, + 656 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/be8e83dcb4ac35768817058468f5090ca1fb439f693a1f5e90ea6beb487a3aee.jpg", + "text": "$$\n{ \\bf b } = \\sum _ { m = 1 } ^ { M } { \\bf d } _ { m } x _ { m } ^ { * } + \\varepsilon = { \\bf D } { \\bf x } ^ { * } + \\varepsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 662, + 611, + 707 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $\\mathbf { b } \\in \\mathbb { R } ^ { N }$ , $\\mathbf { D } = [ \\mathbf { d } _ { 1 } , \\therefore \\cdot \\cdot , \\mathbf { d } _ { M } ] \\in \\mathbb { R } ^ { N \\times M }$ is the dictionary, and $ { \\varepsilon } \\in \\mathbb { R } ^ { N }$ is additive Gaussian white noise. For simplicity, each column of $\\mathbf { D }$ , named as a dictionary kernel, is normalized, that is, $\\lVert \\mathbf { d } _ { m } \\rVert _ { 2 } = \\lVert \\mathbf { D } _ { : , m } \\rVert _ { 2 } \\stackrel { - } { = } 1$ , $m = 1 , 2 , \\cdots , M$ . Typically, we have $N \\ll M$ , so Equation (1) is an under-determined system. ", + "bbox": [ + 174, + 713, + 825, + 772 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, when $\\mathbf { x } ^ { * }$ is sufficiently sparse, it can be recovered faithfully. A popular approach is to solve the LASSO problem below (where $\\lambda$ is a scalar): ", + "bbox": [ + 171, + 777, + 825, + 806 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/f5f00cdff5c499bdfd85c71eb2c9ff22a46f2af5ca8d3df78937bcf20167394f.jpg", + "text": "$$\n\\underset { \\mathbf { x } } { \\mathrm { m i n i m i z e } } \\frac { 1 } { 2 } \\| \\mathbf { b } - \\mathbf { D x } \\| _ { 2 } ^ { 2 } + \\lambda \\| \\mathbf { x } \\| _ { 1 }\n$$", + "text_format": "latex", + "bbox": [ + 385, + 813, + 611, + 843 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "using iterative algorithms such as the iterative shrinkage thresholding algorithm (ISTA): ", + "bbox": [ + 173, + 849, + 750, + 866 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/3856ea88d820bc0a143328c9f55dd94d3696c9c5eefbbf52c2fd6917522e5e60.jpg", + "text": "$$\n\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\lambda / L } \\Big ( \\mathbf { x } ^ { ( k ) } + \\frac { 1 } { L } \\mathbf { D } ^ { T } ( \\mathbf { b } - \\mathbf { D x } ^ { ( k ) } ) \\Big ) , \\quad k = 0 , 1 , 2 , . . .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 871, + 705, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $\\eta _ { \\theta }$ is the soft-thresholding function1 and $L$ is usually taken as the largest eigenvalue of $\\mathbf { D } ^ { T } \\mathbf { D }$ . ", + "bbox": [ + 171, + 102, + 823, + 118 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Inspired by ISTA, the authors of (Gregor $\\&$ LeCun, 2010) proposed to learn the weights in the matrices in ISTA rather than fixing them. Their methods is called Learned ISTA (LISTA) and resembles a recurrent neural network (RNN). If the iteration is truncated to $K$ iterations, LISTA becomes a $K$ -layer feed-forward neural network with side connections. Specifically, LISTA is: ", + "bbox": [ + 174, + 125, + 825, + 181 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/cb8ed69d4a3182a0e4e03697e141169a42c6964b101cc035578723ce107ea8c7.jpg", + "text": "$$\n\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } ( \\mathbf { W } _ { 1 } ^ { ( k ) } \\mathbf { b } + \\mathbf { W } _ { 2 } ^ { ( k ) } \\mathbf { x } ^ { ( k ) } ) , \\quad k = 0 , 1 , \\cdots , K - 1 .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 188, + 705, + 208 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "If we set $\\begin{array} { r } { \\mathbf { W } _ { 1 } ^ { ( k ) } \\equiv \\frac { 1 } { L } \\mathbf { D } ^ { T } } \\end{array}$ , $\\mathbf { W } _ { 2 } ^ { ( k ) } \\equiv \\mathbf { I } - \\mathbf { \\Pi } _ { L } ^ { 1 } \\mathbf { D } ^ { T } \\mathbf { D }$ , $\\theta ^ { ( k ) } \\equiv \\frac { \\ d _ { 1 } } { \\ d { } _ { L } } \\ d \\lambda$ , then LISTA recovers ISTA. Given each pair of sparse vector and its noisy measurements $( \\mathbf { x } ^ { * } , \\mathbf { b } )$ , applying (4) from some initial point $\\mathbf { x } ^ { ( 0 ) }$ and using $\\mathbf { b }$ as the input yields $\\mathbf { x } ^ { ( k ) }$ . Our goal is to choose the parameters $\\begin{array} { r } { \\Theta = \\big \\{ \\mathbf { W } _ { 1 } ^ { ( k ) } , \\mathbf { W } _ { w } ^ { ( k ) } , \\theta ^ { ( k ) } \\big \\} _ { k = 0 , 1 , \\dots , K - 1 } } \\end{array}$ such that $\\mathbf { x } ^ { ( k ) }$ is close to $\\mathbf { x } ^ { * }$ for all sparse $\\mathbf { x } ^ { * }$ following some distribution $\\mathcal { P }$ . Therefore, given the distribution $\\mathcal { P }$ , all parameters in $\\Theta$ are subject to learning: ", + "bbox": [ + 173, + 215, + 825, + 294 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/863e0a2834649200ad6a94de147d85921d4fb84bad33feeb1536829820ddea9f.jpg", + "text": "$$\n\\underset { \\Theta } { \\operatorname* { m i n i m i z e } } \\mathbb { E } _ { \\mathbf { x } ^ { * } , \\mathbf { b } \\sim \\mathcal { P } } \\left\\| \\mathbf { x } ^ { ( K ) } \\Big ( \\Theta , \\mathbf { b } , \\mathbf { x } ^ { ( 0 ) } \\Big ) - \\mathbf { x } ^ { * } \\right\\| _ { 2 } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 299, + 656, + 330 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This problem is approximately solved over a training dataset $\\{ ( \\mathbf { x } _ { i } ^ { * } , \\mathbf { b } _ { i } ) \\} _ { i = 1 } ^ { N }$ sampled from $\\mathcal { P }$ ", + "bbox": [ + 174, + 338, + 779, + 354 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Many empirical results, e.g., (Gregor & LeCun, 2010; Sprechmann et al., 2015; Wang et al., 2016b), show that a trained $K$ -layer LISTA (with $K$ usually set to $1 0 \\sim 2 0 $ ) or its variants can generalize more than well to unseen samples $( \\mathbf { x } ^ { \\prime } , \\mathbf { b } ^ { \\prime } )$ from the same distribution and recover $\\mathbf { x } ^ { \\prime }$ from $\\mathbf { b } ^ { \\prime }$ to the same accuracy within one or two order-of-magnitude fewer iterations than the original ISTA. Additionally, the accuracies of the outputs $\\big \\{ \\mathbf { x } ^ { ( k ) } \\big \\}$ of the layers $k = 1 , . . , K$ gradually improve. However, such networks will generalize worse when the input deviates from the training distribution (e.g., when D varies), in contrast to the classical iterative algorithms such as ISTA that are trainingfree and thus agnostic to the input distribution. The Analysis-Synthesis model (Rubinstein & Elad, 2014; Yang et al., 2016) could also be viewed as a special LISTA model with only one layer $K = 1$ ). ", + "bbox": [ + 174, + 359, + 825, + 487 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "More recently, the convolutional sparse coding (CSC), an extension of the sparse coding (1), gains increasingly attention in the machine learning area. (Sreter & Giryes, 2018) showed that the CSC could be similarly approximated and accelerated by a LISTA-type feed-forward network. (Tolooshams et al., 2018) designed a structure of sparse auto-encoder inspired by multi-layer CSC. (Papyan et al., 2016; Sulam et al., 2017) also revealed CSC as a potentially useful tool for understanding general convolutional neural networks (CNNs). ", + "bbox": [ + 174, + 492, + 825, + 577 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 588, + 331, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Despite the empirical success (Sprechmann et al., 2015; Wang et al., 2016a;b;c;d; Zhang & Ghanem, 2018; Zhou et al., 2018; Ito et al., 2018) in constructing fast trainable regressors for approximating iterative sparse solvers, the theoretical understanding of such approximations remains limited. ", + "bbox": [ + 174, + 607, + 825, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A handful of recent works have been investigating the theory of LISTA. (Moreau & Bruna, 2017) re-factorized the Gram matrix of dictionary, by trying to nearly diagonalize the Gram matrix with a basis, subject to a small $\\ell _ { 1 }$ perturbation. They thus re-parameterized LISTA a new factorized architecture that achieved similar acceleration gain to LISTA, hence ending up with an “indirect” proof. They concluded that LISTA can converge faster than ISTA, but still sublinearly. (Giryes et al., 2018) interpreted LISTA as a projected gradient descent descent (PGD) where the projection step was inaccurate, which enables a trade-off between approximation error and convergence speed. The latest work (Chen et al., 2018) presented the more related results to ours: they introduced necessary conditions for the LISTA weight structure in order to achieve asymptotic linear convergence of LISTA, which also proved to be a theoretical convergence rate upper bound. They also introduced a thresholding scheme for practically improving the convergence speed. Note that, none of the above works extended their discussions to CSC and its similar LISTA-type architectures. ", + "bbox": [ + 174, + 656, + 825, + 823 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Several other works examined the theoretical properties of some sibling architectures to LISTA. (Xin et al., 2016) studied the model proposed by (Wang et al., 2016b), which unfolded/truncated the iterative hard thresholding (IHT) algorithm instead of ISTA, for approximating the solution to $\\ell _ { 0 }$ - minimization. They showed that the learnable fast regressor can be obtained by using a transformed dictionary with improved restricted isometry property (RIP). However, their discussions are not applicable to LISTA directly, although IHT is linearly convergent (Blumensath & Davies, 2009) under rather strong assumptions. Their discussions were also limited to linear sparse coding and resulting fully-connected networks only. (Borgerding et al., 2017; Metzler et al., 2017) studied a similar learning-based model inspired from another LASSO solver, called approximated message passing (AMP). (Borgerding et al., 2017) showed the MMSE-optimality of an AMP-inspired model, but not accompanied with any convergence rate result. Also, the popular assumption in analyzing AMP algorithms (called “state evolution”) does not hold when analyzing ISTA. ", + "bbox": [ + 174, + 830, + 825, + 900 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1.2 MOTIVATION AND CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 178, + 212, + 460, + 226 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This paper presents multi-fold contributions in advancing the theoretical understanding of LISTA, beyond state-of-the-art results. Firstly, we show that the layer-wise weights in LISTA need not being learned from data. That is based on decoupling LISTA training into a data-free analytic optimization stage followed by a lighter-weight data-driven learning stage without compromising the optimal linear convergence rate proved in (Chen et al., 2018). We establish a minimum-coherence criterion between the desired LISTA weights and the dictionary D, which leads to an efficient algorithm that can analytically solve the former from the latter, independent of the distribution of x. The data-driven training is then reduced to learning layer-wise step sizes and thresholds only, which will fit the distribution of x. The new scheme, called Analytic LISTA (ALISTA), provides important insights into the working mechanism of LISTA. Experiments shows ALISTA to perform comparably with previous LISTA models (Gregor & LeCun, 2010; Chen et al., 2018) with much lighter-weight training. Then, we extend the above discussions and conclusions to CSC, and introduce an efficient algorithm to solve the convolutional version of coherence minimization. Further, we introduce a new robust LISTA learning scheme benefiting from the decoupled structure, by adding perturbations to $\\mathbf { D }$ during training. The resulting model is shown to possess much stronger robustness when the input distribution varies, even when $\\mathbf { D }$ changes to some extent, compared to classical LISTA models that learn to (over-)fit one specific $\\mathbf { D }$ . ", + "bbox": [ + 173, + 231, + 825, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 ANALYTIC LISTA: CALCULATING WEIGHTS WITHOUT TRAINING", + "text_level": 1, + "bbox": [ + 176, + 488, + 741, + 503 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We theoretically analyze the LISTA-CPSS model defined in (Chen et al., 2018): ", + "bbox": [ + 178, + 510, + 699, + 525 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/fee2299837a12f089dfca4e228697f147143c7334978af5ef2e3ebb04ac3d2a7.jpg", + "text": "$$\n\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - ( \\mathbf { W } ^ { ( k ) } ) ^ { T } \\big ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 336, + 529, + 656, + 555 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { W } ^ { ( k ) } = [ \\mathbf { w } _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , \\mathbf { w } _ { M } ^ { ( k ) } ] \\in \\mathbb { R } ^ { N \\times M }$ is a linear operator with the same dimensionality with D, $\\mathbf { x } ^ { ( k ) } = \\left[ x _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , x _ { M } ^ { ( k ) } \\right]$ is the $k ^ { \\mathrm { { t h } } }$ layer node. In (6), $\\boldsymbol { \\Theta } = \\{ \\mathbf { W } ^ { ( k ) } , \\boldsymbol { \\theta } ^ { ( k ) } \\} _ { k }$ are parameters to train. Model (6) can be derived from (4) with $\\mathbf { W } _ { 1 } ^ { ( k ) } = ( \\mathbf { W } ^ { ( k ) } ) ^ { T }$ , $\\mathbf { W } _ { 2 } ^ { ( k ) } = \\mathbf { I } - \\mathbf { W } _ { 1 } ^ { ( k ) } \\mathbf { D }$ . (Chen et al., 2018) showed that (6) has the same representation capability with (4) on the sparse recovery problem, with a specifically light weight structure. ", + "bbox": [ + 173, + 558, + 825, + 641 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our theoretical analysis will further define and establish properties of “good” parameters $\\Theta$ in (6), and then discuss how to analytically compute those good parameters rather than relying solely on black-box training. In this way, the LISTA model could be further significantly simplified, with little performance loss. The proofs of all the theorems in this paper are provided in the appendix. ", + "bbox": [ + 173, + 647, + 823, + 704 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 RECOVERY ERROR UPPER BOUND ", + "text_level": 1, + "bbox": [ + 174, + 714, + 450, + 728 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We start with an assumption on the “ground truth” signal $\\mathbf { x } ^ { * }$ and the noise $\\varepsilon$ . ", + "bbox": [ + 173, + 733, + 676, + 748 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Assumption 1 (Basic assumptions). Signal $\\mathbf { x } ^ { * }$ is sampled from the following set: ", + "bbox": [ + 178, + 751, + 702, + 766 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c116b90c9eab8f956031fe56ae6616cf967f9b62f52133eb52994328f2dbf8ef.jpg", + "text": "$$\n\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) \\triangleq \\Big \\{ \\mathbf { x } ^ { * } \\Big | | x _ { i } ^ { * } | \\leq B , \\forall i , \\| \\mathbf { x } ^ { * } \\| _ { 0 } \\leq s \\Big \\} .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 767, + 663, + 795 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In other words, $\\mathbf { x } ^ { * }$ is bounded and $s$ -sparse2 $s \\geq 2 ,$ ). Furthermore, we assume $\\varepsilon = 0$ . ", + "bbox": [ + 173, + 797, + 736, + 814 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The zero-noise assumption is for simplicity of the proofs. Our experiments will show that our models are robust to noisy cases. ", + "bbox": [ + 176, + 823, + 826, + 853 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The mutual coherence of the dictionary $\\mathbf { D }$ is a significant concept in compressive sensing (Donoho & Elad, 2003; Elad, 2007; Lu et al., 2018). A dictionary with small coherence possesses better sparse recovery performance. Motivated by this point, we introduce the following definition. ", + "bbox": [ + 176, + 858, + 825, + 901 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1. Given $\\mathbf { D } \\in \\mathbb { R } ^ { N \\times M }$ with each of its column normalized, we define the generalized mutual coherence: ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/bfdc67a99b7536a7d6d5c7e192b522c6764cd1db5b506af250cfa75d72a6d255.jpg", + "text": "$$\n\\widetilde { \\mu } ( \\mathbf { D } ) = \\operatorname* { i n f } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\bigg \\{ \\operatorname* { m a x } _ { \\substack { 1 \\leq i \\neq j \\leq M } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } \\bigg \\} .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 133, + 684, + 180 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Additionally, We define ${ \\mathcal { W } } ( \\mathbf { D } ) = \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } : \\mathbf { W } \\right.$ attains the infimum given $( \\delta ) \\}$ . A weight matrix W is “good” $f \\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )$ . ", + "bbox": [ + 173, + 185, + 823, + 218 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the above definition, problem (8) is feasible and attainable, i.e., $\\mathcal { W } ( \\mathbf { D } ) \\neq \\mathcal { O }$ , which was proven in Lemma 1 of (Chen et al., 2018). ", + "bbox": [ + 173, + 227, + 823, + 257 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1 (Recovery error upper bound). Take any $\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )$ , any $\\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )$ , and any sequence $\\begin{array} { r } { \\gamma ^ { ( k ) } \\in ( 0 , \\frac { 2 } { 2 \\tilde { \\mu } s - \\tilde { \\mu } + 1 } ) } \\end{array}$ . Using them, define the parameters $\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\}$ : ", + "bbox": [ + 173, + 260, + 823, + 292 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1e163888a17a6494fdc47f8d83e46b078edb5da3846f15609dde3eca81463b60.jpg", + "text": "$$\n\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W } , \\quad \\theta ^ { ( k ) } = \\gamma ^ { ( k ) } \\widetilde { \\mu } ( \\mathbf { D } ) \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) } \\big \\{ \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 1 } \\big \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 267, + 297, + 727, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "while the sequence $\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }$ is generated by (6) using the above parameters and $\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }$ (Note that each $\\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } )$ depends only on $\\theta ^ { ( k - 1 ) } , \\theta ^ { ( k - 2 ) } , \\dots .$ and defines $\\theta ^ { ( k ) }$ ). Let Assumption $^ { l }$ hold with any $B > 0$ and $s < ( 1 + 1 / \\tilde { \\mu } ) / 2$ . Then, we have ", + "bbox": [ + 174, + 334, + 826, + 382 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5d8382f60699f6cf9b7dead385eb0f68a6c1e718355245d422eca06628f8e26e.jpg", + "text": "$$\n\\operatorname { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } , \\quad \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\leq s B \\exp \\Big ( - \\sum _ { \\tau = 0 } ^ { k - 1 } c ^ { ( \\tau ) } \\Big ) , \\quad k = 1 , 2 , \\dots\n$$", + "text_format": "latex", + "bbox": [ + 199, + 386, + 771, + 429 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbb { S }$ is the support of $\\mathbf { x } ^ { * }$ and $c ^ { ( k ) } = - \\log \\left( ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\gamma ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | \\right)$ is a positive constant. ", + "bbox": [ + 173, + 434, + 818, + 453 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In Theorem 1, Eqn. (9) defines the properties of “good” parameters: ", + "bbox": [ + 173, + 462, + 620, + 477 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• The weights $\\mathbf { W } ^ { ( k ) }$ can be separated as the product of a scalar $\\gamma ^ { ( k ) }$ and a matrix $\\mathbf { W }$ independent of layer index $k$ , where $\\mathbf { W }$ has small coherence with $\\mathbf { D }$ . \n• $\\gamma ^ { ( k ) }$ is bounded in an interval. \n• ${ \\theta ^ { ( k ) } } / { \\gamma ^ { ( k ) } }$ is proportional to the $\\ell _ { 1 }$ error of the output of the $k ^ { \\mathrm { { t h } } }$ layer. ", + "bbox": [ + 214, + 486, + 823, + 556 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The factor $c ^ { ( k ) }$ takes the maximum at $\\gamma ^ { ( k ) } = 1$ . If $\\gamma ^ { ( k ) } \\equiv 1$ , the recovery error converges to zero in a linear rate (Chen et al., 2018): ", + "bbox": [ + 173, + 565, + 825, + 597 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4433bcc2d247d7003a2053513a5f0de2924101ec9246786ec18c2ac38a5878f3.jpg", + "text": "$$\n\\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\leq s B \\exp \\big ( - c k \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 602, + 622, + 621 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $c = - \\log ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\geq c ^ { ( k ) }$ . Although $\\gamma ^ { ( k ) } \\equiv 1$ gives the optimal theoretical upper bound if there are infinitely many layers $k = 0 , 1 , 2 , \\cdots$ , it is not the optimal choice for finite $k$ . Practically, there are finitely many layers and $\\gamma ^ { ( k ) }$ obtained by learning is bounded in an interval. ", + "bbox": [ + 173, + 627, + 826, + 672 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 RECOVERY ERROR LOWER BOUND ", + "text_level": 1, + "bbox": [ + 176, + 683, + 455, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this subsection, we introduce a lower bound of the recovery error of LISTA, which illustrates that the parameters analytically given by (9) are optimal in the convergence order (linear). ", + "bbox": [ + 171, + 702, + 823, + 731 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assumption 2. The signal $\\mathbf { x } ^ { * }$ is a random variable following the distribution $P _ { X }$ . Let $\\mathbb { S } ~ =$ suppor $\\mathbf { \\Psi } ^ { * ( \\mathbf { x } ^ { * } ) }$ . $P _ { X }$ satisfies: $2 \\leq | \\mathbb { S } | \\leq s$ ; $\\mathbb { S }$ uniformly distributes on the whole index set; nonzero part ${ \\mathbf { x } } _ { \\mathbb { S } } ^ { * }$ satisfies the uniform distribution with bound $B$ : $| x _ { i } ^ { * } | \\leq B , \\forall i \\in \\mathbb { S }$ . Moreover, the observation noise $\\varepsilon = 0$ . ", + "bbox": [ + 173, + 733, + 825, + 790 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1 tells that an ideal weight $\\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )$ satisfies $\\mathbf { I } - \\mathbf { W } ^ { T } \\mathbf { D } \\approx \\mathbf { 0 }$ . But this cannot be met exactly in the overcomplete $\\mathbf { D }$ case, i.e., $N < M$ . Definition 2 defines the set of matrices W such that $\\dot { \\mathbf { W } } ^ { T } \\mathbf { D }$ is bounded away from the identity I. In Appendix D, we discuss the feasibility of (11). ", + "bbox": [ + 173, + 799, + 825, + 844 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2. Given $\\mathbf { D } \\in \\mathbb { R } ^ { N \\times M } , s \\geq 2 , { \\bar { \\sigma } } _ { \\operatorname* { m i n } } > 0 ,$ , we define a set that $\\mathbf { W } ^ { ( k ) }$ are chosen from: ", + "bbox": [ + 171, + 847, + 795, + 864 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/183550e01fc023d1bb510b3e0d6c0286facc71be690a33aa73005a66750ff772.jpg", + "text": "$$\n\\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } ) = \\Big \\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } \\Big | \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathrm { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathrm { S } } \\Big ) \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } , \\forall \\mathrm { S ~ w i t h } 2 \\leq | \\mathrm { S } | \\leq s \\Big \\} .\n$$", + "text_format": "latex", + "bbox": [ + 179, + 867, + 787, + 895 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Based on Definition 2, we define a set that $\\boldsymbol \\Theta = \\{ \\mathbf W ^ { ( k ) } , \\boldsymbol \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }$ are chosen from: ", + "bbox": [ + 173, + 906, + 717, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 3. Let $\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }$ be generated by (6) with $\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }$ and $\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }$ . Then we define $\\tau$ as the set of parameters that guarantee there is no false positive in $\\mathbf { x } ^ { ( k ) }$ : ", + "bbox": [ + 169, + 102, + 825, + 135 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4e8753ce984d55a7c2a4c1df63f194e6773e7702b44a866715b252e731ba5d12.jpg", + "text": "$$\n{ \\mathcal { T } } = \\Big \\{ \\{ \\mathbf { W } ^ { ( k ) } \\in \\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { m i n } ) , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } , \\forall \\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) , \\forall k \\Big \\}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 140, + 781, + 167 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The conclusion (10) demonstrates that $\\tau$ is nonempty because “support $( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } ^ { \\prime }$ is satisfied as long as $\\theta ^ { ( k - 1 ) }$ large enough. Actually, $\\tau$ contains almost all “good” parameters because considerable false positives lead to large recovery errors. With $\\tau$ defined, we have: ", + "bbox": [ + 174, + 183, + 825, + 228 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 2 (Recovery error lower bound). Let the sequence $\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }$ be generated by (6) with $\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }$ aall $\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }$ . Under Assumption 2, for all parameters ave $\\{ \\mathbf { \\bar { W } } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty } \\in \\mathcal { T }$ )}∞k=0 ∈ T and $\\epsilon > 0$ ", + "bbox": [ + 173, + 233, + 825, + 279 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/510fb2fcf239f765a9cc93a25510fdd8b826ca7b2eb36522b47a02d04bc0aa05.jpg", + "text": "$$\n\\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\geq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\exp ( - \\bar { c } k ) ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 284, + 630, + 304 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "with probability at least $( 1 - \\epsilon s ^ { 3 / 2 } - \\epsilon ^ { 2 } )$ , where $\\bar { c } = s \\log ( 3 ) - \\log ( \\bar { \\sigma } _ { m i n } ) .$ ", + "bbox": [ + 174, + 310, + 661, + 329 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This theorem illustrates that, with high probability, the convergence rate of LISTA cannot be faster than a linear rate. Thus, the parameters given in (9), that leads to the linear convergence if $\\gamma ^ { k }$ is bounded within an interval near 1, are optimal with respect to the order of convergence of LISTA. ", + "bbox": [ + 173, + 338, + 826, + 382 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 ANALYTIC LISTA: LESS PARAMETERS TO LEARN ", + "text_level": 1, + "bbox": [ + 176, + 392, + 553, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Following Theorems 1 and 2, we set $\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W }$ , where $\\gamma ^ { ( k ) }$ is a scalar, and propose Tied ", + "bbox": [ + 178, + 410, + 823, + 426 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/10f52ec24b40d3fd861aecf9edbcb46182777d7bccc51a3fc8408236e2936f9f.jpg", + "text": "$$\n\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - \\gamma ^ { ( k ) } \\mathbf { W } ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } ) \\Big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 428, + 655, + 454 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\boldsymbol { \\Theta } = \\left\\{ \\{ \\gamma ^ { ( k ) } \\} _ { k } , \\{ \\theta ^ { ( k ) } \\} _ { k } , \\mathbf { W } \\right\\}$ are parameters to train. The matrix $\\mathbf { W }$ is tied over all the layers. Further, we notice that the selection of W from $\\mathcal { W } ( \\mathbf { D } )$ depends on $\\mathbf { D }$ only. Hence we propose the analytic LISTA (ALISTA) that decomposes tied-LISTA into two stages: ", + "bbox": [ + 173, + 457, + 825, + 502 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d0d53ca6ce78306068072b685183b5aa3ec02b0a7a2c79f65dd6e2ce27423f09.jpg", + "text": "$$\n\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - \\gamma ^ { ( k ) } \\tilde { \\mathbf { W } } ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } ) \\Big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 508, + 656, + 536 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\tilde { \\mathbf { W } }$ is pre-computed by solving the following problem (Stage 1)3: ", + "bbox": [ + 176, + 545, + 647, + 560 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e243cb1fff86ec715b2ec44172d032059e8dac517e180f6e60abe7efcf3244d6.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { { { \\mathbf { W } } } } \\in \\underset { { \\mathbf { W } } \\in \\mathbb { R } ^ { N \\times M } } { \\arg \\operatorname* { m i n } } \\left\\| \\mathbf { W } ^ { T } { \\mathbf { D } } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( { \\mathbf { W } } _ { : , m } \\right) ^ { T } { \\mathbf { D } } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 565, + 746, + 597 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Then with $\\tilde { \\mathbf { W } }$ fixed, $\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }$ in (15) are learned from end to end (Stage 2). (16) reformulates (8) to minimizing the Frobenius norm of ${ \\bf W } ^ { T } { \\bf D }$ (a quadratic objective), over linear constraints. This is a standard convex quadratic program, which is easier to solve than to solve (8) directly. ", + "bbox": [ + 173, + 604, + 823, + 648 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1: Summary: variants of LISTA and the number of parameters to learn. ", + "bbox": [ + 245, + 651, + 753, + 665 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/ca71257d3e26a0167489a41309483e948a3981082b4ccab6c604c199fb998341.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Vanilla LISTA (4)LISTA-CPSS (6)TiLISTA (14)ALISTA (15)
O(KM²+K+MN)O(KNM+K)O(NM+ K)O(K)
", + "bbox": [ + 233, + 674, + 759, + 707 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 CONVOLUTIONAL ANALYTIC LISTA ", + "text_level": 1, + "bbox": [ + 174, + 723, + 511, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We extend the analytic LISTA to the convolutional case in this section, starting from discussing the convolutional sparse coding (CSC). Many works studied CSC and proposed efficient algorithms for that (Bristow et al., 2013; Heide et al., 2015; Wohlberg, 2014; 2016; Papyan et al., 2017; GarciaCardona & Wohlberg, 2018; Wang et al., 2018; Liu et al., 2017; 2018). In CSC, the general linear transform is replaced by convolutions in order to learn spatially invariant features: ", + "bbox": [ + 173, + 746, + 825, + 818 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/79e6b6b033d97cf03d6d6bef2bd0bf65e72303fcd2b2a426542a828dfd6a59d3.jpg", + "text": "$$\n{ \\bf b } = \\sum _ { m = 1 } ^ { M } { \\bf d } _ { m } * { \\bf x } _ { m } ^ { * } + \\varepsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 418, + 823, + 576, + 866 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where each $\\mathbf { d } _ { m }$ is a dictionary kernel (or filter). $\\lbrace \\mathbf { d } _ { m } \\rbrace _ { m = 1 } ^ { M }$ is the dictionary of filters, $M$ denotes the number of filters. $\\{ \\mathbf { x } _ { m } ^ { * } \\} _ { m = 1 } ^ { M }$ m=1 is the set of coefficient maps that are assumed to have sparse structure, ", + "bbox": [ + 176, + 869, + 826, + 900 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "and $^ *$ is the convolution operator. Now we consider 2D convolution and take4 $\\mathbf b \\in \\mathbb R ^ { N ^ { 2 } } , \\mathbf d _ { m } \\in$ $\\mathbb { R } ^ { D ^ { 2 } } , \\mathbf { x } _ { m } \\in \\mathbb { R } ^ { ( N + D - 1 ) ^ { 2 } }$ . Equation (17) is pointwisely defined $\\mathrm { a s } ^ { 5 }$ : ", + "bbox": [ + 168, + 102, + 825, + 135 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ccc2b54d907f8f7afbb413f003e26baa1ac2ceb047c8a24292b4659a772c8a0f.jpg", + "text": "$$\n\\mathbf { b } ( i , j ) = \\sum _ { k = 0 } ^ { D - 1 } \\sum _ { l = 0 } ^ { D - 1 } \\sum _ { m = 1 } ^ { M } \\mathbf { d } _ { m } ( k , l ) \\mathbf { x } _ { m } ( i + k , j + l ) + \\varepsilon ( i , j ) , \\quad 0 \\leq i , j \\leq N - 1 .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 140, + 764, + 184 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We concatenate $\\mathbf { d } _ { m } \\mathbf { s }$ and $\\mathbf { x } _ { m } \\mathbf { s }$ : $\\mathbf { d } = [ \\mathbf { d } _ { 1 } , \\cdots , \\mathbf { d } _ { M } ] ^ { T }$ , $\\mathbf { x } = [ \\mathbf { x } _ { 1 } , \\cdots , \\mathbf { x } _ { M } ] ^ { T }$ , and rewrite (18) as: ", + "bbox": [ + 171, + 188, + 789, + 207 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/19919637a10b136bfb24e679a4b1f88a364c307d075ff9b4b0030494b3a045e8.jpg", + "text": "$$\n\\boldsymbol { \\mathbf { b } } = \\sum _ { m = 1 } ^ { M } \\mathbf { D } _ { \\mathrm { c o n v } , m } ^ { N } ( \\mathbf { d } _ { m } ) \\mathbf { x } _ { m } + \\varepsilon = \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) \\mathbf { x } + \\varepsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 328, + 210, + 666, + 255 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the matrix $\\mathbf D _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf d ) = [ \\mathbf D _ { \\mathrm { c o n v } , 1 } ^ { N } ( \\mathbf d _ { 1 } ) , \\cdots , \\mathbf D _ { \\mathrm { c o n v } , M } ^ { N } ( \\mathbf d _ { M } ) ] \\in \\mathbb R ^ { N ^ { 2 } \\times ( N + D - 1 ) ^ { 2 } M }$ , depending on the signal size $N$ and the dictionary $\\mathbf { d }$ , is defined in detail in (48) in Appendix C.2. ", + "bbox": [ + 173, + 260, + 825, + 292 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "From (17), the convolutional LISTA becomes a natural extension of the fully-connected LISTA (6): ", + "bbox": [ + 169, + 297, + 821, + 314 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6a41f5b8ec4e924d64e16be5089ef290d15a52a1ee59134fa9f323ec3227516f.jpg", + "text": "$$\n{ \\bf x } _ { m } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\left( { \\bf x } _ { m } ^ { ( k ) } - \\left( { \\bf w } _ { m } ^ { ( k ) } \\right) ^ { \\prime } \\ast \\Big ( \\sum _ { \\bar { m } = 1 } ^ { M } { \\bf d } _ { \\bar { m } } \\ast { \\bf x } _ { \\bar { m } } ^ { ( k ) } - { \\bf b } \\Big ) \\right) , \\quad m = 1 , 2 , \\cdots , M ,\n$$", + "text_format": "latex", + "bbox": [ + 232, + 319, + 764, + 363 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where {w(k)m }Mm=1 share the same sizes with $\\lbrace \\mathbf { d } _ { m } \\rbrace _ { m = 1 } ^ { M }$ and $( \\cdot ) ^ { \\prime }$ means a 180 rotation of the filter (Chalasani et al., 2013). We concatenate the filters together: $\\dot { \\mathbf { w } } ^ { ( k ) } = [ \\mathbf { w } _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , \\mathbf { w } _ { M } ^ { ( k ) } ] ^ { T } \\in \\mathbb { R } ^ { D ^ { 2 } M }$ . Parameters to train are $\\boldsymbol { \\Theta } = \\{ \\mathbf { w } ^ { ( k ) } , \\boldsymbol { \\theta } ^ { ( k ) } \\} _ { k }$ . ", + "bbox": [ + 173, + 369, + 825, + 421 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Let $\\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { ( k ) } )$ be the matrix induced by dictionary $\\mathbf { w } ^ { ( k ) }$ with the same dimensionality as $\\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )$ nv. Since convolution can be written as a matrix form (19), (20) is equivalent to ", + "bbox": [ + 171, + 426, + 823, + 458 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/be0b87973f2b484a025c5e178ff1e406d836e65fe850fb9fb80abe8294b281f8.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - ( \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { ( k ) } ) ) ^ { T } \\big ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) \\mathbf { x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 462, + 710, + 489 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Then by just substituting can be applied to the conv $\\mathbf { D } , \\mathbf { W } ^ { ( k ) }$ with LIST $\\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) , \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { ( k ) } )$ respectively, Theorems 1 and 2 ", + "bbox": [ + 174, + 494, + 821, + 525 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 1. Let $\\mathbf { D } = \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )$ and $\\mathbf { W } ^ { ( k ) } = \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\big ( \\mathbf { w } ^ { ( k ) } \\big )$ . With Assumption 1 and other settings the same with those in Theorem $^ { l }$ , $( I O )$ holds. With Assumption 2 and other settings the same with those in Theorem 2, (13) holds. ", + "bbox": [ + 173, + 529, + 825, + 573 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Similar to the fully connected case (15), based on the results in Proposition 1, we should set $\\mathbf { w } _ { m } ^ { ( k ) } =$ $\\gamma _ { m } ^ { ( k ) } \\tilde { \\mathbf { w } } _ { m }$ , $m = 1 , 2 , \\cdots , M$ , where $\\tilde { \\mathbf { w } } = [ \\tilde { \\mathbf { w } } _ { 1 } , \\cdots , \\tilde { \\mathbf { w } } _ { M } ] ^ { T }$ is chosen from ", + "bbox": [ + 171, + 585, + 821, + 619 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/69facdcab8d97cc0763450ae507fbf8e8a4df1363bc5de378b427a244649f251.jpg", + "text": "$$\n\\tilde { \\mathbf { w } } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\underset { \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , 1 \\leq m \\leq M } { \\arg \\operatorname* { m i n } } \\left\\| \\left( \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) \\right\\| _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 285, + 625, + 712, + 671 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, (22) is not as efficient to solve as (16). To see that, matrices $\\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )$ and $\\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } )$ are both of size $N ^ { 2 } \\times ( N + D - 1 ) ^ { 2 } M$ , the coherence matrix $\\left( \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )$ is thus of size $( N + D - 1 ) ^ { 2 } M \\times ( N + D - 1 ) ^ { 2 } M .$ In the typical application setting of CSC, b is usually an image rather than a small patch. For example, if the image size is $1 0 0 \\times 1 0 0$ , dictionary size is $7 \\times 7 \\times 6 4$ , $N = 1 0 0 , D = 7 , M = 6 4$ , then $( \\dot { N } + D - 1 ) ^ { 2 } \\bar { M } \\times ( N + D - 1 ) ^ { 2 } M \\approx 5 \\times \\dot { 1 } 0 ^ { 1 1 }$ . ", + "bbox": [ + 173, + 678, + 826, + 756 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 CALCULATING CONVOLUTIONAL WEIGHTS ANALYTICALLY AND EFFICIENTLY", + "text_level": 1, + "bbox": [ + 171, + 765, + 751, + 780 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To overcome the computational challenge of solving (22), we exploit the following circular convolution as an efficient approximation: ", + "bbox": [ + 173, + 785, + 825, + 814 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/861f01235b24aed94c7892c63dcbfce62a64bee8193a88a53ee79f3b8d455234.jpg", + "text": "$$\n\\mathbf { b } ( i , j ) = \\sum _ { k = 0 } ^ { D - 1 } \\sum _ { l = 0 } ^ { D - 1 } \\sum _ { m = 1 } ^ { M } \\mathbf { d } _ { m } ( k , l ) \\mathbf { x } _ { m } \\big ( ( i + k ) _ { \\mathrm { m o d } N } , ( j + l ) _ { \\mathrm { m o d } N } \\big ) + \\varepsilon ( i , j ) , \\quad 0 \\leq i , j \\leq N - 1 ,\n$$", + "text_format": "latex", + "bbox": [ + 178, + 818, + 789, + 863 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $ { \\mathbf { b } } \\in \\mathbb { R } ^ { N ^ { 2 } } , \\mathbf { d } _ { m } \\in \\mathbb { R } ^ { D ^ { 2 } } , { \\mathbf { x } } _ { m } \\in \\mathbb { R } ^ { N ^ { 2 } }$ . Similar to (18), we rewrite (23) in a compact way: ", + "bbox": [ + 171, + 101, + 776, + 119 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/c53587267e7b1b7d25ead5cbae04b5a46e2556e095687c3e7af8bc3f806ec10f.jpg", + "text": "$$\n\\mathbf { b } = \\sum _ { m = 1 } ^ { M } \\mathbf { D } _ { \\mathrm { c i r } , m } ^ { N } ( \\mathbf { d } _ { m } ) \\mathbf { x } _ { m } + \\varepsilon = \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\mathbf { x } + \\varepsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 121, + 655, + 164 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where ${ \\bf D } _ { \\mathrm { c i r } } ^ { N } ( { \\bf d } ) : \\mathbb { R } ^ { N ^ { 2 } M } \\mathbb { R } ^ { N ^ { 2 } }$ is a matrix depending on the signal size $N$ and the dictionary $\\mathbf { d }$ Then the coherence minimization with the circular convolution is given by ", + "bbox": [ + 171, + 167, + 821, + 198 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/f66d39ac9ea7aaf918852cb50a3b65ebad36271f3422b92be7cecb37947ed31a.jpg", + "text": "$$\n\\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\underset { \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , \\ 1 \\leq m \\leq M } { \\arg \\operatorname* { m i n } } \\left\\| \\left( \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\right\\| _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 199, + 676, + 247 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The following theorem motivates us to use the solution to (24) to approximate that of (22) ", + "bbox": [ + 171, + 256, + 758, + 271 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 3. The solution sets of (22) and (24) satisfy the following properties: ", + "bbox": [ + 176, + 273, + 691, + 289 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$I . \\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } , \\forall N \\ge 2 D - 1 .$ ", + "bbox": [ + 214, + 296, + 447, + 316 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2. If at least one of the matrices $\\{ \\mathbf { D } _ { \\mathrm { c i r } , 1 } ^ { 2 D - 1 } , \\cdot \\cdot \\cdot , \\mathbf { D } _ { \\mathrm { c i r } , M } ^ { 2 D - 1 } \\}$ is non-singular, $\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ involves only a unique element. Furthermore, ", + "bbox": [ + 209, + 323, + 823, + 356 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/a11b533d34eab49701b03d1d25838ecfe65d4df8c23d7dab8167f49099e409c3.jpg", + "text": "$$\n\\operatorname * { l i m } _ { N \\infty } \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 442, + 357, + 612, + 383 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The solution set $\\mathcal { W } _ { \\mathrm { c i r } } ^ { N }$ is not related with the image size $N$ as long as $N \\geq 2 D - 1$ , thus one can deal with a much smaller-size problem (let $N = 2 D - 1 \\mathrm { \\ Y }$ ). Further, (25) indicates that as $N$ gets (much) larger than $D$ , the boundary condition becomes less important. Thus, one can use $\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 \\breve { D } - 1 }$ to approximate $\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }$ . In Appendix E.2, we introduce the algorithm details of solving (24). ", + "bbox": [ + 173, + 393, + 825, + 457 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Based on Proposition 1 and Theorem 3, we obtain the convolutional ALISTA: ", + "bbox": [ + 174, + 462, + 692, + 477 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/948bc3d2d6deb966f18893c596ef4da05e9991b4b0c679e17e070bb04f48b6dd.jpg", + "text": "$$\n\\mathbf { x } _ { m } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\left( \\mathbf { x } _ { m } ^ { ( k ) } - \\gamma _ { m } ^ { ( k ) } \\left( \\tilde { \\mathbf { w } } _ { m } \\right) ^ { \\prime } \\ast \\bigg ( \\sum _ { \\bar { m } = 1 } ^ { M } \\mathbf { d } _ { \\bar { m } } \\ast \\mathbf { x } _ { \\bar { m } } ^ { ( k ) } - \\mathbf { b } \\bigg ) \\right) , \\quad m = 1 , 2 , \\cdots , M ,\n$$", + "text_format": "latex", + "bbox": [ + 205, + 479, + 763, + 522 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\tilde { \\mathbf { w } } = [ \\tilde { \\mathbf { w } } _ { 1 } , \\cdots , \\tilde { \\mathbf { w } } _ { M } ] ^ { T } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ and $\\Theta = \\{ \\{ \\gamma _ { m } ^ { ( k ) } \\} _ { m , k } , \\{ \\theta ^ { ( k ) } \\} _ { k } \\}$ are the parameters to train. (26) is a simplified form, compared to the empirically unfolded CSC model recently proposed in (Sreter & Giryes, 2018) ", + "bbox": [ + 174, + 525, + 825, + 570 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 ROBUST ALISTA TO MODEL PERTURBATION ", + "text_level": 1, + "bbox": [ + 173, + 583, + 584, + 599 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Many applications, such as often found in surveillance video scenarios (Zhao et al., 2011; Han et al., 2013), can be formulated as sparse coding models whose dictionaries are subject to small dynamic perturbations (e.g, slowly varied over time). Specifically, the linear system model (1) may have uncertain $\\mathbf { D }$ : $\\tilde { \\textbf { D } } = \\textbf { D } + \\varepsilon _ { D }$ , where $\\varepsilon _ { D }$ is some small stochastic perturbation. Classical LISTA entangles the learning of all its parameters, and the trained model is tied to one static D. The important contribution of ALISTA is to decompose fitting W w.r.t. D, from adapting other parameters $\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }$ to training data. ", + "bbox": [ + 173, + 606, + 825, + 708 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we develop a robust variant of ALISTA that is a fast regressor not only for a given $\\mathbf { D }$ , but all its randomly perturbations $\\tilde { \\bf D }$ to some extent. Up to our best knowledge, this approach is new. Robust ALISTA can be sketched as the following empirical routine (at each iteration): ", + "bbox": [ + 176, + 713, + 825, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• Sample a perturbed dictionary $\\tilde { \\bf D }$ . Sample $\\mathbf { x }$ and $\\varepsilon$ to generate b w.r.t. $\\tilde { \\bf D }$ . • Apply Stage 1 of ALISTA w.r.t. $\\tilde { \\bf D }$ and obtain $\\tilde { \\mathbf { W } }$ ; however, instead of an iterative minimization algorithm, we use a neural network that unfolds that algorithm to produce $\\tilde { \\mathbf { W } }$ . • Apply Stage 2 of ALISTA w.r.t. $\\tilde { \\mathbf { W } }$ , D, $\\mathbf { x }$ , and b to obtain $\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }$ . ", + "bbox": [ + 215, + 762, + 823, + 829 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Robust ALISTA above, $\\tilde { \\bf D }$ becomes a part of the data for training the neural network that generates $\\tilde { \\mathbf { W } }$ . This neural network is faster to apply than the minimization algorithm. One might attempt to use $\\tilde { \\bf D }$ in the last step, rather than $\\mathbf { D }$ , but $\\tilde { \\bf D }$ makes training less stable, potentially because of larger weight variations between training iterations due to the random perturbations in $\\tilde { \\bf D }$ . We observe that using $\\mathbf { D }$ stabilizes training better and empirically achieves a good prediction. More details of training Robust ALISTA are given in Appendix G. ", + "bbox": [ + 173, + 834, + 826, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 NUMERICAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 392, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we conduct extensive experiments on both synthesized and real data to demonstrate:6 • We experimentally validate Theorems 1 and 2, and show that ALISTA is as effective as classical LISTA (Gregor & LeCun, 2010; Chen et al., 2018)but is much easier to train. • Similar conclusions can be drawn for convolutional analytic LISTA. • The robust analytic LISTA further shows remarkable robustness in sparse code prediction, given that $\\mathbf { D }$ is randomly perturbed within some extent. ", + "bbox": [ + 171, + 122, + 821, + 137 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 209, + 143, + 825, + 223 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Notation For brevity, we let LISTA denote the vanilla LISTA model (4) in (Gregor & LeCun, 2010); LISTA-CPSS refers to the lately-proposed fast LISTA variant (Chen et al., 2018) with weight coupling and support selection; TiLISTA is the tied LISTA (14); and ALISTA is our proposed Analytic LISTA (15). If the model is for convolutional case, then we add “Conv” as the prefix for model name, such as “Conv ALISTA” that represents the convolutional analytic LISTA. ", + "bbox": [ + 173, + 234, + 825, + 304 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.1 VALIDATION OF THEOREMS 1 AND 2 (ANALYTIC LISTA) ", + "text_level": 1, + "bbox": [ + 174, + 323, + 612, + 338 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We follow the same $N = 2 5 0$ , $M = 5 0 0$ setting as (Chen et al., 2018) by default. We sample the entries of $\\mathbf { D }$ i.i.d. from the standard Gaussian distribution, $\\mathbf { D } _ { i j } \\sim \\mathcal { N } ( 0 , 1 / N )$ and then normalize its columns to have the unit $\\ell _ { 2 }$ norm. We fix a dictionary $\\mathbf { D }$ in this section. To generate sparse vectors $\\mathbf { x } ^ { * }$ , we decide each of its entry to be non-zero following the Bernoulli distribution with $p _ { b } = 0 . 1$ . The values of the non-zero entries are sampled from the standard Gaussian distribution. A test set of 1000 samples generated in the above manner is fixed for all tests in our simulations. The analytic weight $W$ that we use in the ALISTA is obtained by solving (16). ", + "bbox": [ + 173, + 343, + 825, + 443 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "All networks used (vanilla LISTA, LISTA-CPSS, TiLISTA and ALISTA) have the same number of 16 layers. We also include two classical iterative solvers: ISTA and FISTA. We train the networks with four different levels of noises: SNR (Signal-to-Noise Rati $\\mathbf { \\delta } _ { 0 } ) = 2 0 , 3 0 , 4 0$ , and $\\infty$ . While our theory mainly discussed the noise-free case $\\mathrm { S N R } = \\infty$ ), we hope to empirically study the algorithm performance under noise too. As shown in Figure 1, the $\\mathbf { X }$ -axes denotes the indices of layers for the networks, or the number of iterations for the iterative algorithms. The y-axes represent the NMSE (Normalized Mean Squared Error) in the decibel (dB) unit: ", + "bbox": [ + 173, + 448, + 825, + 546 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/b0fa4773ffb90452a0959a32a8d842391f29c771a8da86143d5605a7c9f8831e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { N M S E } _ { \\mathrm { d B } } ( \\hat { \\mathbf { x } } , \\mathbf { x } ^ { * } ) = 1 0 \\log _ { 1 0 } \\left( \\mathbb { E } \\| \\hat { \\mathbf { x } } - \\mathbf { x } ^ { * } \\| ^ { 2 } / \\mathbb { E } \\| \\mathbf { x } ^ { * } \\| ^ { 2 } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 554, + 676, + 573 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\mathbf { x } ^ { * }$ is the ground truth and $\\hat { \\bf x }$ is the estimated one. ", + "bbox": [ + 174, + 580, + 540, + 595 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/971fc90a1cbe56d421357c61684b1c5712be35575d937c498da5ae4642154c58.jpg", + "image_caption": [ + "Figure 1: Justification of Theorems 1 and 2: comparision among LISTA variants. " + ], + "image_footnote": [], + "bbox": [ + 186, + 613, + 831, + 881 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Figure 1 (a) noise-less case, all four learned models apparently converge much faster than two iterative solvers (ISTA/FISTA curves almost overlap in this y-scale, at the small number of iterations). Among the four networks, classical-LISTA is inferior to the other three by an obvious margin. LISTA-CPSS, TiLISTA and ALISTA perform comparably: ALISTA is observed to eventually achieve the lowest NMSE. Figure 1(a) also supports Theorem 2, that all networks have at most linear convergence, regardless of how freely their parameters can be end-to-end learned. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Figure 1 (b) - (d) further show that even in the presence of noise, ALISTA can empirically perform comparably with LISTA-CPSS and TiLISTA, and stay clearly better than LISTA and ISTA/FISTA. Always note that ALISTA the smallest amount of parameters to learn from the end-to-end training (Stage 2). The above results endorse that: i) the optimal LISTA layer-wise weights could be structured as $\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W }$ ; and ii) W could be analytically solved rather than learned from data, without incurring performance loss. We also observe the significant reduction of training time for ALISTA: while LISTA-CPSS of the same depth took ${ \\sim } 1 . 5$ hours to train, ALISTA was trained within only 6 minutes (0.1 hours) to achieve comparable performance, on the same hardware (one 1080 Ti on server). ", + "bbox": [ + 173, + 194, + 825, + 320 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/b4ad8ecd57838661b04ad8c14453ba10544c6080dccc8fdb5991b9989b099d41.jpg", + "image_caption": [ + "Figure 2: Justification of Theorem 1 (noiseless case): parameters obtained by training satisfy (9). " + ], + "image_footnote": [], + "bbox": [ + 179, + 329, + 777, + 452 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We further supply Figures 2 and 3 to justify Theorem 1 from different perspectives. Figure 2 plots the learned parameters $\\{ \\bar { \\gamma ^ { ( k ) } } , \\bar { \\theta ^ { ( k ) } } \\}$ in ALISTA (Stage 2), showing that they satisfy the properties proposed in Theorem 1: $\\gamma ^ { ( k ) }$ bounded; ${ \\theta } ^ { \\bar { ( k ) } }$ and $\\gamma ^ { ( k ) }$ is proportional to $\\begin{array} { r } { \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 1 } ~ ( ^ { * } \\mathrm { s u p } _ { \\mathbf { x } ^ { * } } , } \\end{array}$ is taken over the test set). Figure 3 reports the average magnitude7 of the false positives and the true positives in $\\mathbf { \\Delta x } ^ { k } ( \\mathbf { x } ^ { * } )$ of ALISTA: the “true positives” curve draws the values of $\\mathbb { E } \\{ \\| \\mathbf { x } _ { \\mathbb { S } } ^ { k } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } / \\| \\mathbf { x } ^ { \\hat { k } } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } \\}$ w.r.t. $k$ (the expectation is taken over the test set), while “false positives” for ${ \\mathbb { E } } \\{ \\| \\mathbf { x } _ { \\mathbb { S } ^ { c } } ^ { k } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } / \\| \\mathbf { x } ^ { k } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } \\}$ . False positives take up small proportion over the positives, which supports the Theorem 1 conclusion that support $( \\mathbf { x } ^ { k } ( \\mathbf { x } ^ { \\ast } ) ) \\subset \\mathbb { S }$ . ", + "bbox": [ + 174, + 483, + 549, + 638 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/113a9556b7deff18282564eac7e07763703a959bbb96cb26d5149e31de0c5122.jpg", + "image_caption": [ + "Figure 3: Justification of Theorem 1 (noiseless case): Proportion of false positives vs true positives in $\\mathbf { x } ^ { k } ( \\mathbf { x } ^ { * } )$ . " + ], + "image_footnote": [], + "bbox": [ + 573, + 488, + 803, + 573 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 638, + 823, + 670 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.2 VALIDATION OF THEOREM 3 (CONVOLUTIONAL ANALYTIC LISTA) ", + "text_level": 1, + "bbox": [ + 176, + 683, + 684, + 696 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For convolutional cases, we use real image data to verify Theorem 3. We train a convolutional dictionary $\\mathbf { d }$ with $D = 7 , M = 6 4$ on the BSD500 training set (400 images), using the Algorithm 1 in (Liu et al., 2018). We then use it for problems (22) and (24) and solve them with different $N s$ . ", + "bbox": [ + 174, + 704, + 825, + 746 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Table 2, we take $\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { N }$ , $\\mathbf { w } ^ { * } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 5 0 }$ (consider 50 as large enough) For this example, $\\mathcal { W } _ { \\mathrm { c i r } } ^ { N }$ has only one element. Table 2 shows that $\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } = \\mathbf { w } ^ { * }$ for $N \\geq 1 3$ , i.e., the solution of the problem (24) is independent of if $N \\geq 2 D - 1$ , justifying the first conclusion in Theorem 3. In Table shows valida 3, we take $\\mathbf { w } _ { \\mathrm { c o n v } } ^ { N } \\to \\mathbf { w } ^ { * }$ $\\mathbf { \\bar { w } } _ { \\mathrm { c o n v } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }$ conv cir co, i.e., the solution of the problem (22) d conclusion of Theorem 3. Visualized and $\\mathbf { w ^ { * } } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }$ , where $\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }$ also has only one element. Table 3 o that of (24) as is displayed in increases,pendix F. $\\mathbf { w } ^ { * } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }$ ", + "bbox": [ + 173, + 752, + 825, + 843 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Besides validating Theorem 3, we also present a real image denoising experiment to verify the effectiveness of Conv ALISTA. The detailed settings and results are presented in Appendix $_ \\mathrm { H }$ . ", + "bbox": [ + 178, + 103, + 823, + 132 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/3f8cf78e76af573f2fa2e03635213624c37cd3b53ba2d137dd16960276687fb5.jpg", + "table_caption": [ + "Table 2: Validation of Conclusion 1 in Theorem 3. $D = 7$ . $\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { N }$ and $\\mathbf { w } ^ { * } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 5 0 }$ " + ], + "table_footnote": [], + "table_body": "
lwir- w*||2/1lw*||2
N=10N= 11N = 12N=13N=15N = 20
2.0×10-29.3×10-33.9 ×10-31.4 ×10-128.8×10-135.9×10-13
", + "bbox": [ + 204, + 157, + 795, + 205 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Table 3: Validation of Conclusion 2 in Theorem 3. $D = 7$ . $\\mathbf { w } _ { \\mathrm { c o n v } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }$ and $\\mathbf { w } ^ { * } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }$ ", + "bbox": [ + 196, + 209, + 795, + 227 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/2a35d042ab300ac3570d84bf2062148242d5d7d2be0c28c7648188d5de3a3955.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
|/wconv - w*||2/lw*|2
N=3N=5N= 10N=15N = 20
0.18920.08500.02840.01610.0113
", + "bbox": [ + 320, + 229, + 673, + 276 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/d4471e4ca71aa2a0f014d99d6cc87823b8becfcf52e21856f57b86aa7e1731b3.jpg", + "image_caption": [ + "Figure 4: Validation of Robust ALISTA. " + ], + "image_footnote": [], + "bbox": [ + 209, + 286, + 759, + 421 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5.3 VALIDATION OF ROBUST ALISTA ", + "text_level": 1, + "bbox": [ + 176, + 445, + 450, + 460 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We empirically verify the effectiveness of Robust ALISTA, by sampling the dictionary perturbation $\\varepsilon _ { D }$ entry-wise i.i.d. from another Gaussian distribution $\\mathcal { N } ( \\bar { 0 _ { } } , \\sigma _ { m a x } ^ { 2 } )$ . We choose $\\sigma _ { m a x } = 0 . 0 2$ and 0.03. Other simulation settings are by default the same as in Section 5.1. We then build the Robust ALISTA model, following the strategy in Section 4 and using a 4-layer encoder for approximating its second step (see Appendix G for details). Correspondingly, we compare Robust ALISTA with TiLISTA and ALISTA with specific data augmentation: we straightforwardly augment their training sets, by including all data generated with randomly perturbed $\\breve { \\tilde { \\mathbf { D } } } \\mathbf { s }$ when training Robust ALISTA. We also include the data-free FISTA algorithm into the comparison. ", + "bbox": [ + 173, + 465, + 825, + 579 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Figure 4 plots the results when the trained models are applied on the testing data, generated with the same dictionary and perturbed by $\\mathcal { N } ( 0 , \\sigma _ { t } )$ . We vary $\\sigma _ { t }$ from zero to slightly above $\\sigma _ { m a x }$ . Not surprisingly, FISTA is unaffected, while the other three data-driven models all slight degrade as $\\sigma _ { t }$ increases. Compared to the augmented TiLISTA and ALISTA whose performance are both inferior to FISTA, the proposed Robust ALISTA appears to be much more favorable in improving robustness to model perturbations. In both $\\sigma _ { m a x }$ cases, it consistently achieves much lower NMSE than FISTA, even when $\\sigma _ { t }$ has slightly surpassed $\\sigma _ { m a x }$ . Although the NMSE of ALISTA may decrease faster if $\\sigma _ { t }$ continues growing larger, such decrease could be alleviated by improving $\\sigma _ { m a x }$ in training, e.g., by comparing $\\sigma _ { m a x } = 0 . 0 2$ and 0.03. Robust ALISTA demonstrates remarkable robustness and maintains the best NMSE performance, within at least the $[ 0 , \\sigma _ { m a x } ]$ range. ", + "bbox": [ + 173, + 585, + 825, + 724 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSIONS AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 733, + 503, + 750 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Based on the recent theoretical advances of LISTA, we have made further steps to reduce the training complexity and improve the robustness of LISTA. Specifically, we no longer train any matrix for LISTA but directly use the solution to an analytic minimization problem to solve for its layer-wise weights. Therefore, only two scalar sequences (stepsizes and thresholds) still need to be trained. Excluding the matrix from training is backed by our theoretical upper and lower bounds. The resulting method, Analytic LISTA or ALISTA, is not only faster to train but performs as well as the state-of-the-art variant of LISTA by (Chen et al., 2018). 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", + "bbox": [ + 174, + 140, + 821, + 184 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Joey Tianyi Zhou, Kai Di, Jiawei Du, Xi Peng, Hao Yang, Sinno Jialin Pan, Ivor W Tsang, Yong Liu, Zheng Qin, and Rick Siow Mong Goh. SC2Net: Sparse LSTMs for sparse coding. In AAAI Conference on Artificial Intelligence, 2018. ", + "bbox": [ + 176, + 193, + 823, + 234 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 176, + 101, + 398, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this proof, we use the notion $\\mathbf { x } ^ { ( k ) }$ to replace $\\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } )$ for simplicity. We fix $\\mathbf { D }$ in the proof, $\\tilde { \\mu } ( \\mathbf { D } )$ can be simply written as $\\tilde { \\mu }$ . ", + "bbox": [ + 173, + 131, + 823, + 162 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Before proving Theorem 1, we present and prove a lemma. ", + "bbox": [ + 173, + 167, + 562, + 184 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma 1. With all the settings the same with those in Theorem $^ { l }$ , we have ", + "bbox": [ + 169, + 186, + 668, + 202 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/19a107bb58d0b3b9dd480ba6f4e99e06604e287aa0d68d63bd819bbcdf1bd597.jpg", + "text": "$$\n\\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ) \\subset \\mathbb { S } , \\quad \\forall k .\n$$", + "text_format": "latex", + "bbox": [ + 410, + 208, + 588, + 228 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In another word, there are no false positives in $\\mathbf { x } ^ { ( k ) }$ : $x _ { i } ^ { ( k ) } = 0 , \\forall i \\notin \\mathbb { S } , \\forall k$ ", + "bbox": [ + 174, + 234, + 661, + 253 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. Take arbitrary $\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )$ . We prove Lemma 1 by induction. As $k = 0$ , (27) is satisfied since $\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }$ . Fixing $k$ , and assuming support $( \\mathbf { x } ^ { ( k ) } ) \\subset \\mathbb { S }$ , we have ", + "bbox": [ + 174, + 266, + 823, + 299 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ba67cda30d2837a8a8e87de6941218c4d28e1e10fd9211440af8a7870c6e7ad2.jpg", + "text": "$$\n\\begin{array} { r l } & { x _ { i } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( x _ { i } ^ { ( k ) } - \\gamma ^ { ( k ) } ( \\mathbf { W } _ { : , i } ) ^ { T } \\big ( \\mathbf { D x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) } \\\\ & { \\quad \\quad \\quad = \\eta _ { \\theta ^ { ( k ) } } \\Big ( - \\gamma ^ { ( k ) } \\displaystyle \\sum _ { j \\in \\mathbb { S } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } \\big ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \\big ) \\Big ) , \\quad \\forall i \\notin \\mathbb { S } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 304, + 714, + 369 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By (9), the thresholds are taken as $\\theta ^ { ( k ) } = \\tilde { \\mu } \\gamma ^ { ( k ) } \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\}$ . Also, since $\\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )$ , we have $| ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } | \\leq \\tilde { \\mu }$ for all $j \\neq i$ . Thus, for all $i \\not \\in { \\mathbb S }$ , ", + "bbox": [ + 171, + 375, + 825, + 407 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9f0ca0e63703dd638997fbb9d01c000fe4fc4d10c9b05ef1ee6520d7099f4298.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\theta ^ { ( k ) } \\geq \\widetilde { \\mu } \\gamma ^ { ( k ) } \\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } = \\displaystyle \\sum _ { j \\in \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ) } \\widetilde { \\mu } \\gamma ^ { ( k ) } \\big | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \\big | = \\displaystyle \\sum _ { j \\in \\mathbb { S } } \\widetilde { \\mu } \\gamma ^ { ( k ) } \\big | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \\big | } } \\\\ { { \\displaystyle \\geq \\Big | - \\gamma ^ { ( k ) } \\sum _ { j \\in \\mathbb { S } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) \\Big | , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 218, + 411, + 779, + 488 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "which implies x(ki +1) = 0, ∀i /∈ S by the definition of ηθ(k) , i.e., ", + "bbox": [ + 173, + 494, + 594, + 513 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ae1337fb597aa7696ff4c0ae1323a52c5a47e17c915bad8870b63178ec7596b2.jpg", + "text": "$$\n\\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k + 1 ) } ) \\subset \\mathbb { S }\n$$", + "text_format": "latex", + "bbox": [ + 424, + 520, + 573, + 540 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By induction, (27) is proved. ", + "bbox": [ + 174, + 545, + 362, + 560 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "With Lemma 1, we are able to prove Theorem 1 now. ", + "bbox": [ + 174, + 575, + 524, + 590 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof of Theorem $^ { l }$ . Take arbitrary $\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )$ . For all $i \\in \\mathbb { S }$ , by (27), we obtain ", + "bbox": [ + 176, + 604, + 720, + 622 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/4cff7c479875a57c24cd47501d6448bdb7e4785369071133936df518d3a00baf.jpg", + "text": "$$\n\\begin{array} { r l } & { x _ { i } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\left( x _ { i } ^ { ( k ) } - \\gamma ^ { ( k ) } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } ) \\right) } \\\\ & { \\quad \\quad \\quad \\in x _ { i } ^ { ( k ) } - \\gamma ^ { ( k ) } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } ) - \\theta ^ { ( k ) } \\partial \\ell _ { 1 } ( x _ { i } ^ { ( k + 1 ) } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 627, + 717, + 678 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\partial \\ell _ { 1 } ( x )$ is the sub-gradient of $| x | , x \\in \\mathbb { R }$ : ", + "bbox": [ + 174, + 683, + 483, + 699 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9aa338f0f38bc898d25c979013f5af2cce53777c31791713a286983242e4e040.jpg", + "text": "$$\n\\partial { \\boldsymbol { \\ell } } _ { 1 } ( x ) = { \\left\\{ \\begin{array} { l l } { \\{ \\operatorname { s i g n } ( x ) \\} } & { { \\mathrm { i f ~ } } x \\neq 0 , } \\\\ { [ - 1 , 1 ] } & { { \\mathrm { i f ~ } } x = 0 . } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 385, + 705, + 611, + 741 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The choice of $\\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )$ gives $( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , i } = 1$ . Thus, ", + "bbox": [ + 173, + 747, + 555, + 766 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b0e019ad95b75de9faab7bcef7891705e00259182818d2c5e506887b486a5e2c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad x _ { i } ^ { ( k ) } - \\gamma ^ { ( k ) } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ( { \\mathbf x } _ { \\mathbb { S } } ^ { ( k ) } - { \\mathbf x } _ { \\mathbb { S } } ^ { * } ) } \\\\ & { = x _ { i } ^ { ( k ) } - \\gamma ^ { ( k ) } \\underset { j \\in \\mathbb { S } , j \\neq i } { \\sum } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) - \\gamma ^ { ( k ) } ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) } \\\\ & { = x _ { i } ^ { * } - \\gamma ^ { ( k ) } \\underset { j \\in \\mathbb { S } , j \\neq i } { \\sum } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) + ( 1 - \\gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 771, + 730, + 867 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Then the following inclusion formula holds for all $i \\in \\mathbb S$ , ", + "bbox": [ + 173, + 871, + 547, + 887 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/29a256724b5c73fbb96859fbf4f89c3c0616d291f8d88156fa969348c9710b55.jpg", + "text": "$$\n\\mathbf { \\Phi } _ { i } ^ { ( k + 1 ) } - \\mathbf { \\Phi } _ { x _ { i } ^ { * } } ^ { * } \\in - \\gamma ^ { ( k ) } \\sum _ { \\substack { j \\in \\mathbb { S } , j \\neq i } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) - \\theta ^ { ( k ) } \\partial \\ell _ { 1 } ( x _ { i } ^ { ( k + 1 ) } ) + ( 1 - \\gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 892, + 825, + 929 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By the definition of $\\partial \\ell _ { 1 }$ , every element in $\\partial \\ell _ { 1 } ( x ) , \\forall x \\in \\mathbb { R }$ has a magnitude less than or equal to 1. Thus, for all $i \\in \\mathbb S$ , ", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9d43a100754ec2c8718aa9e66be81b0cd09d44be92bd3db0f78fe7be26b31b72.jpg", + "text": "$$\n\\begin{array} { r l } & { | x _ { i } ^ { ( k + 1 ) } - x _ { i } ^ { * } | \\le \\displaystyle \\sum _ { j \\in \\mathbb S , j \\ne i } \\gamma ^ { ( k ) } \\Big | ( \\mathbf W _ { : , i } ) ^ { T } \\mathbf D _ { : , j } \\Big | | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } | + \\theta ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } | } \\\\ & { \\qquad \\le \\widetilde { \\mu } \\gamma ^ { ( k ) } \\displaystyle \\sum _ { j \\in \\mathbb S , j \\ne i } | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } | + \\theta ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } | . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 133, + 782, + 209 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Equation (27) implies $\\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } = \\| \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } \\| _ { 1 }$ for all $k$ . Then ", + "bbox": [ + 173, + 213, + 622, + 233 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d094400df99d7946ab901830153a0a9d64e91fd8cc4e305777691852d0a61bab.jpg", + "text": "$$\n\\begin{array} { r l } { { \\| { \\mathbf { x } } ^ { ( k + 1 ) } - { \\mathbf { x } } ^ { * } \\| _ { 1 } = \\sum _ { i \\in \\mathbb { S } } \\big | x _ { i } ^ { ( k + 1 ) } - x _ { i } ^ { * } \\big | } } \\\\ & { \\leq \\sum _ { i \\in \\mathbb { S } } \\Big ( \\tilde { \\mu } \\gamma ^ { ( k ) } \\sum _ { j \\in \\mathbb { S } , j \\neq i } \\big | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \\big | + \\theta ^ { ( k ) } + \\big | 1 - \\gamma ^ { ( k ) } \\big | \\big | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } \\big | \\Big ) } \\\\ & { = \\tilde { \\mu } \\gamma ^ { ( k ) } ( \\big | \\mathbb { S } \\big | - 1 ) \\sum _ { i \\in \\mathbb { S } } \\big | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } \\big | + \\theta ^ { ( k ) } \\big | \\mathbb { S } \\big | + \\big | 1 - \\gamma ^ { ( k ) } \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | _ { 1 } } \\\\ & { = \\tilde { \\mu } \\gamma ^ { ( k ) } ( \\big | \\mathbb { S } \\big | - 1 ) \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | _ { 1 } + \\theta ^ { ( k ) } \\big | \\mathbb { S } \\big | + \\big | 1 - \\gamma ^ { ( k ) } \\big | \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 238, + 782, + 367 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Taking supremum of the above inequality over $\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )$ , by $| \\mathbb { S } | \\le s$ , ", + "bbox": [ + 174, + 376, + 658, + 392 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/3f4dbcfb45d8bb4c773696bfd7de6ad9293ce7f02bf6805c23fb63a316ac691c.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k + 1 ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} \\le \\Big ( \\tilde { \\mu } \\gamma ^ { ( k ) } ( s - 1 ) + | 1 - \\gamma ^ { ( k ) } | \\Big ) \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} + \\theta ^ { ( k ) } s .\n$$", + "text_format": "latex", + "bbox": [ + 222, + 396, + 774, + 426 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "By the value of $\\theta ^ { ( k ) }$ given in (9), we have ", + "bbox": [ + 174, + 431, + 447, + 446 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/48cf41424bc52f1412d08d01e14219b965146430650f1bf6c9b2e6f80eb718f8.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k + 1 ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} \\leq \\Big ( \\gamma ^ { ( k ) } ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) + | 1 - \\gamma ^ { ( k ) } | \\Big ) \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 452, + 753, + 481 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Let $c ^ { ( \\tau ) } = - \\log \\left( ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\gamma ^ { ( \\tau ) } + | 1 - \\gamma ^ { ( \\tau ) } | \\right)$ . Then, by induction, ", + "bbox": [ + 173, + 486, + 620, + 503 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/36aa3f087bc511b0b45ce1aba61291a1037243b96fdfc7f8b52289d9a3c816b4.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k + 1 ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} \\leq \\exp \\Big ( - \\sum _ { \\tau = 0 } ^ { k } c ^ { ( \\tau ) } \\Big ) \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( 0 ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} \\leq \\exp \\Big ( - \\sum _ { \\tau = 0 } ^ { k } c ^ { ( \\tau ) } \\Big ) s B .\n$$", + "text_format": "latex", + "bbox": [ + 202, + 508, + 792, + 551 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since $\\| \\mathbf { x } \\| _ { 2 } \\leq \\| \\mathbf { x } \\| _ { 1 }$ for any $\\mathbf { x } \\in \\mathbb { R } ^ { n }$ , we can get the upper bound for $\\ell _ { 2 }$ norm: ", + "bbox": [ + 174, + 555, + 684, + 571 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9caee48fc85da58d1f970a647f3698cd28f386c9e122fc6b28b68ff373507b5d.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k + 1 ) } - \\mathbf { x } ^ { * } \\| _ { 2 } \\} \\leq \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k + 1 ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\} \\leq s B \\exp \\Big ( - \\sum _ { \\tau = 0 } ^ { k } c ^ { ( \\tau ) } \\Big ) .\n$$", + "text_format": "latex", + "bbox": [ + 253, + 575, + 741, + 618 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The assumption $s < ( 1 + 1 / \\tilde { \\mu } ) / 2$ gives $2 \\tilde { \\mu } s - \\tilde { \\mu } < 1$ . If $0 < \\gamma ^ { ( k ) } \\leq 1$ , we have $c ^ { ( k ) } > 0$ . If $1 < \\gamma ^ { ( k ) } < 2 / ( 1 + 2 \\tilde { \\mu } s - \\tilde { \\mu } )$ , we have ", + "bbox": [ + 171, + 622, + 825, + 655 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1e860e15d9457510293b36eb74d8aa36dd05ee98c1ac6806565d13a51bcc3b80.jpg", + "text": "$$\n( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\gamma ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | = ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\gamma ^ { ( k ) } + \\gamma ^ { ( k ) } - 1 < 1 ,\n$$", + "text_format": "latex", + "bbox": [ + 292, + 659, + 702, + 679 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "which implies $c ^ { ( k ) } > 0$ . Theorem 1 is proved. ", + "bbox": [ + 176, + 684, + 473, + 700 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B PROOF OF THEOREM 2 ", + "text_level": 1, + "bbox": [ + 176, + 718, + 400, + 736 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof of Theorem 2. We fix $\\mathbf { D }$ and sample a $\\mathbf { x } ^ { * } \\sim P _ { X }$ . ", + "bbox": [ + 174, + 750, + 535, + 765 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "If we can prove ", + "bbox": [ + 173, + 772, + 279, + 786 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/fb23775c1cd50131b85d7b8feca23f22b133206a84682e331ccca223c326bf86.jpg", + "text": "$$\nP \\Big ( ( 1 3 ) \\mathrm { d o e s ~ n o t ~ h o l d } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) \\leq \\epsilon | \\mathbb { S } | + \\epsilon ^ { | \\mathbb { S } | } ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 784, + 678, + 811 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "then the lower bound (13) in Theorem 2 is proved by ", + "bbox": [ + 173, + 811, + 522, + 827 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/bcf5fe84b2dee4e32c9a288d1ae4ec329f97204729670b1f22d52da9f550ab3a.jpg", + "text": "$$\n\\begin{array} { r l } & { P \\Big ( ( 1 3 ) \\mathrm { h o l d s } \\Big ) = \\displaystyle \\sum _ { \\mathbb { S } , 2 \\leq | \\mathbb { S } | \\leq s } P \\Big ( ( 1 3 ) \\mathrm { h o l d s } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) P \\Big ( \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) } \\\\ & { \\qquad \\geq ( 1 - \\epsilon s ^ { 3 / 2 } - \\epsilon ^ { 2 } ) \\displaystyle \\sum _ { 2 \\leq | \\mathbb { S } | \\leq s } P \\Big ( \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) } \\\\ & { \\qquad = 1 - \\epsilon s ^ { 3 / 2 } - \\epsilon ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 830, + 766, + 929 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now we fix $k$ and prove inequality (28) by three steps: ", + "bbox": [ + 173, + 103, + 534, + 119 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Step 1: If (13) does not hold, then what condition $\\mathbf { x } ^ { * }$ should satisfy? ", + "text_level": 1, + "bbox": [ + 173, + 125, + 647, + 140 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Fixing $k$ , we define a set ${ \\mathcal X } ^ { ( k ) } ( \\epsilon )$ , which involves all the $\\mathbf { x } ^ { * }$ that does not satisfy (13): ", + "bbox": [ + 173, + 146, + 733, + 165 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/7e9b052f8ac4dbdede29470e449b2e9738ec5ba7e20b19a31a168d9714afbc43.jpg", + "text": "$$\n\\mathcal { X } ^ { ( k ) } ( \\epsilon ) = \\{ ( 1 3 ) \\mathrm { d o e s ~ n o t ~ h o l d } \\} = \\Big \\{ \\mathbf { x } ^ { * } \\Big | \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } < \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\Big ( \\frac { \\bar { \\sigma } _ { \\operatorname* { m i n } } } { 3 ^ { s } } \\Big ) ^ { k } \\Big \\} .\n$$", + "text_format": "latex", + "bbox": [ + 233, + 170, + 764, + 199 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Let $\\mathbb { S } = \\operatorname { s u p p o r t } ( \\mathbf { x } ^ { * } )$ . For $\\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( k ) } ( \\epsilon )$ , we consider two cases: ", + "bbox": [ + 173, + 205, + 594, + 223 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "1. $| x _ { i } ^ { * } | > \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } / 3 ^ { s } ) ^ { k } , \\forall i \\in \\mathbb { S } .$ \n2. $| x _ { i } ^ { * } | \\leq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\big ( \\bar { \\sigma } _ { \\operatorname* { m i n } } / 3 ^ { s } \\big ) ^ { k }$ , for some $i \\in \\mathbb S$ . ", + "bbox": [ + 209, + 234, + 514, + 279 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "If case 1 holds, we obtain that the support of $\\mathbf { x } ^ { ( k ) }$ is exactly the same with that of $\\mathbf { x } ^ { * }$ : ", + "bbox": [ + 173, + 291, + 730, + 308 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/11fd443a97c62f14fddee8beb2efa52e0ef6a4b7d1ef6010d5e0c07b28b5fed3.jpg", + "text": "$$\n\\operatorname { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) = \\mathbb { S } .\n$$", + "text_format": "latex", + "bbox": [ + 416, + 313, + 578, + 332 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Then the relationship between $\\mathbf { x } ^ { ( k ) }$ and $\\mathbf { x } ^ { ( k - 1 ) }$ can be reduced to an affine transform: ", + "bbox": [ + 176, + 337, + 733, + 354 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/553ac530fd38e9fc934ae76c4c9a7eefd4d0fa2685500717cb40b3678382e568.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } = \\eta _ { \\theta ^ { ( k ) } } \\left( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k - 1 ) } - \\mathbf { b } ) \\right) } \\\\ & { \\quad \\quad = \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } ) - \\theta ^ { ( k - 1 ) } \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 358, + 723, + 411 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Subtracting $\\mathbf { x } ^ { * }$ from the two sides of (29), we obtain ", + "bbox": [ + 173, + 414, + 519, + 429 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/1b8bb3e79c077fba750c340789a65f705e782cd4ae9e6c47393b38d2e681e28b.jpg", + "text": "$$\n\\Big \\| \\big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\big ) ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } ) - \\theta ^ { ( k - 1 ) } \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } ) \\Big \\| _ { 2 } = \\| \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } \\| _ { 2 } = \\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 183, + 434, + 812, + 462 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the last equality is due to Definition 3. Thus, for all $\\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( k ) } ( \\epsilon )$ , if case 1 holds, we have ", + "bbox": [ + 173, + 468, + 810, + 486 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/08ba25b202523603e954b6085c30456dafe5008acb0b12b99235b4f2d4cfcbc1.jpg", + "text": "$$\n\\| \\big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\big ) ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } ) - \\theta ^ { ( k - 1 ) } \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } ) \\Big \\| _ { 2 } \\leq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\big ( \\bar { \\sigma } _ { \\operatorname* { m i n } } / 3 ^ { s } \\big ) ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 207, + 491, + 759, + 518 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Multiplying both sides of (30) by $( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ) ^ { - 1 }$ , we have ", + "bbox": [ + 173, + 525, + 625, + 545 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/1692139f04706dc803503ef13ef45ce99e3dafb4293b080dbb9ff0005a133c95.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\| \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } - \\theta ^ { ( k - 1 ) } ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ) ^ { - 1 } \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } ) \\| _ { 2 } } \\\\ & { { \\leq } \\| ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ) ^ { - 1 } \\| _ { 2 } \\cdot \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } / 3 ^ { s } ) ^ { k } \\leq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } ) ^ { k - 1 } 3 ^ { - k s } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 549, + 753, + 598 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the last inequality is due to (11). Let $\\tilde { \\mathbf { x } } ^ { ( k - 1 ) }$ denote the bias of $\\mathbf { x } ^ { ( k - 1 ) }$ : ", + "bbox": [ + 173, + 603, + 681, + 619 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/5266d5cfbdcc67c0d0ec09b9f7f2bec2b2c4402966a2503328f5328ac1344f58.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { \\mathbf { x } } ^ { ( k - 1 ) } \\triangleq \\boldsymbol { \\theta } ^ { ( k - 1 ) } ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - 1 ) } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } ) ^ { - 1 } \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 625, + 673, + 647 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "then we get a condition that $\\mathbf { x } ^ { * }$ satisfies if case 1 holds: ", + "bbox": [ + 173, + 651, + 537, + 667 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/40055abc665919d3388ccb33346cd1bfa518f128b2e760972fb1e323f723cfab.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { \\chi } ^ { ( k - 1 ) } ( \\epsilon ) = \\Bigl \\{ \\mathbf { x } ^ { * } \\Big | \\bigl \\| \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 1 ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } _ { \\mathbb { S } } ^ { * } - \\tilde { \\mathbf { x } } ^ { ( k - 1 ) } ( \\mathbf { x } ^ { * } ) \\bigr \\| _ { 2 } \\le \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } ) ^ { k - 1 } \\mathbb { 3 } ^ { - k s } \\Bigr \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 671, + 769, + 700 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "If case 2 holds, $\\mathbf { x } ^ { * }$ belongs to the following set: ", + "bbox": [ + 173, + 704, + 485, + 719 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/357fe5d5d26299b17cd8feb8e7c3d5e29c4ee43be3976a4cd18e089508a94cef.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { \\boldsymbol { \\chi } } ^ { ( k ) } ( \\epsilon ) = \\Bigl \\{ \\mathbf { x } ^ { * } \\Big | | \\boldsymbol { x } _ { i } ^ { * } | \\le \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\big ( \\bar { \\sigma } _ { \\operatorname* { m i n } } / 3 ^ { s } \\big ) ^ { k } , \\mathrm { ~ f o r ~ s o m e ~ } i \\in \\mathbb { S } \\Bigr \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 724, + 697, + 752 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Then for any $\\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( k ) } ( \\epsilon )$ , either $\\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( k - 1 ) } ( \\epsilon )$ or $\\mathbf { x } ^ { * } \\in \\tilde { \\mathcal { X } } ^ { ( k ) } ( \\epsilon )$ holds. In another word, ", + "bbox": [ + 171, + 758, + 774, + 779 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/33d6665fc8d888881a45b3ab9307a22f1e0d131868f6e6f675eb9b48495cc8dc.jpg", + "text": "$$\n\\mathcal { X } ^ { ( k ) } ( \\epsilon ) \\subset \\tilde { \\mathcal { X } } ^ { ( k ) } ( \\epsilon ) \\cup \\mathcal { X } ^ { ( k - 1 ) } ( \\epsilon ) .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 784, + 607, + 806 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Step 2: By imitating the construction of ${ \\mathcal { X } } ^ { ( k ) } ( \\epsilon )$ , we construct ", + "bbox": [ + 171, + 819, + 604, + 837 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/42899ba81a9f7abe96aab7dcf3d8cdb5a37ad3f459a377d9aea14ef738148db9.jpg", + "text": "$$\n\\mathcal { X } ^ { ( k - 2 ) } ( \\epsilon ) , \\mathcal { X } ^ { ( k - 3 ) } ( \\epsilon ) , \\cdot \\cdot \\cdot .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 842, + 589, + 862 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Similar to Step 1, we divide $\\chi ^ { ( k - 1 ) } ( \\epsilon )$ into two sets: $\\tilde { \\boldsymbol { \\chi } } ^ { ( k - 1 ) } ( \\epsilon )$ and $\\chi ^ { ( k - 2 ) } ( \\epsilon )$ , then we divide $\\chi ^ { ( k - 2 ) } ( \\epsilon )$ into $\\tilde { \\chi } ^ { ( k - 2 ) } ( \\epsilon )$ and $\\chi ^ { ( k - 3 ) } ( \\epsilon )$ . Repeating the process, until dividing $\\mathcal { X } ^ { ( 1 ) } ( \\epsilon )$ into $\\tilde { \\chi } ^ { ( 1 ) } ( \\epsilon )$ and ${ \\mathcal X } ^ { ( 0 ) } ( \\epsilon )$ . ", + "bbox": [ + 174, + 869, + 825, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "By induction, we have ", + "bbox": [ + 173, + 103, + 323, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/b16fa607c5b0b85b1f418bc7289b414128c158ab64aac886ab235d5d35d33c61.jpg", + "text": "$$\n\\mathcal { X } ^ { ( k ) } ( \\epsilon ) \\subset \\tilde { \\mathcal { X } } ^ { ( k ) } ( \\epsilon ) \\cup \\tilde { \\mathcal { X } } ^ { ( k - 1 ) } ( \\epsilon ) \\cup \\tilde { \\mathcal { X } } ^ { ( k - 2 ) } ( \\epsilon ) \\cup \\cdots \\cup \\tilde { \\mathcal { X } } ^ { ( 1 ) } ( \\epsilon ) \\cup \\mathcal { X } ^ { ( 0 ) } ( \\epsilon ) ,\n$$", + "text_format": "latex", + "bbox": [ + 258, + 130, + 738, + 154 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the sets are defined as follows for all $j = 0 , 1 , 2 , \\cdots , k$ : ", + "bbox": [ + 173, + 165, + 583, + 181 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/796b84389d6ac4d1878f2ba48e2f9f4cefc11cdea1d51de1eac4351314449d41.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { { \\boldsymbol { \\chi } } } ^ { ( k - j ) } ( \\epsilon ) = \\Bigl \\{ \\mathbf { x } ^ { * } \\Big | | x _ { i } ^ { * } + \\tilde { x } _ { i } ^ { ( k - j ) } ( \\mathbf { x } ^ { * } ) | < \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\big ( \\bar { \\sigma } _ { \\operatorname* { m i n } } \\big ) ^ { k - j } 3 ^ { - k s } , \\mathrm { ~ f o r ~ s o m e ~ } i \\in \\mathbb { S } . \\Bigr \\} , } \\\\ & { { \\boldsymbol { \\chi } } ^ { ( k - j ) } ( \\epsilon ) = \\Bigl \\{ \\mathbf { x } ^ { * } \\Big | \\| \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } _ { \\mathbb { S } } ^ { * } - \\tilde { \\mathbf { x } } ^ { ( k - j ) } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } \\le \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\big ( \\bar { \\sigma } _ { \\operatorname* { m i n } } \\big ) ^ { k - j } 3 ^ { - k s } \\Bigr \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 191, + 766, + 250 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and the bias is defined as following for all $j = 0 , 1 , 2 , \\cdots , k$ : ", + "bbox": [ + 173, + 258, + 575, + 275 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/41dcc44b8237f7173189bf4fdb64bd413bbc217cbffaf6db7b9e598db7732a5d.jpg", + "text": "$$\n\\tilde { \\mathbf { x } } ^ { ( k - j ) } ( \\mathbf { x } ^ { * } ) = \\sum _ { t = 1 } ^ { j } \\left( \\mathbf { I } - \\left( \\mathbf { W } _ { : , \\mathbb { S } } ^ { ( k - j + t - 1 ) } \\right) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\right) ^ { - t } \\boldsymbol { \\theta } ^ { ( k - j + t - 1 ) } \\mathrm { s i g n } \\big ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } ( \\mathbf { x } ^ { * } ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 210, + 286, + 759, + 330 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Step 3: Estimating the probabilities of all the sets in (31). ", + "text_level": 1, + "bbox": [ + 173, + 348, + 570, + 364 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By (31), we have ", + "bbox": [ + 173, + 371, + 289, + 386 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/7425f95c3431e73bb095e7a4645a7a753c79596fed8a9ed72f488cf462262dcc.jpg", + "text": "$$\n\\begin{array} { r l } { { P \\Big ( \\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( k ) } ( \\epsilon ) \\Big | \\operatorname { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) } \\quad } & { } \\\\ & { \\leq \\displaystyle \\sum _ { j = 1 } ^ { k - 1 } P \\Big ( \\mathbf { x } ^ { * } \\in \\tilde { \\mathcal { X } } ^ { ( k - j ) } ( \\epsilon ) \\Big | \\operatorname { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) + P \\Big ( \\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( 0 ) } ( \\epsilon ) \\Big | \\operatorname { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 396, + 779, + 472 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Now we have to prove that each of the above terms is small, then $P ( \\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( k ) } ( \\epsilon ) | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } )$ is small and (28) will be proved. ", + "bbox": [ + 173, + 484, + 823, + 515 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Define a set of $n$ -dimensional sign numbers ", + "bbox": [ + 174, + 520, + 462, + 536 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/89ad8d5ad0636a39a33d3e911125f654c1f015e92ba29e5688e5e59bb1375a4c.jpg", + "text": "$$\n\\operatorname { S i } ( n ) = { \\Big \\{ } ( s _ { 1 } , s _ { 2 } , \\cdots , s _ { n } ) { \\Big | } s _ { i } \\in \\{ 0 , - 1 , 1 \\} , \\forall i = 1 , \\cdots , n { \\Big \\} } .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 547, + 697, + 575 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since $\\mathrm { s i g n } \\big ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } \\big ) \\ \\in \\ \\mathrm { S i } ( | \\mathbb { S } | )$ for all $t = 1 , 2 , \\cdots , j$ , $\\{ \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } ) \\} _ { t = 1 } ^ { j }$ has finitely possible values. Let $\\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } ) = \\mathbf { s } ^ { ( t ) }$ for $t = 1 , 2 , \\cdots , j$ . Then $\\tilde { x } _ { i } ^ { ( k - j ) } ( \\mathbf { x } ^ { * } )$ is independent of $\\mathbf { x } ^ { * }$ and can be written as $\\tilde { x } _ { i } ^ { ( k - j ) } ( \\mathbf { s } ^ { ( 1 ) } , \\mathbf { s } ^ { ( 2 ) } , \\cdot \\cdot \\cdot , \\mathbf { s } ^ { ( j ) } )$ . Thus, we have ", + "bbox": [ + 173, + 587, + 826, + 643 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/e7683e34457615a5dc9e268b823a03a6367db80fcfcfb84a96c62f8ba948b3a1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P ( x ^ { * } \\in \\tilde { \\mathcal { X } } ^ { ( k - \\delta ) } ( e ) \\operatorname* { l a n p p o r } ( x ^ { * } ) = 8 ) } \\\\ & { = \\displaystyle \\sum _ { i \\in \\mathbb { S } } \\displaystyle \\sum _ { s \\in \\{ x \\} \\in \\{ 0 , 1 \\} \\leq n \\leq i \\leq | \\mathbb { S } } \\sum _ { s \\in \\{ 1 \\} | s | \\in \\mathcal { S } } \\sum _ { s \\in \\{ 1 \\} | s | \\in \\mathcal { S } } \\sum _ { s \\in \\{ 1 \\} | s | } } \\\\ & { \\quad P \\Big ( | x _ { s } ^ { * } + \\tilde { x } _ { s } ^ { ( k ) } ( s \\cdot s ) ( x ) | \\times | e | \\mathcal { X } | | \\Big ) \\cdot \\mathrm { ~ e ~ i s ' ~ s h ( \\tilde { x } | \\tilde { x } ) } } \\\\ & \\leq \\displaystyle \\sum _ { i \\in \\mathbb { S } } \\sum _ { s \\in \\{ x \\} \\in \\{ 1 \\} \\leq n \\} \\sum _ { s \\in \\{ 1 \\} \\atop | s \\in \\{ 1 \\} } \\sum _ { s \\in \\{ 1 \\} \\in \\{ 1 \\} } \\sum _ { | s | \\in \\mathcal { S } } \\sum _ { s \\in \\{ 1 \\} | s | } \\Big ( x _ { s } ^ { ( k ) } - \\tilde { x } _ { s } ^ { ( k ) } - s \\cdot k \\cdot s _ { s } \\cdot \\operatorname* { l i g n } ( \\mathbf { x } _ { s } ^ { ( k ) } ) = \\mathbf { s } ^ { ( 1 ) } , \\cdots , \\ s \\operatorname* { l i p } ( \\mathbf { x } _ { s } ^ { ( k - \\delta + 1 ) } ) = \\mathbf { s } ^ { ( \\delta ) } \\Big | s \\Big ) } \\\\ & { \\quad \\times \\displaystyle \\sum _ { i \\in \\mathbb { S } } \\Big ( | x _ { s } ^ { * } + \\tilde { x } _ { s } ^ { ( k ) } ( s \\cdot s ) ( | \\mathbf { x } | ) \\cdot \\ s | ^ { 2 } } \\\\ & { \\quad \\times \\displaystyle \\sum _ { s \\in \\{ x \\} \\leq n \\leq i \\leq | \\mathbb { S } } \\sum _ { s \\in \\{ 1 \\} } \\sum _ { s \\in \\{ 1 \\} } \\sum _ { s \\in \\{ 1 \\} \\leq n \\leq i \\leq | \\mathbb { S } } \\sum _ { s \\in \\{ 1 \\} } E \\Big ( | s | B ( \\sigma _ { \\operatorname* { m i n } } ) ^ { k - \\delta } - s \\cdot | s | \\operatorname* { s u p p o r } ( \\mathbf { x } ^ { * } ) = \\mathbf { s } \\Big ) } \\\\ & \\leq \\displaystyle \\sum _ { s \\in \\{ 1 \\} \\leq n \\leq i \\leq | \\mathbb { S } | } \\sum _ s \\in \\{ 1 \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 165, + 655, + 816, + 887 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the second inequality comes from the uniform distribution of ${ \\mathbf { x } } _ { \\mathbb { S } } ^ { * }$ (Assumption 2), the last inequality comes from $| \\mathbb { S } | \\le s$ . ", + "bbox": [ + 173, + 895, + 825, + 925 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The last term, due to the uniform distribution of ${ \\mathbf { x } } _ { \\mathbb { S } } ^ { * }$ and $\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }$ , can be bounded by ", + "bbox": [ + 173, + 102, + 730, + 119 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7f298af51725517db2bd1d81c4a3430ff15567c26859b0d8bf9f521aa9480d55.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad P ( \\mathbf { x } ^ { * } \\in \\mathcal { X } ^ { ( 0 ) } ( \\epsilon ) | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } ) } \\\\ & { = P \\Big ( \\| \\mathbf { x } ^ { * } + \\tilde { \\mathbf { x } } ^ { ( 0 ) } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } \\leq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } 3 ^ { - k s } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) } \\\\ & { = \\displaystyle \\sum _ { \\mathbf { s } ^ { ( 1 ) } \\in \\mathbb { S } ( | \\mathbb { S } | ) } \\sum _ { \\mathbf { s } ^ { ( 2 ) } \\in \\mathrm { S i } ( | \\mathbb { S } | ) } \\cdot \\sum _ { \\mathbf { s } ^ { ( k ) } \\in \\mathrm { S i } ( | \\mathbb { S } | ) } } \\\\ & { \\quad P \\Big ( \\| \\mathbf { x } ^ { * } + \\tilde { \\mathbf { x } } ^ { ( 0 ) } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } \\leq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } 3 ^ { - k s } , \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( 1 ) } ) = \\mathbf { s } ^ { ( 1 ) } , \\cdots , \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } ) = \\mathbf { s } ^ { ( k ) } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } } \\\\ & { \\leq 3 ^ { k | \\mathbb { S } | } \\Big ( ( \\epsilon 3 ^ { - k s } ) ^ { | \\mathbb { S } | } \\Big ) \\leq \\epsilon ^ { | \\mathbb { S } | } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 168, + 122, + 816, + 265 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Then we obtain ", + "bbox": [ + 174, + 265, + 277, + 279 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/e47d7755e253012a629d6e016ae9dccf4770f99512774a01f790b19220532a5d.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle P ( { \\bf x } ^ { * } \\in \\mathcal { X } ^ { ( k ) } ( \\epsilon ) \\vert \\mathrm { s u p p o r t } ( { \\bf x } ^ { * } ) = \\mathbb { S } ) } \\ ~ } \\\\ { \\displaystyle \\leq \\sum _ { j = 0 } ^ { k - 1 } \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } ) ^ { k - j } \\mathbb { 3 } ^ { ( j - k ) \\vert \\mathbb { S } \\vert } + \\epsilon ^ { \\vert \\mathbb { S } \\vert } = \\sum _ { j = 1 } ^ { k } \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } ) ^ { j } 3 ^ { - j \\vert \\mathbb { S } \\vert } + \\epsilon ^ { \\vert \\mathbb { S } \\vert } } \\ \\\\ { { \\displaystyle = \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } \\frac { \\bar { \\sigma } _ { \\operatorname* { m i n } } 3 ^ { - \\vert \\mathbb { S } \\vert } } { 1 - \\bar { \\sigma } _ { \\operatorname* { m i n } } 3 ^ { - \\vert \\mathbb { S } \\vert } } \\Big ( 1 - ( \\bar { \\sigma } _ { \\operatorname* { m i n } } 3 ^ { - \\vert \\mathbb { S } \\vert } ) ^ { k } \\Big ) + \\epsilon ^ { \\vert \\mathbb { S } \\vert } \\leq \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } + \\epsilon ^ { \\vert \\mathbb { S } \\vert } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 279, + 736, + 383 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Then (28) is proved. ", + "bbox": [ + 174, + 383, + 308, + 398 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C PROOF OF THEOREM 3 ", + "text_level": 1, + "bbox": [ + 174, + 417, + 398, + 434 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "There are two conclusions in Theorem 3. We prove the two conclusions in the following two subsections respectively. ", + "bbox": [ + 176, + 448, + 823, + 477 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C.1 PROOF OF CONCLUSION 1. ", + "text_level": 1, + "bbox": [ + 174, + 492, + 401, + 507 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Before proving Conclusion 1, we analyze the operator $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }$ in detail. ", + "bbox": [ + 173, + 517, + 624, + 535 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The circular convolution (23) is equivalent with: ", + "bbox": [ + 174, + 540, + 491, + 555 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/14416ed8c29f8f3504cb92f8f617571ed9baa25825546fa9c00c4bf4bb4b8107.jpg", + "text": "$$\n\\mathbf { b } ( i , j ) = \\sum _ { k = 0 } ^ { N - 1 } \\sum _ { l = 0 } ^ { N - 1 } \\sum _ { m = 1 } ^ { M } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k , l , m ) \\mathbf { x } _ { m } ( k , l ) , \\quad 0 \\leq i , j \\leq N - 1 ,\n$$", + "text_format": "latex", + "bbox": [ + 259, + 558, + 735, + 603 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the circulant matrix is element-wise defined as: ", + "bbox": [ + 178, + 603, + 526, + 617 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/912be43429246100375c8c183944ff1e4704949743891151764dac2cd2d3f58e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { \\mathfrak { I } } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k , l , m ) = \\left\\{ \\mathbf { d } _ { m } \\left( ( k - i ) _ { \\mathrm { m o d } N } , ( l - j ) _ { \\mathrm { m o d } N } \\right) , \\ : \\ : \\ : 0 \\leq ( k - i ) _ { \\mathrm { m o d } N } , ( l - j ) _ { \\mathrm { m o d } N } \\leq D - 1 \\right. } \\\\ { 0 , \\ : \\ : \\ : \\mathrm { \\ o t h e r s } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 618, + 825, + 655 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Similarly, the corresponding circulant matrix $\\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k , l , m )$ of dictionary w is: ", + "bbox": [ + 176, + 665, + 717, + 683 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/e85be17f2aa8b22a29e56ce32a0b856091bc9de4497b78c34c70ffff8876665a.jpg", + "text": "$$\nN _ { \\mathrm { c i r } } ^ { N } ( i , j ; k , l , m ) = \\left\\{ \\begin{array} { l l } { { \\displaystyle { \\bf w } _ { m } \\left( ( k - i ) _ { \\mathrm { m o d } N } , ( l - j ) _ { \\mathrm { m o d } N } \\right) } , } & { { \\displaystyle 0 \\leq ( k - i ) _ { \\mathrm { m o d } N } , ( l - j ) _ { \\mathrm { m o d } N } \\leq D - 1 } } \\\\ { { 0 , } } & { { \\mathrm { o t h e r s } } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 181, + 685, + 831, + 722 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As we defined in Section 3, $\\mathbf { b }$ is a vector. With $\\mathbf { x } = [ \\mathbf { x } _ { 1 } , \\cdots , \\mathbf { x } _ { M } ] ^ { T }$ , $\\mathbf { x }$ is a vector. Then the operator $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }$ is a matrix, where $( i , j )$ is its row index and $( k , l , m )$ is its column index. ", + "bbox": [ + 174, + 739, + 825, + 770 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Define a function measuring the difference between $i$ and $k$ : ", + "bbox": [ + 173, + 775, + 570, + 790 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/24896e62a752ec10f65447b44a3d501b4d639f7327af7cb8795f0bd76042db4b.jpg", + "text": "$$\nI ( i , k ) \\triangleq ( k - i ) _ { \\mathrm { m o d } N } , \\quad 0 \\leq i , k \\leq N - 1 .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 794, + 647, + 811 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The coherence between $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k , l , m )$ and $\\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k , l , m )$ : $\\mathbf { B } _ { \\mathrm { c o h } } = ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N }$ is elementwise defined by: ", + "bbox": [ + 174, + 814, + 821, + 844 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/428b664be9bfa30165a2f2480078f55455952cf22507003364ab42bf4ff64d77.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) = \\displaystyle \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { j = 0 } ^ { N - 1 } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k _ { 1 } , l _ { 1 } , m _ { 1 } ) \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) } \\\\ & { \\qquad = \\displaystyle \\sum _ { i \\in \\mathbb { Z } ( k _ { 1 } , k _ { 2 } ) } \\sum _ { j \\in \\mathcal { I } ( l _ { 1 } , l _ { 2 } ) } \\mathbf { d } _ { m _ { 1 } } \\big ( I ( i , k _ { 1 } ) , I ( j , l _ { 1 } ) \\big ) \\mathbf { w } _ { m _ { 2 } } \\big ( I ( i , k _ { 2 } ) , I ( j , l _ { 2 } ) \\big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 844, + 816, + 929 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 104, + 217, + 117 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/f6d967411bec56ca235aee7847e9c417607e1338fcb565c44cc9449345fb3e4e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { T } ( k _ { 1 } , k _ { 2 } ) = \\{ i | 0 \\leq i \\leq N - 1 , 0 \\leq I ( i , k _ { 1 } ) \\leq D - 1 , 0 \\leq I ( i , k _ { 2 } ) \\leq D - 1 \\} , } \\\\ & { \\mathcal { I } ( l _ { 1 } , l _ { 2 } ) = \\{ j | 0 \\leq j \\leq N - 1 , 0 \\leq I ( j , l _ { 1 } ) \\leq D - 1 , 0 \\leq I ( j , l _ { 2 } ) \\leq D - 1 \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 121, + 759, + 159 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma 2. Given $N \\geq 2 D - 1$ , it holds that: ", + "bbox": [ + 173, + 162, + 475, + 178 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(a) $\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset$ if and only if “ $0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime \\prime } o r ^ { \\ast } 0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime }$ holds. \n(b) $\\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \\neq \\emptyset$ if and only if “ $0 \\leq ( l _ { 1 } - l _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime \\prime } o r ^ { \\textit { \\infty } } 0 < ( l _ { 2 } - l _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime }$ holds. ", + "bbox": [ + 171, + 188, + 826, + 256 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. Now we prove Conclusion (a). Firstly, we prove “if.” If $0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1$ and $N \\geq 2 D - 1$ , we have ", + "bbox": [ + 171, + 270, + 825, + 299 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/725c17b996fd12d4a91b289e1f1f9c49e273dc5c163cf0343b0d91035ac7946b.jpg", + "text": "$$\n\\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \\big \\{ ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\big | \\delta \\in \\mathbb { Z } , ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq \\delta \\leq D - 1 \\big \\} \\neq \\emptyset .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 304, + 743, + 324 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "If $0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1$ and $N \\geq 2 D - 1$ , we have ", + "bbox": [ + 171, + 329, + 570, + 345 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/2f5a3e62ce951c6abf961cf6e7caeec393dfaf2b6efdc0b75efe8e0d934d82bf.jpg", + "text": "$$\n\\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \\big \\{ ( k _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\big | \\delta \\in \\mathbb { Z } , ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq \\delta \\leq D - 1 \\big \\} \\neq \\emptyset .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 351, + 743, + 371 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Secondly, we prove “only if.” If $\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset$ , we can select an $i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )$ . Let $r _ { 1 } = ( k _ { 1 } -$ $i ) _ { \\mathrm { m o d } N }$ and $r _ { 2 } = ( k _ { 2 } - i ) _ { \\mathrm { m o d } N }$ . By the definition of $\\mathcal { T } ( k _ { 1 } , k _ { 2 } )$ , we have $0 \\leq r _ { 1 } , r _ { 2 } \\leq D - 1$ . Two cases should be considered here. Case 1: $r _ { 1 } \\geq r _ { 2 }$ . Since $0 \\le r _ { 1 } - r _ { 2 } \\le D - 1 \\le N - 1$ , it holds that $r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \\mathrm { m o d } N }$ . Thus, ", + "bbox": [ + 174, + 376, + 825, + 434 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/ca3617944362923737e8fb9e8c72895ab921e7c71fb76c317d388bb5dfd69f77.jpg", + "text": "$$\n\\begin{array} { r l } & { r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \\mathrm { m o d } N } = \\left( ( k _ { 1 } - i ) _ { \\mathrm { m o d } N } - ( k _ { 2 } - i ) _ { \\mathrm { m o d } N } \\right) _ { \\mathrm { m o d } N } } \\\\ & { ~ = \\left( ( k _ { 1 } - i ) - ( k _ { 2 } - i ) \\right) _ { \\mathrm { m o d } N } } \\\\ & { ~ = ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 438, + 722, + 501 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The equality $\\begin{array} { r } { \\mathbf { \\dot { \\mathbf { \\varphi } } } 0 \\leq r _ { 1 } - r _ { 2 } \\leq D - { \\mathbf { \\varphi } } 1 ^ { , } } \\end{array}$ leads to the conclusion $0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\mathfrak { M } }$ . In case 2 where $r _ { 1 } < r _ { 2 }$ , we can obtain $0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1$ with the similar arguments. ", + "bbox": [ + 171, + 505, + 825, + 535 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Conclusion (b) can be proved by the same argument with the proof of (a). Lemma 2 is proved. ", + "bbox": [ + 174, + 540, + 792, + 556 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Now we fix $k _ { 1 } , l _ { 1 }$ and consider what values of $k _ { 2 } , l _ { 2 }$ give $\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset$ and $\\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \\neq \\emptyset$ . Define four index sets given $0 \\leq k _ { 1 } , l _ { 1 } \\leq N - 1$ : ", + "bbox": [ + 174, + 569, + 825, + 599 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c72902ed5871cffecbcfd321dbe10bf14477e7a4292f7acf44ecfed536128f59.jpg", + "text": "$$\n\\begin{array} { l c l } { { } } & { { } } & { { \\displaystyle \\mathcal { K } ( k _ { 1 } ) = \\{ k | 0 \\leq ( k _ { 1 } - k ) _ { \\mathrm { m o d } N } \\leq D - 1 \\} } } \\\\ { { } } & { { } } & { { \\displaystyle \\bar { \\mathcal { K } } ( k _ { 1 } ) = \\{ k | 0 < ( k - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 \\} } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 604, + 640, + 643 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/81f3971f568ba0e7152da654d00d547cbcfa760d752c6ce81ed2ae3436f622e8.jpg", + "text": "$$\n\\mathcal { L } ( l _ { 1 } ) = \\{ l | 0 \\leq ( l _ { 1 } - l ) _ { \\mathrm { m o d } N } \\leq D - 1 \\}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 647, + 632, + 665 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/58a4d217877b7a695f662731f6db1cf9935c8e898601de1f57d41d8682872e04.jpg", + "text": "$$\n\\bar { \\mathcal { L } } ( l _ { 1 } ) = \\{ l | 0 < ( l - l _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 \\}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 667, + 630, + 685 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma 3. If $N \\geq 2 D - 1$ , we have: ", + "bbox": [ + 173, + 686, + 423, + 702 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(a) The cardinality of $\\begin{array} { r } { \\mathcal { K } ( k _ { 1 } ) , \\bar { \\mathcal { K } } ( k _ { 1 } ) \\colon | \\mathcal { K } ( k _ { 1 } ) | = D , | \\bar { \\mathcal { K } } ( k _ { 1 } ) | = D - 1 . } \\end{array}$ ", + "bbox": [ + 174, + 712, + 640, + 731 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(b) $\\mathcal { K } ( k _ { 1 } ) \\cap \\bar { \\mathcal { K } } ( k _ { 1 } ) = \\emptyset .$ . ", + "bbox": [ + 173, + 736, + 341, + 753 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(c) The cardinality of $\\mathcal { L } ( l _ { 1 } ) , \\bar { \\mathcal { L } } ( l _ { 1 } ) \\colon | \\mathcal { L } ( l _ { 1 } ) | = D , | \\bar { \\mathcal { L } } ( l _ { 1 } ) | = D - 1 .$ ", + "bbox": [ + 173, + 758, + 620, + 779 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(d) $\\mathcal { L } ( l _ { 1 } ) \\cap \\bar { \\mathcal { L } } ( l _ { 1 } ) = \\emptyset$ ", + "bbox": [ + 173, + 784, + 331, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. Now we prove Conclusion (a). The set $\\kappa ( k _ { 1 } )$ can be equivalently written as ", + "bbox": [ + 173, + 814, + 720, + 830 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/6d6ea5e3fa7ac38d1f3cc7982a792cd70490824ae24093f8d455dcf814977df7.jpg", + "text": "$$\n\\mathcal { K } ( k _ { 1 } ) = \\{ ( k _ { 1 } - r _ { k } ) _ { \\mathrm { m o d } N } | r _ { k } = 0 , 1 , \\cdots , D - 1 \\}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 835, + 665, + 853 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let $k ( r _ { k } ) = ( k _ { 1 } - r _ { k } ) _ { \\mathrm { m o d } N }$ . We want to show that $k ( r _ { k } ^ { 1 } ) \\neq k ( r _ { k } ^ { 2 } )$ as long as $r _ { k } ^ { 1 } \\neq r _ { k } ^ { 2 }$ . Without loss of generality, we assume $0 \\leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \\leq D - 1$ . By the definition of modulo operation, There exist two integers $q , q ^ { \\prime }$ such that ", + "bbox": [ + 173, + 858, + 825, + 902 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/886496ab9907ee51c6b95ef828e89ce3441d725612365c70494604f39d272243.jpg", + "text": "$$\nk ( r _ { k } ^ { 1 } ) = q N + k _ { 1 } - r _ { k } ^ { 1 } , \\quad k ( r _ { k } ^ { 2 } ) = q ^ { \\prime } N + k _ { 1 } - r _ { k } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 906, + 669, + 925 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Suppose $k ( r _ { k } ^ { 1 } ) = k ( r _ { k } ^ { 2 } )$ . Taking the difference between the above two equations, we obtain $r _ { k } ^ { 2 } \\mathrm { ~ - ~ }$ $r _ { k } ^ { 1 } = ( q ^ { \\prime } - \\overset { \\cdot } { q } ) N$ , i.e, $N$ divides $\\overline { { r _ { k } ^ { 2 } } } - r _ { k } ^ { 1 }$ . However, $0 \\leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \\leq D - \\mathrm { i }$ implies $1 \\leq r _ { k } ^ { 2 } - r _ { k } ^ { 1 } \\leq$ $D - 1 \\leq N - 1$ k k, which contradicts with $^ { \\circ } N$ dividing $r _ { k } ^ { 2 } - r _ { k } ^ { 1 }$ k .” Thus, it holds that $k ( r _ { k } ^ { 1 } ) \\stackrel { \\sim } { = } k ( \\stackrel { \\sim } { r } _ { k } ^ { 2 } )$ . Then we have $| \\kappa ( k _ { 1 } ) | = D$ . ", + "bbox": [ + 173, + 102, + 825, + 160 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In the same way, we have ", + "bbox": [ + 174, + 166, + 343, + 181 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/72ca797a8a77a6ae0def61c6ca53cc7674bce437925b8a81131f08fed5d10454.jpg", + "text": "$$\n\\bar { \\mathcal { K } } ( k _ { 1 } ) = \\{ ( k _ { 1 } + r _ { k } ) _ { \\mathrm { m o d } N } | r _ { k } = 1 , 2 , \\cdot \\cdot \\cdot , D - 1 \\}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 185, + 665, + 203 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and $| \\bar { \\kappa } ( k _ { 1 } ) | = D - 1$ . Conclusion (a) is proved. ", + "bbox": [ + 173, + 205, + 495, + 222 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now we prove Conclusion (b). Suppose ${ \\mathcal { K } } ( k _ { 1 } ) \\cap { \\bar { \\mathcal { K } } } ( k _ { 1 } ) \\neq \\emptyset$ . Pick a $k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } ) \\cap \\bar { \\mathcal { K } } ( k _ { 1 } )$ . Let $r _ { 3 } = ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N }$ and $r _ { 4 } = ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } { N } }$ . Then we have $0 \\leq r _ { 3 } \\leq D - 1$ and $0 < r _ { 4 } \\le D - 1$ . By the definition of modulo operation, There exist two integers $q , q ^ { \\prime }$ such that ", + "bbox": [ + 174, + 227, + 825, + 271 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/5919f9aca37c17a9e9eab41af79e40f681389b4bd3a82e6886da4282d208e8e4.jpg", + "text": "$$\nk _ { 1 } - k _ { 2 } = q N + r _ { 3 } , \\quad k _ { 2 } - k _ { 1 } = q ^ { \\prime } N + r _ { 4 }\n$$", + "text_format": "latex", + "bbox": [ + 351, + 273, + 647, + 291 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "which imply ", + "bbox": [ + 174, + 296, + 258, + 310 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/8ab8cb05aaba5299179504d6a3ba7ff9265e3768b5a9e510e585560028c1e5aa.jpg", + "text": "$$\nr _ { 3 } + r _ { 4 } + ( q + q ^ { \\prime } ) N = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 308, + 584, + 325 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "However, $0 < r _ { 3 } + r _ { 4 } \\le 2 D - 2$ contradicts with $\\dot { \\boldsymbol { q } } \\in \\mathbb { Z } , \\boldsymbol { q } ^ { \\prime } \\in \\mathbb { Z } , N \\in \\mathbb { Z } , N \\geq 2 D - 1$ .” Conclusion (b) is proved. ", + "bbox": [ + 171, + 327, + 823, + 356 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Conclusions (c) and (d) are actually the same with Conclusions (a) and (b) respectively. Thus, it holds that ", + "bbox": [ + 174, + 362, + 823, + 391 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/aa6be485f5d65c78ef5004c13ca4b07409ac8080df3bb351b13a27c498613be5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( l _ { 1 } ) = \\{ ( l _ { 1 } - r _ { l } ) _ { \\mathrm { m o d } N } | r _ { l } = 0 , 1 , \\cdots , D - 1 \\} } \\\\ & { \\bar { \\mathcal { L } } ( l _ { 1 } ) = \\{ ( l _ { 1 } + r _ { l } ) _ { \\mathrm { m o d } N } | r _ { l } = 1 , 2 , \\cdots , D - 1 \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 393, + 655, + 433 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and $| \\mathcal { L } ( l _ { 1 } ) | = D , | \\bar { \\mathcal { L } } ( l _ { 1 } ) | = D - 1$ . Lemma 3 is proved. ", + "bbox": [ + 173, + 435, + 545, + 453 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "With the preparations, we can prove Conclusion 1 of Theorem 3 now. ", + "bbox": [ + 174, + 467, + 629, + 482 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof of Theorem 3, Conclusion $^ { l }$ . Firstly we fix $k _ { 1 } \\in \\{ 0 , 1 , \\cdots , N - 1 \\}$ and consider $k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } )$ . Let $r _ { k } \\stackrel { \\cdot } { = } ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N }$ . Then equation (37) implies that, for any $i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )$ , there exists a $\\delta$ $( r _ { k } \\le \\delta \\le D - 1 )$ such that ", + "bbox": [ + 173, + 496, + 825, + 540 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/d4398c39472066166050e7744f8bcde3f7ef3bb9d313ea89fcaf36b2df577446.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( i , k _ { 1 } ) = \\bigr ( k _ { 1 } - ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta , } \\\\ & { I ( i , k _ { 2 } ) = \\bigr ( k _ { 2 } - ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { k } ) _ { \\mathrm { m o d } N } = \\delta - r _ { k } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 541, + 717, + 585 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now we consider another case for $k _ { 2 }$ : $k _ { 2 } \\in \\bar { \\mathcal { K } } ( k _ { 1 } )$ , $r _ { k } = ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N }$ . Equation (38) implies that, for any $i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )$ , there exists a $\\delta$ $r _ { k } \\le \\delta \\le D - 1 \\}$ ) such that ", + "bbox": [ + 169, + 588, + 825, + 618 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f33ca70194d1c91c25df93f843600b022b8c976b2fb9449f4fc5675c4a3b49cb.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( i , k _ { 1 } ) = \\bigr ( k _ { 1 } - ( k _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { k } ) _ { \\mathrm { m o d } N } = \\delta - r _ { k } , } \\\\ & { I ( i , k _ { 2 } ) = \\bigr ( k _ { 2 } - ( k _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 632, + 717, + 678 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Similarly, for any $l _ { 1 } \\in \\{ 0 , 1 , \\cdots , N - 1 \\}$ and $l _ { 2 } \\in \\mathcal { L } ( l _ { 1 } )$ , we denote $r _ { l } = ( l _ { 1 } - l _ { 2 } ) _ { \\mathrm { m o d } N }$ . For any $j \\in \\mathcal { I } ( l _ { 1 } , l _ { 2 } )$ , there exists a $\\delta$ $( r _ { l } \\le \\delta \\le D - 1 )$ such that ", + "bbox": [ + 173, + 683, + 821, + 712 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/feb573099fd8477293e71c33fad7006cbf042a24fff449d4e806e64fcb190979.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( j , l _ { 1 } ) = \\bigr ( l _ { 1 } - ( l _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta , } \\\\ & { I ( j , l _ { 2 } ) = \\bigr ( l _ { 2 } - ( l _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { l } ) _ { \\mathrm { m o d } N } = \\delta - r _ { l } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 715, + 710, + 758 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Another case for $l _ { 2 }$ : $l _ { 2 } ~ \\in ~ \\bar { \\mathcal { L } } ( l _ { 1 } )$ , $r _ { l } = ( l _ { 2 } - l _ { 1 } ) _ { \\mathrm { m o d } N }$ . For any $j \\in \\mathcal { I } ( l _ { 1 } , l _ { 2 } )$ , there exists a $\\delta$ $( r _ { l } \\le \\delta \\le D - 1 )$ such that ", + "bbox": [ + 173, + 761, + 825, + 791 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/6d06bf20263ae024e876f3c19e74c97bf2dd11af84cea720d145559a4103724f.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( j , l _ { 1 } ) = \\bigr ( l _ { 1 } - ( l _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { l } ) _ { \\mathrm { m o d } N } = \\delta - r _ { l } , } \\\\ & { I ( j , l _ { 2 } ) = \\bigr ( l _ { 2 } - ( l _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 792, + 709, + 838 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now let us consider the following function. By results in Lemmas 2 and 3, we have ", + "bbox": [ + 173, + 844, + 720, + 861 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ff6aeece773c30e62c654efe19299d1727a880f7cf478b0f40c3c31a6c99f85b.jpg", + "text": "$$\n\\begin{array} { r l r } { { f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } ) = \\sum _ { k _ { 2 } = 0 } ^ { N - 1 } \\sum _ { l _ { 2 } = 0 } ^ { N - 1 } ( { \\bf B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) ) ^ { 2 } } } \\\\ & { } & { = f _ { 1 } + f _ { 2 } + f _ { 3 } + f _ { 4 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 863, + 702, + 929 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 104, + 217, + 117 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/fd08ae559e53588942e4af2d7701a9d51b89b66d5531660f5f63a701ecdf5c3e.jpg", + "text": "$$\n\\begin{array} { r l r } { { f _ { 1 } = \\sum _ { k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } ) } \\sum _ { l _ { 2 } \\in \\mathcal { L } ( l _ { 1 } ) } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } } } \\\\ & { } & \\\\ & { } & { f _ { 2 } = \\sum _ { { k _ { 2 } \\in \\widehat { \\mathcal { K } } ( k _ { 1 } ) } } \\sum _ { l _ { 2 } \\in \\mathcal { L } ( l _ { 1 } ) } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } } \\\\ & { } & \\\\ & { } & { f _ { 3 } = \\sum _ { { k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } ) } } \\sum _ { l _ { 2 } \\in \\widehat { \\mathcal { L } } ( l _ { 1 } ) } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } } \\\\ & { } & \\\\ & { } & { f _ { 4 } = \\sum _ { { k _ { 2 } \\in \\widehat { \\mathcal { K } } ( k _ { 1 } ) } } \\sum _ { l _ { 2 } \\in \\widehat { \\mathcal { L } } ( l _ { 1 } ) } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 118, + 679, + 282 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Combining equations (39), (41), (43) and (45), we obtain ", + "bbox": [ + 173, + 286, + 550, + 303 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/790001e0586d81602cc7cd2a390b494bc22037cbc5f249112b6e40786cc8bcd3.jpg", + "text": "$$\nf _ { 1 } = \\sum _ { r _ { k } = 0 } ^ { D - 1 } \\sum _ { r _ { l } = 0 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } , \\delta _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } - r _ { l } ) \\Big ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 309, + 715, + 356 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Combining (40), (41), (44) and (45), we obtain ", + "bbox": [ + 173, + 363, + 482, + 378 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/8fc6e447a2891b3a4d47c636c360b84a030a171d4b12b94efea3a75951f50c73.jpg", + "text": "$$\nf _ { 2 } = \\sum _ { r _ { k } = 1 } ^ { D - 1 } \\sum _ { r _ { l } = 0 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } , \\delta _ { l } - r _ { l } ) \\Big ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 386, + 714, + 431 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Combining (39), (42), (43) and (46), we obtain ", + "bbox": [ + 173, + 439, + 482, + 454 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/fae52290b43843a26fe744429cc44cae4ea1f79a73222d68806b9cab153ce9bf.jpg", + "text": "$$\nf _ { 3 } = \\sum _ { r _ { k } = 0 } ^ { D - 1 } \\sum _ { r _ { l } = 1 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } , \\delta _ { l } - r _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } ) \\Big ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 460, + 714, + 507 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Combining (40), (42), (44) and (46), we obtain ", + "bbox": [ + 173, + 515, + 482, + 530 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/c371ba2864143c47dc29238f9f079c304663375ed10edb60886803e159d666c2.jpg", + "text": "$$\nf _ { 4 } = \\sum _ { r _ { k } = 1 } ^ { D - 1 } \\sum _ { r _ { l } = 1 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } - r _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } , \\delta _ { l } ) \\Big ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 536, + 715, + 583 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "By the above explicit formulas of $f _ { i } , 1 \\le i \\le 4$ , we have $f _ { 1 } , f _ { 2 } , f _ { 3 } , f _ { 4 }$ are all independent of $k _ { 1 } , l _ { 1 }$ and $N$ . They are only related with $m _ { 1 } , m _ { 2 }$ for fixed $\\mathbf { d }$ and $\\mathbf { m }$ . Thus, we are able to denote $f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } )$ as $f ( m _ { 1 } , m _ { 2 } )$ for simplicity. Consequently, ", + "bbox": [ + 174, + 598, + 826, + 642 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/caac35b84b1bd105e0982f9c3b9fae3b13d665df447b4dcf9be820bddfd4233c.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c r i } } ^ { N } \\| _ { F } ^ { 2 } = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { m = 1 } ^ { M } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { N - 1 } \\sum _ { m = 1 } ^ { M } \\sum _ { m = 2 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { M } \\sum _ { \\iota _ { 1 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\cdot N ^ { 2 } \\cdot \\displaystyle \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) = \\sum _ { m _ { 1 } = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 648, + 797, + 833 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Thus, $\\begin{array} { r } { \\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } } \\end{array}$ is dependent of $N$ : ", + "bbox": [ + 173, + 838, + 480, + 858 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/8915c348c645e22d629d2e67dbaab0c3c9923d6d3fd82631bfbb25a8644ab021.jpg", + "text": "$$\n\\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } = \\frac { 1 } { ( 2 D - 1 ) ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\| _ { F } ^ { 2 } , \\quad \\forall N \\geq 2 D - 1 ,\n$$", + "text_format": "latex", + "bbox": [ + 238, + 864, + 756, + 898 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which implies $\\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ , $\\forall N \\geq 2 D - 1$ ", + "bbox": [ + 173, + 906, + 486, + 926 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.2 PROOF OF CONCLUSION 2. ", + "text_level": 1, + "bbox": [ + 174, + 103, + 403, + 118 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Before proving Conclusion 2, let us analyze the relationship between $\\mathbf { D } _ { \\mathrm { { c o n v } } } ^ { N }$ and DN+D−1. ", + "bbox": [ + 173, + 128, + 774, + 147 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Similar to $\\mathbf { D } _ { \\mathrm { c i r } }$ , we use $( i , j )$ as the row index and $( k , l , m )$ as the column index of $\\mathbf { D } _ { \\mathrm { c o n v } }$ . For $0 \\leq i , j \\leq N - 1 , 1 \\leq m \\leq M$ , ", + "bbox": [ + 173, + 151, + 823, + 181 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/16fc7ec23bda097cc935664e6a1177ae52d795156cd0ca1210e1baca7d4cff63.jpg", + "text": "$$\n\\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) = \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( i , j ; k , l , m ) = { \\left\\{ \\begin{array} { l l } { \\mathbf { d } _ { m } ( k - i , l - j ) , } & { 0 \\leq k - i , l - j \\leq D - 1 } \\\\ { 0 , } & { k , l { \\mathrm { ~ t a k e n ~ a s ~ o t h e r s } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 183, + 189, + 812, + 224 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Matrixmatrix $ { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N + D - 1 }$ is of dimensf dimension $( N + D - 1 ) ^ { 2 } \\times ( N + D - 1 ) ^ { 2 } M$ $0 \\leq i , j \\leq N + D - 2$ DNcon v i $( N ) ^ { 2 } \\times ( N + D - 1 ) ^ { 2 } M$ $0 \\leq i , j \\leq N - 1 .$ $\\mathbf { D } _ { \\mathrm { { c o n v } } } ^ { N }$ block in DN+D−1cir , i.e., ", + "bbox": [ + 173, + 234, + 825, + 284 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/e8dce765c0ec2409550713bc5bd222013c9cecd967ae351b6716da257aa24082.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } = \\left[ \\begin{array} { c } { \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } } \\\\ { \\Delta _ { \\mathbf { D } } ^ { N } } \\end{array} \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 284, + 576, + 319 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The matrix $\\Delta _ { \\mathbf { D } } ^ { N }$ is of dimension $( ( N + D - 1 ) ^ { 2 } - N ^ { 2 } ) \\times ( N + D - 1 ) ^ { 2 } M \\colon$ ", + "bbox": [ + 173, + 324, + 678, + 342 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ef11cbb7b67e4296efadc8d42f8747261ed351fc04425817089598a6d76b9123.jpg", + "text": "$$\n\\Delta _ { \\mathbf { D } } ^ { N } = \\left[ \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , \\cdot ) \\right] , \\quad ( i , j ) \\in \\mathcal { T } _ { \\Delta }\n$$", + "text_format": "latex", + "bbox": [ + 349, + 349, + 647, + 371 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 378, + 217, + 392 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/4b056d84ef547d19e14025bea4f92efcc7da5b92cced673b8e950c5b7fa85e23.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { Z } _ { \\Delta } = \\mathcal { Z } _ { 1 } \\cup \\mathcal { Z } _ { 2 } \\cup \\mathcal { Z } _ { 3 } } \\\\ & { \\mathcal { T } _ { 1 } = \\{ ( i , j ) | N \\le i \\le N + D - 2 , 0 \\le j \\le N - 1 \\} } \\\\ & { \\mathcal { T } _ { 2 } = \\{ ( i , j ) | 0 \\le i \\le N - 1 , N \\le j \\le N + D - 2 \\} } \\\\ & { \\mathcal { Z } _ { 3 } = \\{ ( i , j ) | N \\le i \\le N + D - 2 , N \\le j \\le N + D - 2 \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 390, + 696, + 463 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Similarly, ", + "bbox": [ + 173, + 465, + 240, + 481 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/e2ea999af7366ed6a17ad9df51b1ffdc13071896cebe8b19440de9bbf5177aa1.jpg", + "text": "$$\n\\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } = \\left[ \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\right] , \\quad \\Delta _ { \\mathbf { W } } ^ { N } = \\left[ \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot ; \\cdot ) \\right] , \\quad ( i , j ) \\in \\mathcal { T } _ { \\Delta } .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 489, + 743, + 523 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Then, ", + "bbox": [ + 173, + 531, + 214, + 546 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/d3cd9be27e671c38cf93d05e92e587fb28e6ae249c969dabe7c2b6edd1882864.jpg", + "text": "$$\n( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } = ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } + ( \\boldsymbol { \\Delta } _ { \\mathbf { D } } ^ { N } ) ^ { T } \\boldsymbol { \\Delta } _ { \\mathbf { W } } ^ { N } .\n$$", + "text_format": "latex", + "bbox": [ + 300, + 545, + 696, + 565 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma 4. For any $( i , j ) \\in \\mathcal { I } _ { \\Delta }$ , one has ", + "bbox": [ + 173, + 570, + 442, + 585 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/40e97ad7ae049f3b9db903876b87b8c949f1f3ed8fc61e7e134a4b9ef17adde1.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , : ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } , } \\\\ { \\| \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , : ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 592, + 611, + 636 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. Equation (35) implies that, for $( i , j ) \\in \\mathcal { T } _ { 1 } , 1 \\le m \\le M$ , ", + "bbox": [ + 173, + 652, + 594, + 670 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/1055f36a68b10e22f3c6c811e8c82c55a76702b97734d56352642ec04308f4a4.jpg", + "text": "$$\n\\begin{array} { r } { \\mathsf { \\Pi } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) = \\left\\{ \\begin{array} { l l } { \\mathbf { d } _ { m } ( k - i , l - j ) , } & { i \\le k \\le N + D - 2 , j \\le l \\le j + D - 1 } \\\\ { \\mathbf { d } _ { m } ( k - i + N + D - 1 , l - j ) , } & { 0 \\le k \\le i - N , j \\le l \\le j + D - 1 } \\\\ { 0 , } & { k , l \\mathrm { ~ t a k e n ~ a s ~ o t h e r s } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 678, + 838, + 731 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Thus, for any $( i , j ) \\in \\mathcal { T } _ { 1 }$ , ", + "bbox": [ + 174, + 739, + 343, + 756 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/e562a64c2e6428a594f9c99cb9df9ceedf3d426f922b428a61ce02e9def55d15.jpg", + "text": "$$\n\\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , \\cdot ) \\| _ { 2 } ^ { 2 } = \\sum _ { k = 0 } ^ { N + D - 2 } \\sum _ { l = 0 } ^ { N + D - 2 } \\sum _ { m = 1 } ^ { M } \\left| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) \\right| ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 230, + 763, + 769, + 808 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Similarly, ", + "bbox": [ + 173, + 815, + 240, + 830 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/d612a43357729aca2e9e7edf60fd828e809d80e82f3d8cbf62a3890adfea825a.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , \\cdot ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } , \\quad ( i , j ) \\in \\mathcal { T } _ { 2 } \\cup \\mathcal { T } _ { 3 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 830, + 674, + 851 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Equation (50) is proved. With the same argument, equation (51) is also proved. ", + "bbox": [ + 176, + 856, + 691, + 871 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma 5. If $N \\geq 2 D - 1$ , we have ", + "bbox": [ + 174, + 883, + 416, + 900 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/791fd665e4ca093a49aa43f01d01f0226ea1e31fa4f0242f77dbcbd1bc747c3a.jpg", + "text": "$$\n\\begin{array} { r } { \\| ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\| _ { F } ^ { 2 } \\leq \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) ( 2 D - 1 ) ^ { 2 } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 906, + 723, + 926 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. For simplicity, we denote two row vectors: ", + "bbox": [ + 174, + 103, + 503, + 118 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/0ed4d47e034e4e9fb972e11d615dbd40a34bdb797ab77a04de86bcf87addc9d6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { d } _ { i , j } \\triangleq { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\colon , \\colon ) \\in \\mathbb { R } ^ { 1 \\times ( N + D - 1 ) ^ { 2 } M } } \\\\ & { \\mathbf { w } _ { i , j } \\triangleq \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\colon , \\colon ) \\in \\mathbb { R } ^ { 1 \\times ( N + D - 1 ) ^ { 2 } M } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 126, + 656, + 171 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Then, ", + "bbox": [ + 173, + 178, + 215, + 193 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/d7107d113f061c453b1576a574967f28fc42d44d26311d86a98fc2b2d648db9e.jpg", + "text": "$$\n\\Vert ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\Vert _ { F } ^ { 2 } = \\bigg \\Vert \\sum _ { ( i , j ) \\in \\mathcal { Z } _ { \\Delta } } \\mathbf { d } _ { i , j } ^ { T } \\mathbf { w } _ { i , j } \\bigg \\Vert _ { F } ^ { 2 } = \\sum _ { ( i _ { 1 } , j _ { 1 } ) \\in \\mathcal { Z } _ { \\Delta } } \\sum _ { ( i _ { 2 } , j _ { 2 } ) \\in \\mathcal { Z } _ { \\Delta } } \\left. \\mathbf { d } _ { i _ { 1 } , j _ { 1 } } ^ { T } \\mathbf { w } _ { i _ { 1 } , j _ { 1 } } , \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\mathbf { w } _ { i _ { 2 } , j _ { 2 } } \\right. _ { F } ,\n$$", + "text_format": "latex", + "bbox": [ + 184, + 200, + 810, + 244 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 252, + 217, + 267 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/e38464ee4cfe60460468271fb54d7da1b1f5240ee9f64102bfd4e41215b37fe3.jpg", + "text": "$$\n\\mathbf { \\mathop { ' } d } _ { i _ { 1 } , j _ { 1 } } ^ { T } \\mathbf { \\boldsymbol { w } } _ { i _ { 1 } , j _ { 1 } } , \\mathbf { \\boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\mathbf { \\boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } \\bigg \\rangle _ { F } = \\operatorname { t r a c e } \\Bigl ( \\mathbf { w } _ { i _ { 1 } , j _ { 1 } } ^ { T } \\mathbf { \\boldsymbol { d } } _ { i _ { 1 } , j _ { 1 } } \\mathbf { \\boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\mathbf { \\boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } \\Bigr ) = \\bigl ( \\mathbf { \\boldsymbol { d } } _ { i _ { 1 } , j _ { 1 } } \\mathbf { \\boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\bigr ) \\cdot \\bigl ( \\mathbf { \\boldsymbol { w } } _ { i _ { 1 } , j _ { 1 } } \\mathbf { \\boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\bigr ) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 275, + 825, + 301 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Since ", + "bbox": [ + 173, + 309, + 214, + 323 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/36e13485b92043bd67fec88933d31cafc4ca6ab65654fe714aa0f9a6a641d04a.jpg", + "text": "$$\n\\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } = \\sum _ { k = 0 } ^ { N - 1 } \\sum _ { l = 0 } ^ { N - 1 } \\sum _ { m = 1 } ^ { M } = \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i _ { 1 } , j _ { 1 } ; k , l , m ) \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i _ { 2 } , j _ { 2 } ; k , l , m ) ,\n$$", + "text_format": "latex", + "bbox": [ + 228, + 329, + 767, + 375 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "with the same argument in Lemma 2, we have: $\\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\neq 0$ implies ", + "bbox": [ + 174, + 382, + 643, + 400 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/82306b2596bd1a1fe5be753ecbb54253acf905dde5cf2abcbebfe25d5960a295.jpg", + "text": "$$\n\\begin{array} { r l } & { i _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } \\triangleq \\{ i | 0 \\le ( i _ { 1 } - i ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\mathrm { o r } 0 \\le ( i - i _ { 1 } ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\} } \\\\ & { j _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } \\triangleq \\{ j | 0 \\le ( j _ { 1 } - j ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\mathrm { o r } 0 \\le ( j - j _ { 1 } ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 407, + 797, + 462 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 465, + 210, + 479 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/d92bb80359f5e8631b8a6315f945a8187284d1b7d90a99022adf8dda9444807c.jpg", + "text": "$$\n\\begin{array} { r l } { \\| ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\| _ { F } ^ { 2 } = } & { \\displaystyle \\sum _ { ( i _ { 1 } , j _ { 1 } ) \\in \\mathbb { Z } } \\displaystyle \\sum _ { \\Delta } \\sum _ { i _ { 2 } \\in \\mathbb { Z } _ { \\Delta } ^ { \\prime } } \\displaystyle \\sum _ { j _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } } ( \\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } ) \\cdot ( \\mathbf { w } _ { i _ { 1 } , j _ { 1 } } \\mathbf { w } _ { i _ { 2 } , j _ { 2 } } ^ { T } ) } \\\\ { \\leq } & { \\displaystyle \\sum _ { ( i _ { 1 } , j _ { 1 } ) \\in \\mathbb { Z } } \\displaystyle \\sum _ { \\Delta } \\sum _ { i _ { 2 } \\in \\mathbb { Z } _ { \\Delta } ^ { \\prime } } \\displaystyle \\sum _ { j _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } } \\\\ { } & { = | \\mathcal { Z } _ { \\Delta } | \\cdot | \\mathcal { I } _ { \\Delta } ^ { \\prime } | \\cdot | \\mathcal { I } _ { \\Delta } ^ { \\prime } | \\cdot \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } } \\\\ { } & { = \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) ( 2 D - 1 ) ^ { 2 } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 477, + 736, + 594 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where the inequality in the second line follows from (50) and (51). Inequality (52) is proved. ", + "bbox": [ + 166, + 598, + 779, + 614 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "With these preparations, we can prove Theorem 3, Conclusion 2 now. ", + "bbox": [ + 174, + 635, + 629, + 650 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof of Theorem 3, Conclusion 2. Define set ", + "bbox": [ + 173, + 671, + 477, + 688 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/1c7e9261265d7f20fbf76185a230fef6e2087f505c3a22a44a7e5df76270c7ea.jpg", + "text": "$$\n\\mathcal { W } _ { \\mathrm { n o r m a l } } = \\Big \\{ \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } \\Big | \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , \\forall m = 1 , \\cdots , M \\Big \\} .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 695, + 699, + 723 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Since $\\mathbf { d } \\in \\mathcal { W } _ { \\mathrm { n o r m a l } }$ , the set is nonempty: ", + "bbox": [ + 174, + 731, + 444, + 746 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/b8bfd965b18f857304ea2f27899df09bb7aa1b7065c5f92938d88d1483aa80b4.jpg", + "text": "$$\n\\mathcal { W } _ { \\mathrm { n o r m a l } } \\neq \\emptyset .\n$$", + "text_format": "latex", + "bbox": [ + 447, + 753, + 549, + 771 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Define functions $F _ { \\mathrm { c o n v } } ^ { N } : \\mathbb { R } ^ { D ^ { 2 } M } \\to \\mathbb { R } , F _ { \\mathrm { c i r } } ^ { N } : \\mathbb { R } ^ { D ^ { 2 } M } \\to \\mathbb { R } .$ ", + "bbox": [ + 173, + 779, + 552, + 797 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/1f8895a3cb9be8c1cba7feca9ff43ea2fa66de39856dbc3aab3b62f1186b7ebe.jpg", + "text": "$$\n\\begin{array} { r l } & { F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) = \\displaystyle \\frac { 1 } { N + D - 1 } \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) \\big ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) \\Big \\| _ { F } + \\imath \\gamma _ { \\mathrm { n o r m a l } } ( \\mathbf { w } ) } \\\\ & { ~ F _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) = \\displaystyle \\frac { 1 } { N } \\Big \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\Big \\| _ { F } + \\imath \\gamma _ { \\mathrm { n o r m a l } } ( \\mathbf { w } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 805, + 730, + 869 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "By the definitions of $\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } , \\mathcal { W } _ { \\mathrm { c i r } } ^ { N }$ , we have ", + "bbox": [ + 174, + 875, + 460, + 893 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/ef36f2230078fc726f29e72ee7c12bd901b75fa0d7335ea1dbbe64fa7800f15f.jpg", + "text": "$$\n\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\underset { \\mathbf { w } } { \\arg \\operatorname* { m i n } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) , \\quad \\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\underset { \\mathbf { w } } { \\arg \\operatorname* { m i n } } F _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } )\n$$", + "text_format": "latex", + "bbox": [ + 310, + 901, + 686, + 928 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Step 1: Proving $F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } )$ uniformly converges to $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } )$ on $X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } }$ for any compact set X ⊂ RD2M. ", + "bbox": [ + 168, + 101, + 825, + 135 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We arbitrarily choose such a compact set $X$ . Based on (47), (49) and (52), one has, for all w $\\in$ X ∩ Wnormal, ", + "bbox": [ + 171, + 141, + 823, + 172 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/455c5dd273976382938b323bc7f54acb1ae96b25d32ffbc54054d87de9ab1e8a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\| = | F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) - F _ { \\mathrm { c i r } } ^ { N + D - 1 } ( \\mathbf { w } ) | } \\\\ & { \\qquad = \\displaystyle \\frac { 1 } { N + D - 1 } \\Big | \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } \\Big \\| _ { F } - \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } \\big ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\Big \\| _ { F } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { 1 } { N + D - 1 } \\Big \\| ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\Big \\| _ { F } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { \\sqrt { \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) } ( 2 D - 1 ) } { N + D - 1 } \\| \\mathbf { d } \\| _ { 2 } \\| \\mathbf { w } \\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { ( 2 D - 1 ) \\sqrt { 2 ( D - 1 ) } } { \\sqrt { N + D - 1 } } \\| \\mathbf { d } \\| _ { 2 } \\| \\mathbf { w } \\| _ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 174, + 813, + 343 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Thus, there exists a constant $B > 0$ , which is independent of $N$ , such that ", + "bbox": [ + 174, + 345, + 658, + 361 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/4832443d38512316027642c072a501b7325bbfa969bc9aae1687ee7ea6b5dcd9.jpg", + "text": "$$\n| F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) | \\leq \\frac { B } { \\sqrt { N } } \\operatorname* { s u p } _ { \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } } \\| \\mathbf { w } \\| , \\quad \\forall \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } .\n$$", + "text_format": "latex", + "bbox": [ + 238, + 364, + 758, + 400 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Step 2: Proving $F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } )$ epigraphically converges8 to $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } )$ ", + "bbox": [ + 173, + 412, + 632, + 431 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We want to show, at each point w it holds that ", + "bbox": [ + 173, + 436, + 478, + 450 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/8f39f36af8809eab1def7e771bb9b2e78731f936aef06d8780023334d1939231.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { N \\infty } { \\operatorname* { l i m } \\operatorname* { i n f } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) \\geq F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\quad \\mathrm { f o r ~ e v e r y ~ s e q u e n c e ~ } \\mathbf { w } ^ { N } \\mathbf { w } } \\\\ & { \\underset { N \\infty } { \\operatorname* { l i m } \\operatorname* { s u p } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) \\leq F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\quad \\mathrm { f o r ~ s o m e ~ s e q u e n c e ~ } \\mathbf { w } ^ { N } \\mathbf { w } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 277, + 454, + 720, + 511 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Firstly, we prove (56). We arbitrarily pick a sequence $\\{ \\mathbf { w } ^ { N } \\} _ { N = 0 } ^ { \\infty }$ such that $\\mathbf { w } ^ { N } \\to \\mathbf { w }$ ", + "bbox": [ + 174, + 522, + 735, + 540 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "$\\notin \\mathcal { W } _ { \\mathrm { n o r m a l } }$ , o $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) = + \\infty$ . Since s, one h $\\mathcal { W } _ { \\mathrm { n o r m a l } }$ et, therfor all , $N ^ { + }$ such that, $\\mathbf { w } ^ { N } \\notin \\mathcal { W } _ { \\mathrm { n o r m a l } }$ $N \\geq N ^ { + }$ $F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) = + \\infty$ $N \\geq N ^ { + }$ ", + "bbox": [ + 176, + 545, + 826, + 579 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/8d41c772478cb00f31875d5369a85cce1ab56b9e27d18f455389d595d9c7c4ee.jpg", + "text": "$$\n\\operatorname* { l i m } _ { N \\to \\infty } \\operatorname* { i n f } _ { C \\mathrm { c o n v } } ( \\mathbf { w } ^ { N } ) = F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) = + \\infty .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 582, + 637, + 608 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "If $\\mathbf { v } \\in \\mathcal { W } _ { \\mathrm { n o r m a l } }$ , two cases should be considered. The first case is that any subsequences of $\\{ \\mathbf { w } ^ { N } \\} _ { N = 0 } ^ { \\infty }$ re not kept within . Then we have $\\mathcal { W } _ { \\mathrm { n o r m a l } }$ , i.e., there exists a $N ^ { + }$ such that $\\mathbf { w } ^ { \\mathbf { \\bar { \\boldsymbol { N } } } } \\notin \\mathcal { W } _ { \\mathrm { n o r m a l } }$ for $N \\geq N ^ { + }$ ", + "bbox": [ + 173, + 619, + 826, + 661 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/2128501bdf156f8e0bfef95cf8eaa7479b6736a5783ecd0b1a17d5f0c8213e69.jpg", + "text": "$$\n\\operatorname* { l i m } _ { N \\to \\infty } \\operatorname* { i n f } _ { C \\mathrm { c o n v } } ( \\mathbf { w } ^ { N } ) = + \\infty > F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 665, + 638, + 690 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "The second case is that there exists a subsequence $\\{ \\mathbf { w } ^ { N _ { k } } \\} _ { k = 0 } ^ { \\infty } \\subset \\{ \\mathbf { w } ^ { N } \\} _ { N = 0 } ^ { \\infty }$ such that ", + "bbox": [ + 173, + 695, + 735, + 713 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/154b776a6b0c46bf150971871c6c0dc9032b23846e9aadd348849eade1f2c752.jpg", + "text": "$$\n\\mathbf { w } ^ { N _ { k } } \\in { \\mathcal { W } } _ { \\mathrm { n o r m a l } } , \\quad \\forall k = 0 , 1 , 2 , \\cdots .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 718, + 625, + 736 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Since $\\mathbf { w } ^ { N }$ converges to $\\mathbf { w }$ , any subsequences should be Cauchy. Given any Cauchy sequence $\\{ \\mathbf { w } ^ { N _ { k } } \\} _ { k = 0 } ^ { \\infty }$ in finite dimensional Euclidean space, there exists a compact set $X$ such that ", + "bbox": [ + 173, + 741, + 825, + 771 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/064eea5194094bba723a9eefca86b266b0c0abe88bc131b720832801e4a24a57.jpg", + "text": "$$\n\\mathbf { w } ^ { N _ { k } } \\in X , \\quad \\forall k = 0 , 1 , 2 , \\cdot \\cdot \\cdot\n$$", + "text_format": "latex", + "bbox": [ + 398, + 776, + 598, + 795 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Let $B ^ { \\prime } = \\operatorname* { s u p } _ { \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } } \\left\\| \\mathbf { w } \\right\\|$ . By (55), we obtain ", + "bbox": [ + 173, + 799, + 521, + 818 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/6c2b6f500848a9480763a6e407221e6b12b40d0266e2f32ceac222eed272e7c8.jpg", + "text": "$$\n\\begin{array} { r l } { \\big | F _ { \\mathrm { c o n v } } ^ { N _ { k } } \\big ( \\mathbf { w } ^ { N _ { k } } \\big ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\big | \\le \\big | F _ { \\mathrm { c o n v } } ^ { N _ { k } } ( \\mathbf { w } ^ { N _ { k } } ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\big ( \\mathbf { w } ^ { N _ { k } } \\big ) \\big | + \\big | F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\big ( \\mathbf { w } ^ { N _ { k } } \\big ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\big ( \\mathbf { w } \\big ) \\big | } & { } \\\\ { \\le \\displaystyle \\frac { B B ^ { \\prime } } { \\sqrt { N _ { k } } } + \\big | F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\big ( \\mathbf { w } ^ { N _ { k } } \\big ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\big ( \\mathbf { w } \\big ) \\big | . } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 819, + 812, + 878 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For any \u000f > 0, by tinuity o F 2D−1cir the con for all f . ,a are able tosuch that $K > 0$ hat . T $| F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ^ { N _ { k } } ) -$ $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf w ) | \\ < \\ \\epsilon$ $\\operatorname* { m a x } ( K , K ^ { \\prime } )$ , we have $| F _ { \\mathrm { c o n v } } ^ { N _ { k } } ( \\mathbf { w } ^ { N _ { k } } ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) | < 2 \\epsilon$ $k \\geq K$ $K ^ { \\prime }$ , i.e., $N _ { K ^ { \\prime } } \\geq ( B B ^ { \\prime } / \\epsilon ) ^ { 2 }$ $k \\geq$ ", + "bbox": [ + 173, + 101, + 825, + 152 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/23a4d5941a54fdbc604d8b421137c364304d52836dd40ef652cd267b6f20d5d4.jpg", + "text": "$$\n\\operatorname* { l i m } _ { k \\infty } F _ { \\mathrm { c o n v } } ^ { N _ { k } } ( \\mathbf { w } ^ { N _ { k } } ) = F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 156, + 609, + 183 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "mulation point of The above conclusion holds for all subsequences $+ \\infty$ because $F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) = F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) = + \\infty$ k=0 normal. All the other accumulation points of for all $\\{ \\mathbf { w } ^ { N _ { k } } \\} _ { k = 0 } ^ { \\infty } \\subset \\mathcal { W } _ { \\mathrm { n o r m a l } }$ $\\mathbf { w } \\notin \\mathcal { W } _ { \\mathrm { n o r m a l } }$ . Thus, . $\\{ \\bar F _ { \\mathrm { c o n v } } ^ { \\bar { N } } ( \\mathbf { w } ^ { N } ) \\} _ { N = 0 } ^ { \\infty }$ $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } )$ is an accu- must ", + "bbox": [ + 173, + 189, + 825, + 238 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/874a1ff21ab2ba224ad5a9709a25eddfe9016eef6982e49611b41da05bcaae7c.jpg", + "text": "$$\n\\operatorname* { l i m } _ { N \\to \\infty } \\operatorname* { i n f } _ { C \\mathrm { c o n v } } ( \\mathbf { w } ^ { N } ) = F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) < + \\infty .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 242, + 638, + 267 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Secondly, we prove (57). We set $\\mathbf { w } ^ { N } = \\mathbf { w }$ for all $N = 0 , 1 , 2 , \\cdots$ . Then (57) is a direct result of (55). ", + "bbox": [ + 173, + 280, + 823, + 311 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Step 3: proving (25). Define ", + "bbox": [ + 173, + 325, + 380, + 340 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/da043a519a3b7a06d3db139c755df78f74c01ac4f2a8530ee78d3374df231094.jpg", + "text": "$$\nG ( \\mathbf { w } ) = \\left. ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\right. _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 345, + 611, + 368 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We want to show that $G ( \\mathbf { w } )$ is strongly convex. ", + "bbox": [ + 173, + 373, + 485, + 388 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "$\\tilde { \\mathbf { w } } _ { i } \\in \\mathbb { R } ^ { ( 2 D - 1 ) ^ { 2 } }$ be the $i ^ { \\mathrm { { t h } } }$ column of $\\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ ", + "bbox": [ + 173, + 395, + 529, + 414 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/d1542d8758b01561e0ad8f8175933d76c8775efef3523e7b835eaa335c3f2a09.jpg", + "text": "$$\n\\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } = \\left[ \\tilde { \\mathbf { w } } _ { 1 } , \\tilde { \\mathbf { w } } _ { 2 } , \\cdots , \\tilde { \\mathbf { w } } _ { ( 2 D - 1 ) ^ { 2 } M } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 367, + 419, + 629, + 446 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 450, + 210, + 465 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/f7fcb09c93c52506b8b5726c39a0dd94663b1e368d45dd6a33802c3649cf1e65.jpg", + "text": "$$\nG ( \\mathbf { w } ) = \\sum _ { i = 1 } ^ { ( 2 D - 1 ) ^ { 2 } M } ( \\tilde { \\mathbf { w } } _ { i } ) ^ { T } \\Big ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \\Big ) \\tilde { \\mathbf { w } } _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 462, + 666, + 507 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "$\\tilde { \\mathbf { w } } \\in \\mathbb { R } ^ { ( 2 D - 1 ) ^ { 4 } M }$ vectorize $\\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ ", + "bbox": [ + 173, + 510, + 467, + 527 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/fb60a26a284023482f376a31ced673659156e7be161d1b04739db51d55cb940a.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { { \\bf w } } = \\left[ ( \\tilde { \\bf w } _ { 1 } ) ^ { T } , ( \\tilde { \\bf w } _ { 2 } ) ^ { T } , \\cdots , ( \\tilde { \\bf w } _ { ( 2 D - 1 ) ^ { 2 } M } ) ^ { T } \\right] ^ { T } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 344, + 534, + 650, + 564 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Then $G ( \\mathbf { w } )$ can be written as a quadratic form of w˜ : ", + "bbox": [ + 173, + 569, + 519, + 584 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/0b4842ca5ea763c18dc300d7e995d5bca03d097a1c2f145ad40eddf4b3ae8cbb.jpg", + "text": "$$\nG ( \\mathbf { w } ) = \\tilde { \\mathbf { w } } ^ { T } Q \\tilde { \\mathbf { w } } ,\n$$", + "text_format": "latex", + "bbox": [ + 437, + 589, + 557, + 608 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 614, + 217, + 628 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/8623af2e47d753576f07a687ca7dab84157ae35c4cbeba6d91e2d349fd0a22e7.jpg", + "text": "$$\n\\begin{array} { r } { Q = \\left[ \\begin{array} { l l l } { \\left( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \\right) } & & \\\\ & { \\cdots } & \\\\ & & { \\left( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \\right) } \\end{array} \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 623, + 699, + 691 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "As long as at least one of the matrices {D2D−1cir,0 , $\\{ \\mathbf { D } _ { \\mathrm { c i r } , 0 } ^ { 2 D - 1 } , \\cdot \\cdot \\cdot , \\mathbf { D } _ { \\mathrm { c i r } , M - 1 } ^ { 2 D - 1 } \\}$ in on-singular, is positive d $\\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ is full row rank, which implies that D2D−1cir ( $\\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T }$ $Q$ ", + "bbox": [ + 168, + 712, + 826, + 747 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "The transform between w and $\\tilde { \\mathbf { w } }$ is linear. We denote the transform as $T$ , i.e., ", + "bbox": [ + 174, + 753, + 683, + 768 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/404fb526d4323b7bf96dc0bcc6d891f769afc1e29984af2637a0bb11230302a5.jpg", + "text": "$$\n\\tilde { \\mathbf { w } } = T \\mathbf { w } .\n$$", + "text_format": "latex", + "bbox": [ + 462, + 775, + 532, + 789 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "It’s trivial that $\\| \\tilde { \\mathbf { w } } \\| _ { 2 } ^ { 2 } = 0$ implies $\\| \\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\| _ { F } ^ { 2 } = 0 .$ . By the definition of $\\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ , $\\| \\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\| _ { F } ^ { 2 } = 0$ implies $\\| \\mathbf { w } \\| _ { 2 } ^ { 2 } = 0$ . Thus, linear operator $T$ is full column rank. Thus, $T ^ { T } Q T$ is positive definite, and ", + "bbox": [ + 173, + 795, + 826, + 839 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/6076e8bf61d6618cb3caba4ff468ef4f50d7ee57138ad5dcffb1307b652389ca.jpg", + "text": "$$\nG ( \\mathbf { w } ) = \\mathbf { w } ^ { T } ( T ^ { T } Q T ) \\mathbf { w }\n$$", + "text_format": "latex", + "bbox": [ + 416, + 837, + 580, + 856 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "is strongly convex. Then $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( { \\bf w } ) = \\sqrt { G ( { \\bf w } ) } + \\iota _ { \\mathcal { W } _ { \\mathrm { n o r m a l } } } ( { \\bf w } )$ has only one minimizer, i.e., $\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ involves only a unique element. ", + "bbox": [ + 174, + 858, + 825, + 888 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Now we check the conditions of Propositions 7.32(c) and 7.33 in (Rockafellar & Wets, 2009) to apply them. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "1. F N $F _ { \\mathrm { c o n v } } ^ { N } \\xrightarrow [ ] { \\mathrm { e } } F _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ . This is proved in Step 2. ", + "bbox": [ + 212, + 101, + 513, + 121 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "2. $F _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ vel bounded. Since must be level bounde $G ( \\mathbf { w } )$ is strongly convex, $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) ~ = ~ \\sqrt { G ( \\mathbf { w } ) } ~ +$ $\\iota _ { \\mathcal { W } _ { \\mathrm { n o r m a l } } } ( \\mathbf { w } )$ ", + "bbox": [ + 202, + 127, + 823, + 160 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "3. $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\not \\equiv + \\infty$ . Since $\\mathcal { W } _ { \\mathrm { n o r m a l } }$ is nonempty (54), dom $F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\neq \\emptyset$ , $F _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ s not con- i $+ \\infty$ ", + "bbox": [ + 209, + 166, + 820, + 198 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "$F _ { \\mathrm { c o n v } } ^ { N }$ $F _ { \\mathrm { c o n v } } ^ { N }$ ", + "bbox": [ + 205, + 205, + 816, + 223 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "5 . F 2D−1 and F Ncon are all lower semi-continuous and proper. This condition follows from the fact that the functions F 2D−1 and F N are all continuous functions defined on a nonempty closed convex domain $\\mathcal { W } _ { \\mathrm { n o r m a l } }$ . ", + "bbox": [ + 210, + 231, + 825, + 279 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Applying Proposition 7.32(c), we have $\\{ F _ { \\mathrm { c o n v } } ^ { N } \\}$ is eventually level bounded. If we arbitrarily pick a $\\mathbf { w } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }$ and let $\\mathbf { w } _ { \\mathrm { c i r } }$ convbe the unique point in $\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }$ . Applying Proposition 7.33, we have $\\mathbf { w } ^ { N } \\to \\mathbf { w } _ { \\mathrm { c i r } }$ . Bsets o: ts, 2009), we obtain the convergence of the. $\\{ \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } \\}$ $\\begin{array} { r } { \\operatorname* { l i m } _ { N \\to \\infty } \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } } \\end{array}$ ", + "bbox": [ + 173, + 289, + 826, + 354 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "D DISCUSSION OF DEFINITION 2 (11) ", + "text_level": 1, + "bbox": [ + 173, + 371, + 504, + 388 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In this section, we want to numerically show that, given typical $\\mathbf { D }$ and $s$ , there is a $\\bar { \\sigma } _ { \\operatorname* { m i n } } > 0$ such that a random generated matrix $\\mathbf { W } \\in \\mathbf { \\bar { \\mathcal { W } } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } )$ . However, given $\\mathbf { D }$ and $\\mathbf { W }$ , it’s intractable to completely check (11): ", + "bbox": [ + 176, + 401, + 826, + 445 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/07141b11e27bd14a85e864e17e896eb6aeae5c72a3f3d77710ec420be89f17eb.jpg", + "text": "$$\n\\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } , \\forall \\mathbb { S } \\mathrm { ~ w i t h ~ } 2 \\leq | \\mathbb { S } | \\leq s .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 446, + 687, + 474 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The reason is that there are extremely large amount of possible $\\mathbb { S }$ s. For example, we take $M =$ $2 5 0 , N = 5 0 0 , s = 5 0$ . There are totally ", + "bbox": [ + 178, + 477, + 821, + 505 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/f6643be7dea0c01988edc5faba3ed58b1fd06155a47ecc0ffae96d841bf2d70d.jpg", + "text": "$$\n\\binom { 5 0 0 } { 5 0 } + \\binom { 5 0 0 } { 4 9 } + \\cdot \\cdot \\cdot + \\binom { 5 0 0 } { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 385, + 507, + 612, + 542 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "possible $\\mathbb { S } s$ satisfying $2 \\leq | \\mathbb { S } | \\leq s$ . It’s impossible to check (11) on all possible $\\mathbb { S } s$ ", + "bbox": [ + 173, + 545, + 715, + 560 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Instead of checking all possible $\\mathbb { S } s$ , we sample $5 0 0 0 \\mathbb { S } \\mathrm { s }$ from the whole set: ", + "bbox": [ + 173, + 565, + 663, + 580 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/860264e3b1bc307d4d39e54d56156661aa9da6effedd3a98b89505a6f08bb178.jpg", + "text": "$$\n\\mathcal S ^ { \\prime } \\subset S = \\{ \\mathbb S : \\mathbb S \\subset \\{ 1 , 2 , \\cdots , 5 0 0 \\} | 2 \\leq | S | \\leq s \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 583, + 665, + 602 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where $S ^ { \\prime }$ is the set of all the samples. Then we estimate $\\bar { \\sigma } _ { \\mathrm { m i n } }$ with the following quantity: ", + "bbox": [ + 169, + 603, + 753, + 618 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/93eb1491bde8aa9fe638becf5b448d70ec1795a5b8a231562e871594acb35687.jpg", + "text": "$$\n\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } ) = \\operatorname* { m i n } _ { \\mathbb { S } \\in \\mathcal { S } ^ { \\prime } } \\left\\{ \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 621, + 655, + 648 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Furthermore, we use the same $\\mathbf { D }$ as that in Section 5 and generate $1 0 0 0 ~ \\mathbf { W } \\mathbf { s }$ with each entry i.i.d sampled from the normal distribution. Then we normalize each column of the generated Ws. This technique is commonly used in sparse coding. Finally, we report the distribution of $\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )$ with the fixed $\\mathbf { D }$ and the 1000 sampled Ws in Figure 5. ", + "bbox": [ + 173, + 651, + 825, + 708 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Figure 5 demonstrates that, with the fixed $\\mathbf { D }$ , most of the random generated Ws have a $\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )$ within the interval [0.25, 0.35]. Thus, the numerical results support our claim: with high probability, a random generated W satisfies ", + "bbox": [ + 173, + 713, + 825, + 756 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/5710d2058752b6b4c88c514c9dd9973f4677cbad7589f17042c8a0c2e8d5fd21.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbb { S } \\in \\mathcal { S } } \\left\\{ \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\right\\} \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } > 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 344, + 758, + 650, + 787 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "that is, $\\mathbf { W } \\in \\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } )$ ", + "bbox": [ + 173, + 791, + 361, + 808 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "E EFFICIENT ALGORITHM TO CALCULATE ANALYTIC WEIGHTS ", + "text_level": 1, + "bbox": [ + 171, + 827, + 705, + 843 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "E.1 AN EFFICIENT ALGORITHM TO SOLVE (16) ", + "text_level": 1, + "bbox": [ + 173, + 856, + 511, + 872 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In this section, we introduce an algorithm to solve (16) (we copy (16) below to facilitate reading): ", + "bbox": [ + 169, + 882, + 810, + 898 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/54e994429acc9adf0525b7d249640894e7f774f59fc7639483fecff56ad39cdf.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\left\\| \\mathbf { W } ^ { T } \\mathbf { D } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( \\mathbf { W } _ { : , m } \\right) ^ { T } \\mathbf { D } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M ,\n$$", + "text_format": "latex", + "bbox": [ + 267, + 900, + 725, + 928 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/212353f30a28fcf8c4d3a4fb864f91d09298479d20268f8523dd485628469f57.jpg", + "image_caption": [ + "Figure 5: Discussion of Definition 2: distribution of $\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )$ on random generated Ws. " + ], + "image_footnote": [], + "bbox": [ + 250, + 104, + 750, + 250 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "By the definition of the Frobenius norm, it holds that ", + "bbox": [ + 173, + 300, + 522, + 315 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/3521703d63a855830bfac0099139782dc210146e4a0443fcf2b2a55ea229b331.jpg", + "text": "$$\n\\| \\mathbf { W } ^ { T } \\mathbf { D } \\| _ { F } ^ { 2 } = \\| ( \\mathbf { W } ^ { T } \\mathbf { D } ) ^ { T } \\| _ { F } ^ { 2 } = \\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 320, + 643, + 340 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Thus, the above problem is equivalent with ", + "bbox": [ + 176, + 345, + 459, + 361 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/fd51096bf293bfd8357e5592ab63420954a8148a0a1dd7f7661ed556112c8b3e.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\left\\| \\mathbf { D } ^ { T } \\mathbf { W } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( \\mathbf { D } _ { : , m } \\right) ^ { T } \\mathbf { W } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 364, + 727, + 393 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We apply projected gradient descent (PGD) to solve the above problem. The gradient of $\\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 }$ is $\\nabla \\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 } = \\mathbf { D } \\mathbf { D } ^ { T } \\mathbf { W }$ . Denote the set by ", + "bbox": [ + 173, + 400, + 823, + 431 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/b041b6efdf8d30dd617d402f804c675234db7280653b230fc9008f9cf73b613e.jpg", + "text": "$$\n\\mathcal { W } = \\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } | ( \\mathbf { D } _ { : , m } ) ^ { T } \\mathbf { W } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M . \\}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 436, + 709, + 457 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Then the projection onto $\\mathcal { W }$ can be calculated by ", + "bbox": [ + 174, + 462, + 493, + 476 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "$\\mathrm { \\mathrm { \\mathrm { ~ \\ p ~ } } } _ { \\mathrm { r o j } _ { \\mathcal { W } } } ( { \\bf W } ) = { \\bf W } + \\Delta { \\bf W } , \\Delta { \\bf W } = \\left[ ( 1 - ( { \\bf D } _ { : , 1 } ) ^ { T } { \\bf W } _ { : , 1 } ) { \\bf W } _ { : , 1 } , \\ \\cdots , \\ ( 1 - ( { \\bf D } _ { : , M } ) ^ { T } { \\bf W } _ { : , M } ) { \\bf W } _ { : , M } \\right]$ With these formulas, we are able to write down the PGD, which is listed in Algorithm 1. ", + "bbox": [ + 176, + 479, + 823, + 520 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Algorithm 1: Projected gradient descent for solving (16) ", + "text_level": 1, + "bbox": [ + 173, + 537, + 549, + 553 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Input: Dictionary D ∈ RN×M . \nInitialize: Let $\\dot { \\mathbf { W } } ^ { 0 } = \\mathbf { D }$ . ", + "bbox": [ + 173, + 556, + 385, + 584 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "1 for $j = 0 , 1 , 2 , \\ldots$ until convergence do ", + "bbox": [ + 163, + 585, + 437, + 599 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "3 end ", + "text_level": 1, + "bbox": [ + 160, + 623, + 202, + 636 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Output: $\\mathbf { W } ^ { J }$ , where $J$ is the last iterate. ", + "bbox": [ + 173, + 638, + 441, + 652 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In each step, calculating the gradient has the complexity of $O ( N ^ { 2 } M )$ because $\\mathbf { D } \\mathbf { D } ^ { T }$ can be precomputed. Calculating the projection takes $O ( N M )$ time consumptions. Due to the objective function to minimize in (16) is restricted strongly convex, Algorithm 1 is linear convergent (Zhang & Cheng, 2015). To get an $\\epsilon$ -accurate solution, PGD takes ${ \\cal O } ( \\log ( 1 / \\epsilon ) )$ steps. Thus, the complexity of Algorithm 1 is ${ \\cal O } ( \\log ( 1 / \\epsilon ) N ^ { 2 } M )$ . We should note that the bounds given in Table 1 are the number of parameters to train, not the training complexity. The training complexity can be estimated by “Number of iterations $\\times$ complexity of back-propagation”, i.e., $\\bar { O } ( I B \\bar { K } N \\bar { M } )$ ,where $I$ is the number of iterations for training, $B$ is the batch size , and $K$ is the number of layers. Actually, Algorithm 1 (Stage 1) only takes a few seconds on an example of $\\mathbf { D } : 2 5 0 \\times 5 0 0$ , while the training process (Stage 2) of, for example, ALISTA, takes around 0.1 hours. ", + "bbox": [ + 173, + 680, + 825, + 821 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "E.2 AN EFFICIENT ALGORITHM TO SOLVE (24) ", + "text_level": 1, + "bbox": [ + 173, + 837, + 511, + 852 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In this section, we introduce an algorithm to solve (24) (we copy (24) below to facilitate reading): ", + "bbox": [ + 171, + 862, + 810, + 878 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/9fe9c001e54b9c083124e44b29b653f316878f14140e9920e36660cd1f03207d.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } } \\Big \\| \\big ( \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\big ) ^ { T } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\Big \\| _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 883, + 648, + 928 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Similarly, by (58), the above problem is equivalent with ", + "bbox": [ + 173, + 103, + 540, + 118 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/3fd8c908a02a39c82d31ea67e04659f56160e67538ffb9a7fbed32ea7475ca81.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } } \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\big ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\Big \\| _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 125, + 650, + 170 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Since the circular convolution is very efficient to calculate in the frequency domain, we consider solving (59) utilizing the fast Fourier transform (FFT). ", + "bbox": [ + 174, + 174, + 825, + 203 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Firstly, we introduce the operators $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) , \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } )$ in the frequency domain. To simplify the notation, we denote the operators as $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }$ and $\\mathbf { W } _ { \\mathrm { c i r } } ^ { N }$ respectively. Let $\\mathcal { F }$ be the FFT operator. Thus, ${ \\bf b } = { \\bf D } _ { \\mathrm { c i r } } ^ { N } { \\bf x }$ is equivalent with ", + "bbox": [ + 173, + 208, + 825, + 255 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/039ad5f5277ea925816fb197cc37b4eca4e1c4dc9e0e4bae721f0c81c9e90458.jpg", + "text": "$$\n\\mathcal { F } \\mathbf { b } = \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\mathcal { F } \\mathbf { x } .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 253, + 578, + 272 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Let $\\hat { \\mathbf { b } } = \\mathcal { F } \\mathbf { b } , \\hat { \\mathbf { x } } = \\mathcal { F } \\mathbf { x }$ be the frequency domain signals, let $\\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H }$ be the frequency domain operator. The above equation is: ", + "bbox": [ + 171, + 276, + 825, + 306 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/9d261e7075b84fbff77469776a876f02d5da1d4123f2b75134300775b6f96731.jpg", + "text": "$$\n\\hat { \\mathbf { b } } = \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } \\hat { \\mathbf { x } } .\n$$", + "text_format": "latex", + "bbox": [ + 457, + 313, + 539, + 333 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "The frequency domain operator $\\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N }$ is much cheaper to calculate than the operator $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }$ in the spacial domain because it is block diagonal (Wohlberg, 2016). Specifically, we zero pad d to $N \\times N$ and do FFT: $\\hat { \\mathbf { d } } _ { m } = \\mathrm { F F T } \\left( \\mathrm { z e r o p a d } ( \\mathbf { d } _ { m } , N - D ) \\right)$ , then the above operator can be explicitly written as: ", + "bbox": [ + 173, + 339, + 825, + 398 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/12be63c474cd1c81de195338a6598529dc8968cca41de5de787ffc3f770437e2.jpg", + "text": "$$\n\\hat { \\mathbf { b } } = \\sum _ { m = 1 } ^ { M } \\overline { { \\hat { \\mathbf { d } } _ { m } } } \\odot \\hat { \\mathbf { x } } _ { m } ,\n$$", + "text_format": "latex", + "bbox": [ + 429, + 395, + 565, + 438 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "where ¯· means complex conjugate. This is due to $\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }$ is actually cross-correlation, not convolution (see (18)). Cross-correlation is equal to the transpose of convolution. Thus, there should be complex conjugate in the frequency domain. ", + "bbox": [ + 174, + 440, + 825, + 484 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Further, since ", + "bbox": [ + 173, + 491, + 266, + 505 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/e12680ad09d0d0f98d4aedbd125c2b7f2850f2946a995d425347c5adbe4b1daa.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| ( \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } ) ^ { H } \\hat { \\mathbf { W } } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } = \\| ( \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } ) ^ { H } \\mathcal { F } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\| _ { F } ^ { 2 } = \\| \\mathcal { F } ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\| _ { F } ^ { 2 } } \\\\ & { \\qquad = \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 222, + 510, + 776, + 574 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "problem (59) is equivalent with ", + "bbox": [ + 173, + 578, + 382, + 592 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/b4cea4b4e9874943a856d3f0ab81f0a268ba37f82054674be17328ed0b36d38c.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } } \\Big \\| \\big ( \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } \\big ) ^ { H } \\hat { \\mathbf { W } } _ { \\mathrm { c i r } } ^ { N } \\Big \\| _ { F } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 598, + 624, + 643 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "which can be efficiently solved by the frequency domain ISTA in (Liu et al., 2017). The details are outlined in Algorithm 2. ", + "bbox": [ + 173, + 648, + 825, + 678 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "F VISUALIZATION OF THE ANALYTIC CONVOLUTIONAL WEIGHTS ", + "text_level": 1, + "bbox": [ + 173, + 698, + 730, + 714 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Fig. 6 visualizes the dictionary d A-LISTA simulation of Section 5 $( 7 \\times 7 \\times 6 4 )$ and the weights ned by Algorith $\\tilde { \\mathbf { w } } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 1 3 }$ , used in the convolutionalendix E.2. ", + "bbox": [ + 169, + 727, + 825, + 758 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "G ALGORITHM DETAILS OF TRAINING ROBUST ALISTA ", + "text_level": 1, + "bbox": [ + 174, + 776, + 663, + 794 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "G.1 MODEL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 174, + 809, + 388, + 824 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Inspired by the a similar unrolling and truncating fashion in LISTA, we can approximately solve the coherence minimization problem (16) using a similar finite-layer neural network that is unfolded from iterative algorithms. Because the linear constraints in (16) are hard to enforce in deep neural networks, we first relax it to the following form: ", + "bbox": [ + 174, + 834, + 825, + 891 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/caac860c850f61fc6948bd272dd22b53e02f403f87c096e91c207278bd3fb79c.jpg", + "text": "$$\n\\underset { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } { \\arg \\operatorname* { m i n } } \\left\\| \\mathbf { Q } \\odot ( \\mathbf { D } ^ { T } \\mathbf { W } - \\pmb { I } _ { M } ) \\right\\| _ { F } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 897, + 617, + 928 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Algorithm 2: Frequency-domain ISTA for solving (24) ", + "text_level": 1, + "bbox": [ + 174, + 107, + 540, + 122 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Input: Dictionary $\\mathbf { d } = [ \\mathbf { d } _ { 1 } , \\boldsymbol { \\cdot \\cdot \\cdot } , \\mathbf { d } _ { M } ] ^ { T } , \\mathbf { d } _ { m } \\in \\mathbb { R } ^ { D ^ { 2 } } , m = 1 , 2 , \\cdots , M .$ Initialize: Let $\\mathbf { w } ^ { \\bar { 0 } } = \\mathbf { d }$ . \nfor $j = 0 , 1 , 2 , \\ldots$ until convergence do ", + "bbox": [ + 169, + 126, + 648, + 170 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "2 ", + "bbox": [ + 160, + 172, + 169, + 183 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Zeropad and FFT: ", + "bbox": [ + 197, + 171, + 318, + 184 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/09df2bd333921c5d313f19743176f0c414022ae4227ba9af26f40218a6c20151.jpg", + "text": "$$\n\\hat { \\mathbf { w } } _ { m } ^ { j } = \\mathrm { F F T } \\Big ( \\mathrm { z e r o p a d } \\big ( \\mathbf { w } _ { m } ^ { j } , N - D \\big ) \\Big ) , \\quad m = 1 , \\cdots , M .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 193, + 700, + 219 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "3 Compute frequency domain gradient: ", + "bbox": [ + 155, + 227, + 446, + 242 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/e9b52c96d73939aab4bd1d75425f36193dbe88a514d4e478ea983641adab8fa0.jpg", + "text": "$$\n( \\nabla f ) _ { m } = \\Big ( \\sum _ { m = 1 } ^ { M } \\hat { \\mathbf { d } } _ { m } \\odot \\bar { \\hat { \\mathbf { d } } } _ { m } \\Big ) \\odot \\hat { \\mathbf { w } } _ { m } ^ { j } , \\quad m = 1 , \\cdots , M ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 248, + 696, + 292 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where ¯· represents the conjugate of a complex number. ", + "bbox": [ + 207, + 299, + 563, + 313 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "4 ", + "bbox": [ + 160, + 314, + 169, + 325 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Compute the next iterate: ", + "bbox": [ + 196, + 313, + 366, + 325 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/ea655e912f8e0a8ebc4ea511abcc242099bf9a353a7fbb6a750f5ccd1a5eefa7.jpg", + "text": "$$\n\\mathbf { w } _ { m } ^ { j + 1 } = \\mathrm { P r o j } _ { \\mathcal { W } _ { \\mathrm { n o r m a l } } } \\Big ( \\mathrm { I F F T } \\big ( \\hat { \\mathbf { w } } _ { m } ^ { j } - \\boldsymbol { \\eta } ( \\nabla f ) _ { m } \\big ) \\Big ) , \\quad m = 1 , \\cdots , M ,\n$$", + "text_format": "latex", + "bbox": [ + 294, + 334, + 730, + 362 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where the set $\\mathcal { W } _ { \\mathrm { n o r m a l } }$ is defined in (53). ", + "bbox": [ + 205, + 368, + 477, + 383 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "5 end ", + "text_level": 1, + "bbox": [ + 160, + 382, + 202, + 393 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Output: $\\mathbf { w } ^ { J }$ , where $J$ is the last iterate. ", + "bbox": [ + 173, + 397, + 436, + 411 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/e098e2cc3cfc2da2fdab50ba85dada72320489488e18bbff5cca05da392227dc.jpg", + "image_caption": [ + "Figure 6: A visualization of convolutional kernels d and $\\tilde { \\mathbf { w } }$ , which is obtained by Algorithm 2 and used in the convolutional A-LISTA. w˜ keeps the high-frequency texture in d. The support of w is small, most of the pixels in w are zeros. Then the coherence between shifted $\\mathbf { d }$ and w is nearly 0. " + ], + "image_footnote": [], + "bbox": [ + 245, + 453, + 787, + 643 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where $\\odot$ is the Hadamard product and $\\mathbf { Q }$ is a weight matrix that put more penalty on errors on diagonals, because entries on the diagonal will be far smaller than off-diagonal. The above relaxed coherence minimization can be solved using the gradient descent algorithm: ", + "bbox": [ + 174, + 733, + 825, + 776 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/2d610c3086158f8858e233e0c8c14250777efaf99dea3ba0fd386dad87548ef9.jpg", + "text": "$$\n\\mathbf { W } ^ { ( k + 1 ) } = \\mathbf { W } ^ { ( k ) } - \\gamma ^ { ( k ) } \\mathbf { D } ( \\mathbf { Q } ^ { 2 } \\odot ( \\mathbf { D } ^ { T } \\mathbf { W } ^ { ( k ) } - \\pmb { I } _ { M } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 786, + 678, + 806 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "By unfolding (61) and truncate to $K$ steps, and considering the $\\gamma ^ { ( k ) } \\mathbf { D } ^ { T }$ outside the residual as learnable parameters $\\mathbf { B }$ , we will have a deep neural network $\\mathbf { W } = E ( \\mathbf { D } )$ as a coherence minimizer. We call it a Stage 1 encoder as it encodes a dictionary $\\mathbf { D }$ into a weight matrix, that can be used in the Stage 2 of ALISTA, refered as a decoder. One layer of this model is shown in Fig. 7(a). ", + "bbox": [ + 174, + 818, + 825, + 876 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "The illustration of the whole feed-forward robust model is shown in Fig. 7(b). The two parts, the encoder and the decoder, can be jointly trained to gain the most from data-driven learning. We further adopt pre-training and curriculum learning to stabilize training, as to be discussed below. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/6776e69f5cec6be82f55156bb9af67711806163f340570dab157b388fb069086.jpg", + "image_caption": [ + "Figure 7: Feed-Forward Analytic LISTA. " + ], + "image_footnote": [], + "bbox": [ + 191, + 104, + 816, + 257 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "G.2 MODEL TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 316, + 348, + 330 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "To stabilize the training process, we train the model in two stages: the pre-training stage and curriculum (joint) training stage. ", + "bbox": [ + 176, + 344, + 823, + 372 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Pre-Training Stage. We first pre-train the encoder and the decoder individually. The pre-training of the decoder, e.g., ALISTA, follows the standard training procedure in Section 5.1, without bothering the perturbations of $_ { D }$ . On the other hand, the encoder will always see perturbed dictionaries $\\tilde { \\cal D } = { \\cal D } + \\varepsilon _ { \\cal D }$ , where $\\varepsilon _ { D }$ ’s entries are sampled from i.i.d. normal distribution with zero mean and $\\sigma _ { p r e } ^ { 2 }$ variance, and update its weight to minimize loss function defined by (60). The $\\sigma _ { p r e }$ is a hyperparameter that we manually select for the pre-training stage, with a default value of 0.01. We use an exponentially decaying learning rate for encoder pre-training with an initial value $\\alpha _ { p r e } = 1 0 ^ { - 4 }$ . ", + "bbox": [ + 173, + 390, + 825, + 491 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Curriculum (Joint) Training Stage. After the pre-training stage, we concatenate these two parts and do joint training. However, a direct end-to-end tuning was observed to cause much instability, due to the randomness in weights. Inspired by the curriculum learning technique, we first perturb the dictionaries with smaller standard deviations and gradually increase the perturbation level during training. Specifically, starting from a small standard deviation $\\sigma ^ { t } = \\sigma ^ { 0 }$ , the curriculum joint training procedure repeats the routine below: ", + "bbox": [ + 174, + 508, + 825, + 592 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "• First uniformly sample a batch of standard deviations $\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { B _ { D } }$ from $[ 0 , \\sigma ^ { t } ]$ , where $B _ { D }$ is the batch size for perturbations of the original dictionary $\\mathbf { D }$ . Use the sampled standard deviations to sample $B _ { D }$ perturbations and apply to $\\mathbf { D }$ and then normalized them to get $\\{ \\tilde { \\bf D } \\} _ { i = 1 } ^ { B _ { D } }$ . \n• Sample a batch of sparse codes $\\{ \\mathbf { x } _ { j } \\} _ { j = 1 } ^ { B _ { x } }$ from a pre-defined Gaussian-Bernoulli distribution; the supports of the sparse codes are decided i.i.d. by a Bernoulli distribution to have around $1 0 \\%$ non-zero entries; and the magnitudes are sampled from i.i.d. standard Gaussian. $B _ { x }$ is the batch size for sparse codes in training. \n• Measure $\\mathbf { y } _ { i , j } = \\tilde { \\mathbf { D } } _ { i } \\mathbf { x } _ { j }$ . Then $( \\mathbf { y } _ { i , j } , \\mathbf { x } _ { j } , \\tilde { \\mathbf { D } } _ { i } )$ forms a tripelet of training sample. Note that only Robust ALISTA needs $\\tilde { \\bf D } _ { i }$ as part of the training samples. \n• Feed in the data and update the encoder and decoder with learning rates $\\alpha _ { e }$ and $\\alpha _ { d }$ , using the Adam Optimizer, respectively. \n• Increase $\\sigma ^ { t }$ to the next larger value, after $C$ training batches. \n• Repeat the above steps, until the value of $\\sigma ^ { t }$ exceeds the pre-defined $\\sigma _ { m a x }$ , that represents the maximal standard deviation to sample the dictionary perturbation. ", + "bbox": [ + 214, + 607, + 825, + 866 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "In the experiment, we have $B _ { D } = 4$ , $B _ { x } = 1 6$ , $C = 5 0 0 0 0$ , $\\alpha _ { e } = 1 0 ^ { - 6 }$ , $\\alpha _ { d } = 1 0 ^ { - 4 }$ and $\\sigma _ { m a x } \\in$ $\\{ 0 . 0 2 , 0 . { \\bar { 0 } } 3 \\}$ and $\\sigma ^ { t }$ is obtained by linearly interpolating between $[ 0 , \\sigma _ { m a x } ]$ for $L - 1$ times. We choose $L = 5$ ; hence $\\sigma ^ { t }$ takes $\\scriptstyle { \\frac { i } { 5 } } \\sigma _ { m a x } , i = 1 , 2 , \\ldots , 5$ in order. ", + "bbox": [ + 174, + 880, + 825, + 925 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "H RESULTS OF NATURAL IMAGE DENOISING USING CONV ALISTA ", + "text_level": 1, + "bbox": [ + 171, + 102, + 756, + 118 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "The natural image denoising experiment is conducted on the same BSD 500 dataset using the 400- image training set, 50-image validation set and 50-image test set. We convert them all to grayscale, and then add $\\sigma = 2 0$ Gaussian i.i.d. noise. We train both Conv LISTA (i.e., model (20)) and Conv ALISTA (i.e., model (26)). Both networks have 5 layers, with the same dictionary D obtained from the training set by solving (24). We reconstruct the denoised images using by convolving the learned feature maps with the original dictionary D. The mean-square-error (MSE) between denoised and clean images are adopted as the network training loss, as inspired by (Zhou et al., 2018). ", + "bbox": [ + 174, + 133, + 825, + 232 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Six popular benchmark images (adding $\\sigma = 2 0$ noise) are tested and reported in Table 4. The APSNR denotes the average PSNR over all images and the A-Times represents the average inference time (in seconds) for denoising one image. We compare Conv LISTA and Conv ALISTA, as well as the classical KSVD denoising algorithm (Elad & Aharon, 2006) and the recent CSC denoising algorithm with gradient regularization (CSC-GR) (Wohlberg, 2018). The results show that Conv LISTA and Conv ALISTA (without heavy tuning done for their optimal performance) can perform comparably with KSVD and outperforms CSC-GR, but with tremendously faster inference speeds than KSVD/CSC-GR. More importantly, Conv LISTA and Conv ALISTA only have marginal performance differences, validating again the analytic weights in convolutional cases. ", + "bbox": [ + 174, + 238, + 825, + 364 + ], + "page_idx": 30 + }, + { + "type": "table", + "img_path": "images/b9dcd65253224f7655bd515c11fb3be3a3377e006ba984c701ffcc6da5443721.jpg", + "table_caption": [ + "Table 4: Peak Signal to Noise Ratio (PSNR) Comparision between Conv LISTA and Conv ALISTA. " + ], + "table_footnote": [], + "table_body": "
ModelImage PSNR (dB)A-PSNRA-Time
LennaHousePepperCoupleBoatsBarbara
KSVD31.0333.2430.9731.7131.0030.4731.4024.70
CSC-GR28.4129.1127.3929.3128.3527.1928.297.56
Conv LISTA31.2632.7731.0031.8930.7829.5331.210.012
Conv ALISTA31.0132.4630.8131.8530.5829.7231.070.014
", + "bbox": [ + 173, + 411, + 825, + 501 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "I RESULTS OF ABLATION STUDIES IN ROBUSTNESS EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 535, + 740, + 553 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "As one anonymous reviewer kindly pointed out, Robust ALISTA has larger parameter space over TiLISTA and ALISTA trained. Therefore, they suggested that we increased the number of layers in TiLISTA, ALISTA and the baseline model LISTA-CPSS in (Chen et al., 2018) to see if their performance in this above evaluation setting can be improved in that way. Note that LISTA-CPSS has tens of layers, hence actually containing more parameters than robust ALISTA. In addition, the reviewers also suggested a set of ablation studies to investigate whether more layers in the encoder can endorse the model better adaptivity to higher level perturbations. We conduct the suggested experiments and present the results in this section. ", + "bbox": [ + 173, + 568, + 825, + 679 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "I.1 NUMBER OF LAYERS IN TILISTA, ALISTA AND LISTA-CPSS ", + "text_level": 1, + "bbox": [ + 174, + 696, + 650, + 712 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "As the reviewers pointed out, the comparison we present in Section. 5.3 might be unfair because robust ALISTA contains much more parameters (because it contains a 4-layer encoder) comparing to ALISTA, which only learns two series of scalars, and TiLISTA which has just one more matrix weight than ALISTA. Therefore, we add the following experiments to consolidate our claim on the effectiveness of the robust ALISTA model: ", + "bbox": [ + 174, + 724, + 825, + 794 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "• we increase the number of layers of TiLISTA and ALISTA which are then trained in the same data augmentation setting as we do in Section. 5.3, to see if they could yield comparitive robustness against dictionary perturbations; we also compare the robust ALISTA with the baseline LISTA-CPSS model in Chen et al. (2018), which contains tens of layers of independent weight matrices, thus having even more parameters than robust ALISTA. This comparison could consolidate our claim that the outstanding adaptiveness to dictionary perturbations of robust ALISTA is brought by its encoder-decoder structure rather than its learning capacity alone. ", + "bbox": [ + 215, + 806, + 825, + 924 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "The results are shown in Table. 5, where the performances are measured with NMSE in dB, which is defined in Section 5.1. The “Augmented” prefix means the models are trained in the data augmentation setting. $\\sigma$ is the standard deviation of the Gaussian distribution that is used to generate the dictioanry perturbations. $T$ stands for the number of layers (in the case of robust ALISTA it means the nubmer of layers of the ALISTA decoder, with a 4-layer encoder). We follow the training strategy and settings explained in Appendix G, with $\\sigma _ { m a x } = 0 . 0 2$ during training. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "On one hand, the comparison of performances of ALISTA, TiLISTA and LISTA-CPSS shows results that are consistent to the intuition that larger parameter space yields larger learning capacity, and therefore, better adaptiveness (LISTA-CPSS $>$ TiLISTA $>$ ALISTA). On the other hand, we can also find that ALISTA with more layers has worse performance. We think this observation is also reasonable for two reasons: 1) adding more layers in ALISTA does not enlarge the parameter volume significantly because it has only two scalar parameters in each layer; noting that ALISTA uses a fixed, analytically solved weight matrix, if this weight matrix is not compatible with the perturbed dictionary, more layers can even hurt the performance instead of improving. Lastly, it’s clearly shown that robust ALISTA outperforms LISTA-CPSS, even if it contains less parameters. This proves that the encoding process that adaptively transforms the perturbed dictiories is necessary to achieve good robustness against perturbations in dictionaries. ", + "bbox": [ + 174, + 194, + 825, + 347 + ], + "page_idx": 31 + }, + { + "type": "table", + "img_path": "images/9e51f42941c85eec4bc501a0e6e57efe6443b7d15da3f37f93a74e0a94a2358f.jpg", + "table_caption": [], + "table_footnote": [ + "Table 5: The results (recovery NMSE in dB) of ablation study on the influence of model capacity towards the model robustness against dictionary perturbations. " + ], + "table_body": "
σ of perturbationsduring testing0.00010.0010.010.0150.020.025
AugmentedALISTAT=16-26.58-25.87-15.49-11.71-8.84-6.74
T=20-24.43-24.46-15.39-11.77-8.94-6.82
T=24-24.12-24.00-15.45-11.68-8.81-6.70
AugmentedTiLISTAT=16-27.76-27.18-16.83-12.95-9.81-7.55
T=20-28.13-28.54-17.15-12.98-9.83-7.58
T=24-26.08-27.27-17.34-13.14-9.91-7.61
AugmentedLISTA-CPSST=16-27.93-27.18-16.96-12.99-9.93-7.70
T=20-28.17-27.33-16.95-13.00-9.94-7.71
T=24-30.30-29.24-16.86-12.97-9.94-7.70
Robust ALISTAT=16-62.47-62.41-62.02-61.50-60.67-45.00
", + "bbox": [ + 217, + 361, + 779, + 531 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "I.2 ABLATION STUDY ON THE DEPTHS OF ENCODERS IN ROBUST ALISTA ", + "text_level": 1, + "bbox": [ + 174, + 595, + 704, + 611 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Another constructive suggestion from the reviewers is to design an ablation study to investigate the influence of the depth of encoders in robust ALISTA on its adaptivity to dictionary perturbations. A natural intuition is that, adding more layers to the encoder can increase its ability to sustain larger perturbation levels. But is this true? ", + "bbox": [ + 174, + 622, + 825, + 678 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Therefore, we train another two robust ALISTA models, with a 5-layer and a 6-layer encoders respectively and 16-layer ALISTA decoders for both, and compare them with the originally reported robust ALISTA model with a 4-layer encoder and a 16-layer ALISTA decoders. All three models use one pretrained decoder, and pretrain their encoders using the same method (see Appendix G). We only use $\\sigma _ { m a x } = 0 . 0 2$ during training. One thing to notice is that we observe unstable training process if we use default initial learning rates $\\alpha _ { p r e } = 1 0 ^ { - 4 }$ in the pre-training stage and $\\alpha _ { e } = 1 0 ^ { - } \\overline { { 6 } }$ in the joint training stage when encoders have 5 or 6 layers. Therefore, we use $\\alpha _ { p r e } ^ { \\prime } = 1 0 ^ { - 5 }$ for the 6-layer encoder in the pre-training stage, and in the joint training stage use a decreased and uniform initial learning rate $\\alpha _ { e } ^ { \\prime } = 1 0 ^ { - 9 }$ for the three encoders while keeping the default initial learning rate $\\alpha _ { d } = 1 0 ^ { - 4 }$ for decoders. The other settings remain the same. ", + "bbox": [ + 173, + 684, + 825, + 825 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Results are shown in Table. 6. The performances are measured with NMSE in dB, defined in Section 5.1. From the table we can see that encoders do show better robustness when they have more layers, i.e. larger learning capacity. ", + "bbox": [ + 174, + 832, + 823, + 875 + ], + "page_idx": 31 + }, + { + "type": "table", + "img_path": "images/a307eaddd20854b14eefd776a6c2f72b38e953b1403d43011fdfc01e5a888ad9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
# Encoder Layerso of perturbations during testing
0.00010.0010.010.0150.020.025
4-68.57-68.56-67.94-66.86-64.84-56.63
5-69.34-69.34-69.02-68.49-67.20-65.55
6-70.38-70.33-69.92-69.22-67.72-65.60
", + "bbox": [ + 241, + 454, + 758, + 529 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Table 6: The results of ablation study on the influence of the depths of encoder towards the model robustness against dictionary perturbations. 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The Analysis-Synthesis model (Rubinstein & Elad,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 469, + 387 + ], + "score": 1.0, + "content": "2014; Yang et al., 2016) could also be viewed as a special LISTA model with only one layer", + "type": "text" + }, + { + "bbox": [ + 469, + 374, + 498, + 385 + ], + "score": 0.85, + "content": "K = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 373, + 505, + 387 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "More recently, the convolutional sparse coding (CSC), an extension of the sparse coding (1),", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "gains increasingly attention in the machine learning area. (Sreter & Giryes, 2018) showed that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "the CSC could be similarly approximated and accelerated by a LISTA-type feed-forward network.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "(Tolooshams et al., 2018) designed a structure of sparse auto-encoder inspired by multi-layer CSC.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "(Papyan et al., 2016; Sulam et al., 2017) also revealed CSC as a potentially useful tool for under-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 446, + 333, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 333, + 458 + ], + "score": 1.0, + "content": "standing general convolutional neural networks (CNNs).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 203, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 205, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 205, + 479 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "Despite the empirical success (Sprechmann et al., 2015; Wang et al., 2016a;b;c;d; Zhang & Ghanem,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "2018; Zhou et al., 2018; Ito et al., 2018) in constructing fast trainable regressors for approximating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 483, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 483, + 516 + ], + "score": 1.0, + "content": "iterative sparse solvers, the theoretical understanding of such approximations remains limited.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "A handful of recent works have been investigating the theory of LISTA. (Moreau & Bruna, 2017)", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "re-factorized the Gram matrix of dictionary, by trying to nearly diagonalize the Gram matrix with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 217, + 555 + ], + "score": 1.0, + "content": "a basis, subject to a small", + "type": "text" + }, + { + "bbox": [ + 218, + 542, + 228, + 553 + ], + "score": 0.85, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "perturbation. They thus re-parameterized LISTA a new factorized", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "architecture that achieved similar acceleration gain to LISTA, hence ending up with an “indirect”", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "proof. They concluded that LISTA can converge faster than ISTA, but still sublinearly. (Giryes et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "2018) interpreted LISTA as a projected gradient descent descent (PGD) where the projection step", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "was inaccurate, which enables a trade-off between approximation error and convergence speed. The", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "latest work (Chen et al., 2018) presented the more related results to ours: they introduced necessary", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "conditions for the LISTA weight structure in order to achieve asymptotic linear convergence of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "LISTA, which also proved to be a theoretical convergence rate upper bound. They also introduced", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "a thresholding scheme for practically improving the convergence speed. Note that, none of the above", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 437, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 437, + 654 + ], + "score": 1.0, + "content": "works extended their discussions to CSC and its similar LISTA-type architectures.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 670 + ], + "score": 1.0, + "content": "Several other works examined the theoretical properties of some sibling architectures to LISTA.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 668, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 682 + ], + "score": 1.0, + "content": "(Xin et al., 2016) studied the model proposed by (Wang et al., 2016b), which unfolded/truncated the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 490, + 693 + ], + "score": 1.0, + "content": "iterative hard thresholding (IHT) algorithm instead of ISTA, for approximating the solution to", + "type": "text" + }, + { + "bbox": [ + 490, + 680, + 501, + 691 + ], + "score": 0.83, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "minimization. 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The Analysis-Synthesis model (Rubinstein & Elad,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 469, + 387 + ], + "score": 1.0, + "content": "2014; Yang et al., 2016) could also be viewed as a special LISTA model with only one layer", + "type": "text" + }, + { + "bbox": [ + 469, + 374, + 498, + 385 + ], + "score": 0.85, + "content": "K = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 373, + 505, + 387 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 285, + 506, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "More recently, the convolutional sparse coding (CSC), an extension of the sparse coding (1),", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "gains increasingly attention in the machine learning area. (Sreter & Giryes, 2018) showed that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "the CSC could be similarly approximated and accelerated by a LISTA-type feed-forward network.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "(Tolooshams et al., 2018) designed a structure of sparse auto-encoder inspired by multi-layer CSC.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "(Papyan et al., 2016; Sulam et al., 2017) also revealed CSC as a potentially useful tool for under-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 446, + 333, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 333, + 458 + ], + "score": 1.0, + "content": "standing general convolutional neural networks (CNNs).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 390, + 506, + 458 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 203, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 205, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 205, + 479 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "Despite the empirical success (Sprechmann et al., 2015; Wang et al., 2016a;b;c;d; Zhang & Ghanem,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "2018; Zhou et al., 2018; Ito et al., 2018) in constructing fast trainable regressors for approximating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 483, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 483, + 516 + ], + "score": 1.0, + "content": "iterative sparse solvers, the theoretical understanding of such approximations remains limited.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 480, + 505, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "A handful of recent works have been investigating the theory of LISTA. (Moreau & Bruna, 2017)", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "re-factorized the Gram matrix of dictionary, by trying to nearly diagonalize the Gram matrix with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 217, + 555 + ], + "score": 1.0, + "content": "a basis, subject to a small", + "type": "text" + }, + { + "bbox": [ + 218, + 542, + 228, + 553 + ], + "score": 0.85, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "perturbation. They thus re-parameterized LISTA a new factorized", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "architecture that achieved similar acceleration gain to LISTA, hence ending up with an “indirect”", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "proof. They concluded that LISTA can converge faster than ISTA, but still sublinearly. (Giryes et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "2018) interpreted LISTA as a projected gradient descent descent (PGD) where the projection step", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "was inaccurate, which enables a trade-off between approximation error and convergence speed. The", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "latest work (Chen et al., 2018) presented the more related results to ours: they introduced necessary", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "conditions for the LISTA weight structure in order to achieve asymptotic linear convergence of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "LISTA, which also proved to be a theoretical convergence rate upper bound. They also introduced", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "a thresholding scheme for practically improving the convergence speed. Note that, none of the above", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 437, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 437, + 654 + ], + "score": 1.0, + "content": "works extended their discussions to CSC and its similar LISTA-type architectures.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 519, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 670 + ], + "score": 1.0, + "content": "Several other works examined the theoretical properties of some sibling architectures to LISTA.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 668, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 682 + ], + "score": 1.0, + "content": "(Xin et al., 2016) studied the model proposed by (Wang et al., 2016b), which unfolded/truncated the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 490, + 693 + ], + "score": 1.0, + "content": "iterative hard thresholding (IHT) algorithm instead of ISTA, for approximating the solution to", + "type": "text" + }, + { + "bbox": [ + 490, + 680, + 501, + 691 + ], + "score": 0.83, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "minimization. They showed that the learnable fast regressor can be obtained by using a transformed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 702, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 506, + 715 + ], + "score": 1.0, + "content": "dictionary with improved restricted isometry property (RIP). However, their discussions are not", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "applicable to LISTA directly, although IHT is linearly convergent (Blumensath & Davies, 2009)", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "under rather strong assumptions. Their discussions were also limited to linear sparse coding and", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "resulting fully-connected networks only. (Borgerding et al., 2017; Metzler et al., 2017) studied", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "a similar learning-based model inspired from another LASSO solver, called approximated message", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "passing (AMP). (Borgerding et al., 2017) showed the MMSE-optimality of an AMP-inspired model,", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "but not accompanied with any convergence rate result. Also, the popular assumption in analyzing", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 425, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 425, + 161 + ], + "score": 1.0, + "content": "AMP algorithms (called “state evolution”) does not hold when analyzing ISTA.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 658, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "applicable to LISTA directly, although IHT is linearly convergent (Blumensath & Davies, 2009)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "under rather strong assumptions. Their discussions were also limited to linear sparse coding and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "resulting fully-connected networks only. (Borgerding et al., 2017; Metzler et al., 2017) studied", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "a similar learning-based model inspired from another LASSO solver, called approximated message", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "passing (AMP). (Borgerding et al., 2017) showed the MMSE-optimality of an AMP-inspired model,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "but not accompanied with any convergence rate result. Also, the popular assumption in analyzing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 425, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 425, + 161 + ], + "score": 1.0, + "content": "AMP algorithms (called “state evolution”) does not hold when analyzing ISTA.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 109, + 168, + 282, + 179 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 284, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 284, + 181 + ], + "score": 1.0, + "content": "1.2 MOTIVATION AND CONTRIBUTIONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "This paper presents multi-fold contributions in advancing the theoretical understanding of LISTA,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "beyond state-of-the-art results. Firstly, we show that the layer-wise weights in LISTA need not", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 221 + ], + "score": 1.0, + "content": "being learned from data. That is based on decoupling LISTA training into a data-free analytic op-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "timization stage followed by a lighter-weight data-driven learning stage without compromising the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "optimal linear convergence rate proved in (Chen et al., 2018). We establish a minimum-coherence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "score": 1.0, + "content": "criterion between the desired LISTA weights and the dictionary D, which leads to an efficient algo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 251, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 504, + 263 + ], + "score": 1.0, + "content": "rithm that can analytically solve the former from the latter, independent of the distribution of x. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "data-driven training is then reduced to learning layer-wise step sizes and thresholds only, which will", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "fit the distribution of x. The new scheme, called Analytic LISTA (ALISTA), provides important in-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "sights into the working mechanism of LISTA. Experiments shows ALISTA to perform comparably", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 308 + ], + "score": 1.0, + "content": "with previous LISTA models (Gregor & LeCun, 2010; Chen et al., 2018) with much lighter-weight", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "training. Then, we extend the above discussions and conclusions to CSC, and introduce an efficient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "algorithm to solve the convolutional version of coherence minimization. Further, we introduce a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "score": 1.0, + "content": "new robust LISTA learning scheme benefiting from the decoupled structure, by adding perturba-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 138, + 351 + ], + "score": 1.0, + "content": "tions to", + "type": "text" + }, + { + "bbox": [ + 138, + 338, + 149, + 348 + ], + "score": 0.36, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "during training. The resulting model is shown to possess much stronger robustness when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 271, + 362 + ], + "score": 1.0, + "content": "the input distribution varies, even when", + "type": "text" + }, + { + "bbox": [ + 271, + 349, + 281, + 359 + ], + "score": 0.35, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "changes to some extent, compared to classical LISTA", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 360, + 289, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 275, + 372 + ], + "score": 1.0, + "content": "models that learn to (over-)fit one specific", + "type": "text" + }, + { + "bbox": [ + 276, + 360, + 286, + 370 + ], + "score": 0.41, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 360, + 289, + 372 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 387, + 454, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 457, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 457, + 401 + ], + "score": 1.0, + "content": "2 ANALYTIC LISTA: CALCULATING WEIGHTS WITHOUT TRAINING", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 109, + 404, + 428, + 416 + ], + "lines": [ + { + "bbox": [ + 107, + 404, + 427, + 419 + ], + "spans": [ + { + "bbox": [ + 107, + 404, + 427, + 419 + ], + "score": 1.0, + "content": "We theoretically analyze the LISTA-CPSS model defined in (Chen et al., 2018):", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 419, + 402, + 440 + ], + "lines": [ + { + "bbox": [ + 206, + 419, + 402, + 440 + ], + "spans": [ + { + "bbox": [ + 206, + 419, + 402, + 440 + ], + "score": 0.92, + "content": "\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - ( \\mathbf { W } ^ { ( k ) } ) ^ { T } \\big ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) ,", + "type": "interline_equation", + "image_path": "fee2299837a12f089dfca4e228697f147143c7334978af5ef2e3ebb04ac3d2a7.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 206, + 419, + 402, + 440 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 442, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 441, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 133, + 458 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 442, + 284, + 457 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { ( k ) } = [ \\mathbf { w } _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , \\mathbf { w } _ { M } ^ { ( k ) } ] \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 441, + 506, + 459 + ], + "score": 1.0, + "content": "is a linear operator with the same dimensionality with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 120, + 475 + ], + "score": 1.0, + "content": "D,", + "type": "text" + }, + { + "bbox": [ + 120, + 457, + 217, + 473 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( k ) } = \\left[ x _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , x _ { M } ^ { ( k ) } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 456, + 241, + 473 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 242, + 459, + 255, + 470 + ], + "score": 0.85, + "content": "k ^ { \\mathrm { { t h } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 456, + 329, + 473 + ], + "score": 1.0, + "content": "layer node. In (6),", + "type": "text" + }, + { + "bbox": [ + 329, + 458, + 410, + 471 + ], + "score": 0.92, + "content": "\\boldsymbol { \\Theta } = \\{ \\mathbf { W } ^ { ( k ) } , \\boldsymbol { \\theta } ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 456, + 505, + 473 + ], + "score": 1.0, + "content": "are parameters to train.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 469, + 507, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 263, + 489 + ], + "score": 1.0, + "content": "Model (6) can be derived from (4) with", + "type": "text" + }, + { + "bbox": [ + 263, + 471, + 338, + 487 + ], + "score": 0.73, + "content": "\\mathbf { W } _ { 1 } ^ { ( k ) } = ( \\mathbf { W } ^ { ( k ) } ) ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 469, + 341, + 489 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 341, + 472, + 425, + 487 + ], + "score": 0.67, + "content": "\\mathbf { W } _ { 2 } ^ { ( k ) } = \\mathbf { I } - \\mathbf { W } _ { 1 } ^ { ( k ) } \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 469, + 507, + 489 + ], + "score": 1.0, + "content": ". (Chen et al., 2018)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "showed that (6) has the same representation capability with (4) on the sparse recovery problem, with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 252, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 252, + 509 + ], + "score": 1.0, + "content": "a specifically light weight structure.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 467, + 526 + ], + "score": 1.0, + "content": "Our theoretical analysis will further define and establish properties of “good” parameters", + "type": "text" + }, + { + "bbox": [ + 468, + 514, + 477, + 524 + ], + "score": 0.8, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "in (6),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "and then discuss how to analytically compute those good parameters rather than relying solely on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 547 + ], + "score": 1.0, + "content": "black-box training. In this way, the LISTA model could be further significantly simplified, with little", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 473, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 473, + 559 + ], + "score": 1.0, + "content": "performance loss. The proofs of all the theorems in this paper are provided in the appendix.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 107, + 566, + 276, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 277, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 277, + 578 + ], + "score": 1.0, + "content": "2.1 RECOVERY ERROR UPPER BOUND", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 414, + 593 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 414, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 335, + 595 + ], + "score": 1.0, + "content": "We start with an assumption on the “ground truth” signal", + "type": "text" + }, + { + "bbox": [ + 336, + 582, + 348, + 592 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 580, + 404, + 595 + ], + "score": 1.0, + "content": "and the noise", + "type": "text" + }, + { + "bbox": [ + 404, + 584, + 410, + 591 + ], + "score": 0.73, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 580, + 414, + 595 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 109, + 595, + 430, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 433, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 283, + 609 + ], + "score": 1.0, + "content": "Assumption 1 (Basic assumptions). Signal", + "type": "text" + }, + { + "bbox": [ + 283, + 596, + 295, + 605 + ], + "score": 0.86, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 594, + 433, + 609 + ], + "score": 1.0, + "content": "is sampled from the following set:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 608, + 406, + 630 + ], + "lines": [ + { + "bbox": [ + 204, + 608, + 406, + 630 + ], + "spans": [ + { + "bbox": [ + 204, + 608, + 406, + 630 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) \\triangleq \\Big \\{ \\mathbf { x } ^ { * } \\Big | | x _ { i } ^ { * } | \\leq B , \\forall i , \\| \\mathbf { x } ^ { * } \\| _ { 0 } \\leq s \\Big \\} .", + "type": "interline_equation", + "image_path": "c116b90c9eab8f956031fe56ae6616cf967f9b62f52133eb52994328f2dbf8ef.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 204, + 608, + 406, + 630 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 451, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 452, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 169, + 646 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 169, + 634, + 181, + 643 + ], + "score": 0.81, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 632, + 246, + 646 + ], + "score": 1.0, + "content": "is bounded and", + "type": "text" + }, + { + "bbox": [ + 246, + 635, + 252, + 643 + ], + "score": 0.31, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 632, + 289, + 646 + ], + "score": 1.0, + "content": "-sparse2", + "type": "text" + }, + { + "bbox": [ + 290, + 633, + 315, + 645 + ], + "score": 0.78, + "content": "s \\geq 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 632, + 423, + 646 + ], + "score": 1.0, + "content": "). Furthermore, we assume", + "type": "text" + }, + { + "bbox": [ + 423, + 634, + 448, + 643 + ], + "score": 0.87, + "content": "\\varepsilon = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 632, + 452, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 652, + 506, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "The zero-noise assumption is for simplicity of the proofs. Our experiments will show that our models", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 664, + 209, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 209, + 677 + ], + "score": 1.0, + "content": "are robust to noisy cases.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 267, + 694 + ], + "score": 1.0, + "content": "The mutual coherence of the dictionary", + "type": "text" + }, + { + "bbox": [ + 267, + 681, + 277, + 691 + ], + "score": 0.57, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "is a significant concept in compressive sensing (Donoho", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "& Elad, 2003; Elad, 2007; Lu et al., 2018). A dictionary with small coherence possesses better", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 477, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 477, + 715 + ], + "score": 1.0, + "content": "sparse recovery performance. Motivated by this point, we introduce the following definition.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 342, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 343, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 163, + 733 + ], + "score": 1.0, + "content": "2A signal is", + "type": "text" + }, + { + "bbox": [ + 164, + 723, + 169, + 730 + ], + "score": 0.38, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 720, + 274, + 733 + ], + "score": 1.0, + "content": "-sparse if it has no more than", + "type": "text" + }, + { + "bbox": [ + 274, + 725, + 279, + 730 + ], + "score": 0.5, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 720, + 343, + 733 + ], + "score": 1.0, + "content": "non-zero entries.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 104, + 82, + 505, + 161 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 109, + 168, + 282, + 179 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 284, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 284, + 181 + ], + "score": 1.0, + "content": "1.2 MOTIVATION AND CONTRIBUTIONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "This paper presents multi-fold contributions in advancing the theoretical understanding of LISTA,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "beyond state-of-the-art results. Firstly, we show that the layer-wise weights in LISTA need not", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 221 + ], + "score": 1.0, + "content": "being learned from data. That is based on decoupling LISTA training into a data-free analytic op-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "timization stage followed by a lighter-weight data-driven learning stage without compromising the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "optimal linear convergence rate proved in (Chen et al., 2018). We establish a minimum-coherence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "score": 1.0, + "content": "criterion between the desired LISTA weights and the dictionary D, which leads to an efficient algo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 251, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 504, + 263 + ], + "score": 1.0, + "content": "rithm that can analytically solve the former from the latter, independent of the distribution of x. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "data-driven training is then reduced to learning layer-wise step sizes and thresholds only, which will", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "fit the distribution of x. The new scheme, called Analytic LISTA (ALISTA), provides important in-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "sights into the working mechanism of LISTA. Experiments shows ALISTA to perform comparably", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 308 + ], + "score": 1.0, + "content": "with previous LISTA models (Gregor & LeCun, 2010; Chen et al., 2018) with much lighter-weight", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "training. Then, we extend the above discussions and conclusions to CSC, and introduce an efficient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "algorithm to solve the convolutional version of coherence minimization. Further, we introduce a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 504, + 339 + ], + "score": 1.0, + "content": "new robust LISTA learning scheme benefiting from the decoupled structure, by adding perturba-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 138, + 351 + ], + "score": 1.0, + "content": "tions to", + "type": "text" + }, + { + "bbox": [ + 138, + 338, + 149, + 348 + ], + "score": 0.36, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "during training. The resulting model is shown to possess much stronger robustness when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 271, + 362 + ], + "score": 1.0, + "content": "the input distribution varies, even when", + "type": "text" + }, + { + "bbox": [ + 271, + 349, + 281, + 359 + ], + "score": 0.35, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "changes to some extent, compared to classical LISTA", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 360, + 289, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 275, + 372 + ], + "score": 1.0, + "content": "models that learn to (over-)fit one specific", + "type": "text" + }, + { + "bbox": [ + 276, + 360, + 286, + 370 + ], + "score": 0.41, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 360, + 289, + 372 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 182, + 506, + 372 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 387, + 454, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 457, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 457, + 401 + ], + "score": 1.0, + "content": "2 ANALYTIC LISTA: CALCULATING WEIGHTS WITHOUT TRAINING", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 109, + 404, + 428, + 416 + ], + "lines": [ + { + "bbox": [ + 107, + 404, + 427, + 419 + ], + "spans": [ + { + "bbox": [ + 107, + 404, + 427, + 419 + ], + "score": 1.0, + "content": "We theoretically analyze the LISTA-CPSS model defined in (Chen et al., 2018):", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 107, + 404, + 427, + 419 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 419, + 402, + 440 + ], + "lines": [ + { + "bbox": [ + 206, + 419, + 402, + 440 + ], + "spans": [ + { + "bbox": [ + 206, + 419, + 402, + 440 + ], + "score": 0.92, + "content": "\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - ( \\mathbf { W } ^ { ( k ) } ) ^ { T } \\big ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) ,", + "type": "interline_equation", + "image_path": "fee2299837a12f089dfca4e228697f147143c7334978af5ef2e3ebb04ac3d2a7.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 206, + 419, + 402, + 440 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 442, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 441, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 133, + 458 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 442, + 284, + 457 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { ( k ) } = [ \\mathbf { w } _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , \\mathbf { w } _ { M } ^ { ( k ) } ] \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 441, + 506, + 459 + ], + "score": 1.0, + "content": "is a linear operator with the same dimensionality with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 120, + 475 + ], + "score": 1.0, + "content": "D,", + "type": "text" + }, + { + "bbox": [ + 120, + 457, + 217, + 473 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( k ) } = \\left[ x _ { 1 } ^ { ( k ) } , \\cdot \\cdot \\cdot , x _ { M } ^ { ( k ) } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 456, + 241, + 473 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 242, + 459, + 255, + 470 + ], + "score": 0.85, + "content": "k ^ { \\mathrm { { t h } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 456, + 329, + 473 + ], + "score": 1.0, + "content": "layer node. In (6),", + "type": "text" + }, + { + "bbox": [ + 329, + 458, + 410, + 471 + ], + "score": 0.92, + "content": "\\boldsymbol { \\Theta } = \\{ \\mathbf { W } ^ { ( k ) } , \\boldsymbol { \\theta } ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 456, + 505, + 473 + ], + "score": 1.0, + "content": "are parameters to train.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 469, + 507, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 263, + 489 + ], + "score": 1.0, + "content": "Model (6) can be derived from (4) with", + "type": "text" + }, + { + "bbox": [ + 263, + 471, + 338, + 487 + ], + "score": 0.73, + "content": "\\mathbf { W } _ { 1 } ^ { ( k ) } = ( \\mathbf { W } ^ { ( k ) } ) ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 469, + 341, + 489 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 341, + 472, + 425, + 487 + ], + "score": 0.67, + "content": "\\mathbf { W } _ { 2 } ^ { ( k ) } = \\mathbf { I } - \\mathbf { W } _ { 1 } ^ { ( k ) } \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 469, + 507, + 489 + ], + "score": 1.0, + "content": ". (Chen et al., 2018)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "showed that (6) has the same representation capability with (4) on the sparse recovery problem, with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 252, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 252, + 509 + ], + "score": 1.0, + "content": "a specifically light weight structure.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 441, + 507, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 467, + 526 + ], + "score": 1.0, + "content": "Our theoretical analysis will further define and establish properties of “good” parameters", + "type": "text" + }, + { + "bbox": [ + 468, + 514, + 477, + 524 + ], + "score": 0.8, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "in (6),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "and then discuss how to analytically compute those good parameters rather than relying solely on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 547 + ], + "score": 1.0, + "content": "black-box training. In this way, the LISTA model could be further significantly simplified, with little", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 473, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 473, + 559 + ], + "score": 1.0, + "content": "performance loss. The proofs of all the theorems in this paper are provided in the appendix.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 513, + 506, + 559 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 566, + 276, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 277, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 277, + 578 + ], + "score": 1.0, + "content": "2.1 RECOVERY ERROR UPPER BOUND", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 414, + 593 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 414, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 335, + 595 + ], + "score": 1.0, + "content": "We start with an assumption on the “ground truth” signal", + "type": "text" + }, + { + "bbox": [ + 336, + 582, + 348, + 592 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 580, + 404, + 595 + ], + "score": 1.0, + "content": "and the noise", + "type": "text" + }, + { + "bbox": [ + 404, + 584, + 410, + 591 + ], + "score": 0.73, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 580, + 414, + 595 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 580, + 414, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 595, + 430, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 433, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 283, + 609 + ], + "score": 1.0, + "content": "Assumption 1 (Basic assumptions). Signal", + "type": "text" + }, + { + "bbox": [ + 283, + 596, + 295, + 605 + ], + "score": 0.86, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 594, + 433, + 609 + ], + "score": 1.0, + "content": "is sampled from the following set:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 594, + 433, + 609 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 608, + 406, + 630 + ], + "lines": [ + { + "bbox": [ + 204, + 608, + 406, + 630 + ], + "spans": [ + { + "bbox": [ + 204, + 608, + 406, + 630 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) \\triangleq \\Big \\{ \\mathbf { x } ^ { * } \\Big | | x _ { i } ^ { * } | \\leq B , \\forall i , \\| \\mathbf { x } ^ { * } \\| _ { 0 } \\leq s \\Big \\} .", + "type": "interline_equation", + "image_path": "c116b90c9eab8f956031fe56ae6616cf967f9b62f52133eb52994328f2dbf8ef.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 204, + 608, + 406, + 630 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 451, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 452, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 169, + 646 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 169, + 634, + 181, + 643 + ], + "score": 0.81, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 632, + 246, + 646 + ], + "score": 1.0, + "content": "is bounded and", + "type": "text" + }, + { + "bbox": [ + 246, + 635, + 252, + 643 + ], + "score": 0.31, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 632, + 289, + 646 + ], + "score": 1.0, + "content": "-sparse2", + "type": "text" + }, + { + "bbox": [ + 290, + 633, + 315, + 645 + ], + "score": 0.78, + "content": "s \\geq 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 632, + 423, + 646 + ], + "score": 1.0, + "content": "). Furthermore, we assume", + "type": "text" + }, + { + "bbox": [ + 423, + 634, + 448, + 643 + ], + "score": 0.87, + "content": "\\varepsilon = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 632, + 452, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 632, + 452, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 652, + 506, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "The zero-noise assumption is for simplicity of the proofs. Our experiments will show that our models", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 664, + 209, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 209, + 677 + ], + "score": 1.0, + "content": "are robust to noisy cases.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 653, + 505, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 267, + 694 + ], + "score": 1.0, + "content": "The mutual coherence of the dictionary", + "type": "text" + }, + { + "bbox": [ + 267, + 681, + 277, + 691 + ], + "score": 0.57, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "is a significant concept in compressive sensing (Donoho", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "& Elad, 2003; Elad, 2007; Lu et al., 2018). A dictionary with small coherence possesses better", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 477, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 477, + 715 + ], + "score": 1.0, + "content": "sparse recovery performance. Motivated by this point, we introduce the following definition.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 680, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 192, + 97 + ], + "score": 1.0, + "content": "Definition 1. Given", + "type": "text" + }, + { + "bbox": [ + 192, + 81, + 247, + 93 + ], + "score": 0.91, + "content": "\\mathbf { D } \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 79, + 506, + 97 + ], + "score": 1.0, + "content": "with each of its column normalized, we define the generalized", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 183, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 183, + 106 + ], + "score": 1.0, + "content": "mutual coherence:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 106, + 419, + 143 + ], + "lines": [ + { + "bbox": [ + 191, + 106, + 419, + 143 + ], + "spans": [ + { + "bbox": [ + 191, + 106, + 419, + 143 + ], + "score": 0.93, + "content": "\\widetilde { \\mu } ( \\mathbf { D } ) = \\operatorname* { i n f } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\bigg \\{ \\operatorname* { m a x } _ { \\substack { 1 \\leq i \\neq j \\leq M } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } \\bigg \\} .", + "type": "interline_equation", + "image_path": "bfdc67a99b7536a7d6d5c7e192b522c6764cd1db5b506af250cfa75d72a6d255.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 106, + 419, + 124.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 191, + 124.5, + 419, + 143.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 504, + 173 + ], + "lines": [ + { + "bbox": [ + 104, + 144, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 204, + 163 + ], + "score": 1.0, + "content": "Additionally, We define", + "type": "text" + }, + { + "bbox": [ + 204, + 147, + 335, + 162 + ], + "score": 0.89, + "content": "{ \\mathcal { W } } ( \\mathbf { D } ) = \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } : \\mathbf { W } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 144, + 441, + 163 + ], + "score": 1.0, + "content": "attains the infimum given", + "type": "text" + }, + { + "bbox": [ + 441, + 148, + 459, + 162 + ], + "score": 0.86, + "content": "( \\delta ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 144, + 506, + 163 + ], + "score": 1.0, + "content": ". A weight", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 256, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 194, + 174 + ], + "score": 1.0, + "content": "matrix W is “good”", + "type": "text" + }, + { + "bbox": [ + 195, + 161, + 252, + 173 + ], + "score": 0.88, + "content": "f \\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 159, + 256, + 174 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 504, + 204 + ], + "lines": [ + { + "bbox": [ + 104, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 179, + 375, + 194 + ], + "score": 1.0, + "content": "In the above definition, problem (8) is feasible and attainable, i.e.,", + "type": "text" + }, + { + "bbox": [ + 375, + 181, + 425, + 193 + ], + "score": 0.92, + "content": "\\mathcal { W } ( \\mathbf { D } ) \\neq \\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 179, + 505, + 194 + ], + "score": 1.0, + "content": ", which was proven", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 248, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 248, + 203 + ], + "score": 1.0, + "content": "in Lemma 1 of (Chen et al., 2018).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 504, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 325, + 221 + ], + "score": 1.0, + "content": "Theorem 1 (Recovery error upper bound). Take any", + "type": "text" + }, + { + "bbox": [ + 326, + 206, + 386, + 218 + ], + "score": 0.93, + "content": "\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 204, + 409, + 221 + ], + "score": 1.0, + "content": ", any", + "type": "text" + }, + { + "bbox": [ + 410, + 206, + 465, + 218 + ], + "score": 0.9, + "content": "\\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 204, + 505, + 221 + ], + "score": 1.0, + "content": ", and any", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 214, + 434, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 145, + 237 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 217, + 228, + 233 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\gamma ^ { ( k ) } \\in ( 0 , \\frac { 2 } { 2 \\tilde { \\mu } s - \\tilde { \\mu } + 1 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 214, + 373, + 237 + ], + "score": 1.0, + "content": ". Using them, define the parameters", + "type": "text" + }, + { + "bbox": [ + 373, + 218, + 427, + 231 + ], + "score": 0.92, + "content": "\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 214, + 434, + 237 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 236, + 445, + 261 + ], + "lines": [ + { + "bbox": [ + 164, + 236, + 445, + 261 + ], + "spans": [ + { + "bbox": [ + 164, + 236, + 445, + 261 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W } , \\quad \\theta ^ { ( k ) } = \\gamma ^ { ( k ) } \\widetilde { \\mu } ( \\mathbf { D } ) \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) } \\big \\{ \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 1 } \\big \\} ,", + "type": "interline_equation", + "image_path": "1e163888a17a6494fdc47f8d83e46b078edb5da3846f15609dde3eca81463b60.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 164, + 236, + 445, + 261 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 506, + 303 + ], + "lines": [ + { + "bbox": [ + 103, + 263, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 103, + 263, + 187, + 281 + ], + "score": 1.0, + "content": "while the sequence", + "type": "text" + }, + { + "bbox": [ + 187, + 265, + 248, + 279 + ], + "score": 0.92, + "content": "\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 263, + 464, + 281 + ], + "score": 1.0, + "content": "is generated by (6) using the above parameters and", + "type": "text" + }, + { + "bbox": [ + 464, + 265, + 504, + 277 + ], + "score": 0.92, + "content": "\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 275, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 172, + 293 + ], + "score": 1.0, + "content": "(Note that each", + "type": "text" + }, + { + "bbox": [ + 172, + 278, + 209, + 291 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 275, + 280, + 293 + ], + "score": 1.0, + "content": "depends only on", + "type": "text" + }, + { + "bbox": [ + 280, + 278, + 353, + 290 + ], + "score": 0.89, + "content": "\\theta ^ { ( k - 1 ) } , \\theta ^ { ( k - 2 ) } , \\dots .", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 275, + 405, + 293 + ], + "score": 1.0, + "content": "and defines", + "type": "text" + }, + { + "bbox": [ + 405, + 278, + 422, + 289 + ], + "score": 0.87, + "content": "\\theta ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 275, + 497, + 293 + ], + "score": 1.0, + "content": "). 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Then, we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 306, + 472, + 340 + ], + "lines": [ + { + "bbox": [ + 122, + 306, + 472, + 340 + ], + "spans": [ + { + "bbox": [ + 122, + 306, + 472, + 340 + ], + "score": 0.93, + "content": "\\operatorname { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } , \\quad \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\leq s B \\exp \\Big ( - \\sum _ { \\tau = 0 } ^ { k - 1 } c ^ { ( \\tau ) } \\Big ) , \\quad k = 1 , 2 , \\dots", + "type": "interline_equation", + "image_path": "5d8382f60699f6cf9b7dead385eb0f68a6c1e718355245d422eca06628f8e26e.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 122, + 306, + 472, + 317.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 122, + 317.3333333333333, + 472, + 328.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 122, + 328.66666666666663, + 472, + 339.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 501, + 359 + ], + "lines": [ + { + "bbox": [ + 104, + 341, + 503, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 341, + 132, + 361 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 346, + 140, + 356 + ], + "score": 0.52, + "content": "\\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 341, + 208, + 361 + ], + "score": 1.0, + "content": "is the support of", + "type": "text" + }, + { + "bbox": [ + 208, + 346, + 219, + 356 + ], + "score": 0.84, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 341, + 239, + 361 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 344, + 411, + 359 + ], + "score": 0.92, + "content": "c ^ { ( k ) } = - \\log \\left( ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\gamma ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 341, + 503, + 361 + ], + "score": 1.0, + "content": "is a positive constant.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 380, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 380, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 380, + 380 + ], + "score": 1.0, + "content": "In Theorem 1, Eqn. 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If", + "type": "text" + }, + { + "bbox": [ + 309, + 449, + 345, + 462 + ], + "score": 0.93, + "content": "\\gamma ^ { ( k ) } \\equiv 1", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 447, + 506, + 464 + ], + "score": 1.0, + "content": ", the recovery error converges to zero in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 460, + 236, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 236, + 474 + ], + "score": 1.0, + "content": "a linear rate (Chen et al., 2018):", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 477, + 381, + 492 + ], + "lines": [ + { + "bbox": [ + 228, + 477, + 381, + 492 + ], + "spans": [ + { + "bbox": [ + 228, + 477, + 381, + 492 + ], + "score": 0.91, + "content": "\\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\leq s B \\exp \\big ( - c k \\big ) ,", + "type": "interline_equation", + "image_path": "4433bcc2d247d7003a2053513a5f0de2924101ec9246786ec18c2ac38a5878f3.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 228, + 477, + 381, + 492 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 506, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 133, + 511 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 497, + 246, + 510 + ], + "score": 0.91, + "content": "c = - \\log ( 2 \\tilde { \\mu } s - \\tilde { \\mu } ) \\geq c ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 496, + 290, + 511 + ], + "score": 1.0, + "content": ". 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The signal", + "type": "text" + }, + { + "bbox": [ + 223, + 582, + 235, + 592 + ], + "score": 0.85, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 581, + 441, + 594 + ], + "score": 1.0, + "content": "is a random variable following the distribution", + "type": "text" + }, + { + "bbox": [ + 441, + 582, + 455, + 593 + ], + "score": 0.88, + "content": "P _ { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 581, + 482, + 594 + ], + "score": 1.0, + "content": ". 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Given", + "type": "text" + }, + { + "bbox": [ + 190, + 671, + 311, + 684 + ], + "score": 0.86, + "content": "\\mathbf { D } \\in \\mathbb { R } ^ { N \\times M } , s \\geq 2 , { \\bar { \\sigma } } _ { \\operatorname* { m i n } } > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 668, + 393, + 687 + ], + "score": 1.0, + "content": ", we define a set that", + "type": "text" + }, + { + "bbox": [ + 394, + 671, + 417, + 683 + ], + "score": 0.89, + "content": "\\mathbf { W } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 668, + 489, + 687 + ], + "score": 1.0, + "content": "are chosen from:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 687, + 482, + 709 + ], + "lines": [ + { + "bbox": [ + 110, + 687, + 482, + 709 + ], + "spans": [ + { + "bbox": [ + 110, + 687, + 482, + 709 + ], + "score": 0.9, + "content": "\\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } ) = \\Big \\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } \\Big | \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathrm { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathrm { S } } \\Big ) \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } , \\forall \\mathrm { S ~ w i t h } 2 \\leq | \\mathrm { S } | \\leq s \\Big \\} .", + "type": "interline_equation", + "image_path": "183550e01fc023d1bb510b3e0d6c0286facc71be690a33aa73005a66750ff772.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 110, + 687, + 482, + 709 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 718, + 439, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 716, + 441, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 716, + 277, + 736 + ], + "score": 1.0, + "content": "Based on Definition 2, we define a set that", + "type": "text" + }, + { + "bbox": [ + 278, + 719, + 368, + 733 + ], + "score": 0.93, + "content": "\\boldsymbol \\Theta = \\{ \\mathbf W ^ { ( k ) } , \\boldsymbol \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 716, + 441, + 736 + ], + "score": 1.0, + "content": "are chosen from:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 192, + 97 + ], + "score": 1.0, + "content": "Definition 1. Given", + "type": "text" + }, + { + "bbox": [ + 192, + 81, + 247, + 93 + ], + "score": 0.91, + "content": "\\mathbf { D } \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 79, + 506, + 97 + ], + "score": 1.0, + "content": "with each of its column normalized, we define the generalized", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 183, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 183, + 106 + ], + "score": 1.0, + "content": "mutual coherence:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 79, + 506, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 106, + 419, + 143 + ], + "lines": [ + { + "bbox": [ + 191, + 106, + 419, + 143 + ], + "spans": [ + { + "bbox": [ + 191, + 106, + 419, + 143 + ], + "score": 0.93, + "content": "\\widetilde { \\mu } ( \\mathbf { D } ) = \\operatorname* { i n f } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\bigg \\{ \\operatorname* { m a x } _ { \\substack { 1 \\leq i \\neq j \\leq M } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } \\bigg \\} .", + "type": "interline_equation", + "image_path": "bfdc67a99b7536a7d6d5c7e192b522c6764cd1db5b506af250cfa75d72a6d255.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 106, + 419, + 124.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 191, + 124.5, + 419, + 143.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 504, + 173 + ], + "lines": [ + { + "bbox": [ + 104, + 144, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 204, + 163 + ], + "score": 1.0, + "content": "Additionally, We define", + "type": "text" + }, + { + "bbox": [ + 204, + 147, + 335, + 162 + ], + "score": 0.89, + "content": "{ \\mathcal { W } } ( \\mathbf { D } ) = \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } : \\mathbf { W } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 144, + 441, + 163 + ], + "score": 1.0, + "content": "attains the infimum given", + "type": "text" + }, + { + "bbox": [ + 441, + 148, + 459, + 162 + ], + "score": 0.86, + "content": "( \\delta ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 144, + 506, + 163 + ], + "score": 1.0, + "content": ". A weight", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 256, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 194, + 174 + ], + "score": 1.0, + "content": "matrix W is “good”", + "type": "text" + }, + { + "bbox": [ + 195, + 161, + 252, + 173 + ], + "score": 0.88, + "content": "f \\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 159, + 256, + 174 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 144, + 506, + 174 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 504, + 204 + ], + "lines": [ + { + "bbox": [ + 104, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 179, + 375, + 194 + ], + "score": 1.0, + "content": "In the above definition, problem (8) is feasible and attainable, i.e.,", + "type": "text" + }, + { + "bbox": [ + 375, + 181, + 425, + 193 + ], + "score": 0.92, + "content": "\\mathcal { W } ( \\mathbf { D } ) \\neq \\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 179, + 505, + 194 + ], + "score": 1.0, + "content": ", which was proven", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 248, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 248, + 203 + ], + "score": 1.0, + "content": "in Lemma 1 of (Chen et al., 2018).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 179, + 505, + 203 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 504, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 325, + 221 + ], + "score": 1.0, + "content": "Theorem 1 (Recovery error upper bound). Take any", + "type": "text" + }, + { + "bbox": [ + 326, + 206, + 386, + 218 + ], + "score": 0.93, + "content": "\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 204, + 409, + 221 + ], + "score": 1.0, + "content": ", any", + "type": "text" + }, + { + "bbox": [ + 410, + 206, + 465, + 218 + ], + "score": 0.9, + "content": "\\mathbf { W } \\in \\mathcal { W } ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 204, + 505, + 221 + ], + "score": 1.0, + "content": ", and any", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 214, + 434, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 145, + 237 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 217, + 228, + 233 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\gamma ^ { ( k ) } \\in ( 0 , \\frac { 2 } { 2 \\tilde { \\mu } s - \\tilde { \\mu } + 1 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 214, + 373, + 237 + ], + "score": 1.0, + "content": ". 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The signal", + "type": "text" + }, + { + "bbox": [ + 223, + 582, + 235, + 592 + ], + "score": 0.85, + "content": "\\mathbf { x } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 581, + 441, + 594 + ], + "score": 1.0, + "content": "is a random variable following the distribution", + "type": "text" + }, + { + "bbox": [ + 441, + 582, + 455, + 593 + ], + "score": 0.88, + "content": "P _ { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 581, + 482, + 594 + ], + "score": 1.0, + "content": ". 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But this cannot be met", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 219, + 658 + ], + "score": 1.0, + "content": "exactly in the overcomplete", + "type": "text" + }, + { + "bbox": [ + 220, + 647, + 230, + 657 + ], + "score": 0.54, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 646, + 271, + 658 + ], + "score": 1.0, + "content": "case, i.e.,", + "type": "text" + }, + { + "bbox": [ + 271, + 647, + 307, + 657 + ], + "score": 0.9, + "content": "N < M", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 646, + 505, + 658 + ], + "score": 1.0, + "content": ". 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Given", + "type": "text" + }, + { + "bbox": [ + 190, + 671, + 311, + 684 + ], + "score": 0.86, + "content": "\\mathbf { D } \\in \\mathbb { R } ^ { N \\times M } , s \\geq 2 , { \\bar { \\sigma } } _ { \\operatorname* { m i n } } > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 668, + 393, + 687 + ], + "score": 1.0, + "content": ", we define a set that", + "type": "text" + }, + { + "bbox": [ + 394, + 671, + 417, + 683 + ], + "score": 0.89, + "content": "\\mathbf { W } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 668, + 489, + 687 + ], + "score": 1.0, + "content": "are chosen from:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 668, + 489, + 687 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 687, + 482, + 709 + ], + "lines": [ + { + "bbox": [ + 110, + 687, + 482, + 709 + ], + "spans": [ + { + "bbox": [ + 110, + 687, + 482, + 709 + ], + "score": 0.9, + "content": "\\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } ) = \\Big \\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } \\Big | \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathrm { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathrm { S } } \\Big ) \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } , \\forall \\mathrm { S ~ w i t h } 2 \\leq | \\mathrm { S } | \\leq s \\Big \\} .", + "type": "interline_equation", + "image_path": "183550e01fc023d1bb510b3e0d6c0286facc71be690a33aa73005a66750ff772.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 110, + 687, + 482, + 709 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 718, + 439, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 716, + 441, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 716, + 277, + 736 + ], + "score": 1.0, + "content": "Based on Definition 2, we define a set that", + "type": "text" + }, + { + "bbox": [ + 278, + 719, + 368, + 733 + ], + "score": 0.93, + "content": "\\boldsymbol \\Theta = \\{ \\mathbf W ^ { ( k ) } , \\boldsymbol \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 716, + 441, + 736 + ], + "score": 1.0, + "content": "are chosen from:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 716, + 441, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 178, + 97 + ], + "score": 1.0, + "content": "Definition 3. Let", + "type": "text" + }, + { + "bbox": [ + 179, + 81, + 240, + 95 + ], + "score": 0.92, + "content": "\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 79, + 340, + 97 + ], + "score": 1.0, + "content": "be generated by (6) with", + "type": "text" + }, + { + "bbox": [ + 340, + 81, + 409, + 95 + ], + "score": 0.92, + "content": "\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 79, + 428, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 81, + 465, + 93 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 79, + 506, + 97 + ], + "score": 1.0, + "content": ". Then we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 435, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 133, + 108 + ], + "score": 1.0, + "content": "define", + "type": "text" + }, + { + "bbox": [ + 133, + 95, + 142, + 105 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 93, + 410, + 108 + ], + "score": 1.0, + "content": "as the set of parameters that guarantee there is no false positive in", + "type": "text" + }, + { + "bbox": [ + 411, + 94, + 429, + 105 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 93, + 435, + 108 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 111, + 478, + 133 + ], + "lines": [ + { + "bbox": [ + 116, + 111, + 478, + 133 + ], + "spans": [ + { + "bbox": [ + 116, + 111, + 478, + 133 + ], + "score": 0.92, + "content": "{ \\mathcal { T } } = \\Big \\{ \\{ \\mathbf { W } ^ { ( k ) } \\in \\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { m i n } ) , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } , \\forall \\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) , \\forall k \\Big \\}", + "type": "interline_equation", + "image_path": "4e8753ce984d55a7c2a4c1df63f194e6773e7702b44a866715b252e731ba5d12.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 116, + 111, + 478, + 133 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 265, + 159 + ], + "score": 1.0, + "content": "The conclusion (10) demonstrates that", + "type": "text" + }, + { + "bbox": [ + 266, + 147, + 275, + 156 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 145, + 400, + 159 + ], + "score": 1.0, + "content": "is nonempty because “support", + "type": "text" + }, + { + "bbox": [ + 401, + 145, + 470, + 158 + ], + "score": 0.81, + "content": "( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 145, + 505, + 159 + ], + "score": 1.0, + "content": "is satis-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 155, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 169, + 173 + ], + "score": 1.0, + "content": "fied as long as", + "type": "text" + }, + { + "bbox": [ + 170, + 157, + 197, + 168 + ], + "score": 0.9, + "content": "\\theta ^ { ( k - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 155, + 300, + 173 + ], + "score": 1.0, + "content": "large enough. Actually,", + "type": "text" + }, + { + "bbox": [ + 300, + 159, + 310, + 169 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 155, + 506, + 173 + ], + "score": 1.0, + "content": "contains almost all “good” parameters because", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 169, + 441, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 357, + 182 + ], + "score": 1.0, + "content": "considerable false positives lead to large recovery errors. With", + "type": "text" + }, + { + "bbox": [ + 357, + 170, + 367, + 180 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 169, + 441, + 182 + ], + "score": 1.0, + "content": "defined, we have:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 345, + 200 + ], + "score": 1.0, + "content": "Theorem 2 (Recovery error lower bound). Let the sequence", + "type": "text" + }, + { + "bbox": [ + 345, + 184, + 407, + 198 + ], + "score": 0.91, + "content": "\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 183, + 506, + 200 + ], + "score": 1.0, + "content": "be generated by (6) with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 191, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 107, + 196, + 176, + 210 + ], + "score": 0.92, + "content": "\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 191, + 186, + 222 + ], + "score": 1.0, + "content": "aall", + "type": "text" + }, + { + "bbox": [ + 195, + 197, + 231, + 208 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 191, + 397, + 222 + ], + "score": 1.0, + "content": ". Under Assumption 2, for all parameters ave", + "type": "text" + }, + { + "bbox": [ + 397, + 197, + 486, + 211 + ], + "score": 0.93, + "content": "\\{ \\mathbf { \\bar { W } } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty } \\in \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 196, + 507, + 212 + ], + "score": 1.0, + "content": ")}∞k=0 ∈ T and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 186, + 210, + 209, + 219 + ], + "spans": [ + { + "bbox": [ + 186, + 210, + 209, + 219 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 225, + 386, + 241 + ], + "lines": [ + { + "bbox": [ + 223, + 225, + 386, + 241 + ], + "spans": [ + { + "bbox": [ + 223, + 225, + 386, + 241 + ], + "score": 0.91, + "content": "\\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\geq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\exp ( - \\bar { c } k ) ,", + "type": "interline_equation", + "image_path": "510fb2fcf239f765a9cc93a25510fdd8b826ca7b2eb36522b47a02d04bc0aa05.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 223, + 225, + 386, + 241 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 405, + 261 + ], + "lines": [ + { + "bbox": [ + 104, + 243, + 403, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 205, + 264 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 205, + 246, + 271, + 260 + ], + "score": 0.91, + "content": "( 1 - \\epsilon s ^ { 3 / 2 } - \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 243, + 302, + 264 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 247, + 403, + 261 + ], + "score": 0.87, + "content": "\\bar { c } = s \\log ( 3 ) - \\log ( \\bar { \\sigma } _ { m i n } ) .", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 506, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "This theorem illustrates that, with high probability, the convergence rate of LISTA cannot be faster", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 278, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 482, + 293 + ], + "score": 1.0, + "content": "than a linear rate. Thus, the parameters given in (9), that leads to the linear convergence if", + "type": "text" + }, + { + "bbox": [ + 482, + 279, + 494, + 291 + ], + "score": 0.89, + "content": "\\gamma ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 278, + 506, + 293 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 290, + 498, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 498, + 304 + ], + "score": 1.0, + "content": "bounded within an interval near 1, are optimal with respect to the order of convergence of LISTA.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 311, + 339, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 342, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 342, + 324 + ], + "score": 1.0, + "content": "2.3 ANALYTIC LISTA: LESS PARAMETERS TO LEARN", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 109, + 325, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 261, + 341 + ], + "score": 1.0, + "content": "Following Theorems 1 and 2, we set", + "type": "text" + }, + { + "bbox": [ + 262, + 325, + 331, + 338 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 322, + 364, + 341 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 365, + 325, + 383, + 338 + ], + "score": 0.89, + "content": "\\gamma ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 322, + 506, + 341 + ], + "score": 1.0, + "content": "is a scalar, and propose Tied", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 339, + 401, + 360 + ], + "lines": [ + { + "bbox": [ + 208, + 339, + 401, + 360 + ], + "spans": [ + { + "bbox": [ + 208, + 339, + 401, + 360 + ], + "score": 0.86, + "content": "\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - \\gamma ^ { ( k ) } \\mathbf { W } ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } ) \\Big ) ,", + "type": "interline_equation", + "image_path": "10f52ec24b40d3fd861aecf9edbcb46182777d7bccc51a3fc8408236e2936f9f.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 208, + 339, + 401, + 360 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 133, + 379 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 362, + 250, + 377 + ], + "score": 0.93, + "content": "\\boldsymbol { \\Theta } = \\left\\{ \\{ \\gamma ^ { ( k ) } \\} _ { k } , \\{ \\theta ^ { ( k ) } \\} _ { k } , \\mathbf { W } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 362, + 390, + 379 + ], + "score": 1.0, + "content": "are parameters to train. The matrix", + "type": "text" + }, + { + "bbox": [ + 391, + 364, + 404, + 374 + ], + "score": 0.34, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 362, + 506, + 379 + ], + "score": 1.0, + "content": "is tied over all the layers.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 300, + 389 + ], + "score": 1.0, + "content": "Further, we notice that the selection of W from", + "type": "text" + }, + { + "bbox": [ + 301, + 376, + 329, + 388 + ], + "score": 0.9, + "content": "\\mathcal { W } ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 375, + 378, + 389 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 379, + 376, + 389, + 386 + ], + "score": 0.58, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "only. Hence we propose the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 401, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 401, + 400 + ], + "score": 1.0, + "content": "analytic LISTA (ALISTA) that decomposes tied-LISTA into two stages:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 403, + 402, + 425 + ], + "lines": [ + { + "bbox": [ + 208, + 403, + 402, + 425 + ], + "spans": [ + { + "bbox": [ + 208, + 403, + 402, + 425 + ], + "score": 0.92, + "content": "\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - \\gamma ^ { ( k ) } \\tilde { \\mathbf { W } } ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } ) \\Big ) ,", + "type": "interline_equation", + "image_path": "d0d53ca6ce78306068072b685183b5aa3ec02b0a7a2c79f65dd6e2ce27423f09.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 208, + 403, + 402, + 425 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 432, + 396, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 398, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 133, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 431, + 146, + 442 + ], + "score": 0.83, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 430, + 398, + 447 + ], + "score": 1.0, + "content": "is pre-computed by solving the following problem (Stage 1)3:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 448, + 457, + 473 + ], + "lines": [ + { + "bbox": [ + 153, + 448, + 457, + 473 + ], + "spans": [ + { + "bbox": [ + 153, + 448, + 457, + 473 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { { { \\mathbf { W } } } } \\in \\underset { { \\mathbf { W } } \\in \\mathbb { R } ^ { N \\times M } } { \\arg \\operatorname* { m i n } } \\left\\| \\mathbf { W } ^ { T } { \\mathbf { D } } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( { \\mathbf { W } } _ { : , m } \\right) ^ { T } { \\mathbf { D } } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M , } \\end{array}", + "type": "interline_equation", + "image_path": "e243cb1fff86ec715b2ec44172d032059e8dac517e180f6e60abe7efcf3244d6.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 153, + 448, + 457, + 473 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 504, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 150, + 495 + ], + "score": 1.0, + "content": "Then with", + "type": "text" + }, + { + "bbox": [ + 150, + 479, + 164, + 491 + ], + "score": 0.74, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 478, + 190, + 495 + ], + "score": 1.0, + "content": "fixed,", + "type": "text" + }, + { + "bbox": [ + 191, + 479, + 243, + 493 + ], + "score": 0.93, + "content": "\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 478, + 506, + 495 + ], + "score": 1.0, + "content": "in (15) are learned from end to end (Stage 2). (16) reformulates", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 269, + 505 + ], + "score": 1.0, + "content": "(8) to minimizing the Frobenius norm of", + "type": "text" + }, + { + "bbox": [ + 269, + 491, + 297, + 502 + ], + "score": 0.9, + "content": "{ \\bf W } ^ { T } { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "(a quadratic objective), over linear constraints. This", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 503, + 465, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 465, + 515 + ], + "score": 1.0, + "content": "is a standard convex quadratic program, which is easier to solve than to solve (8) directly.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 150, + 516, + 461, + 527 + ], + "lines": [ + { + "bbox": [ + 149, + 514, + 461, + 530 + ], + "spans": [ + { + "bbox": [ + 149, + 514, + 461, + 530 + ], + "score": 1.0, + "content": "Table 1: Summary: variants of LISTA and the number of parameters to learn.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "table", + "bbox": [ + 143, + 534, + 465, + 560 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 143, + 534, + 465, + 560 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 534, + 465, + 560 + ], + "spans": [ + { + "bbox": [ + 143, + 534, + 465, + 560 + ], + "score": 0.962, + "html": "
Vanilla LISTA (4)LISTA-CPSS (6)TiLISTA (14)ALISTA (15)
O(KM²+K+MN)O(KNM+K)O(NM+ K)O(K)
", + "type": "table", + "image_path": "ca71257d3e26a0167489a41309483e948a3981082b4ccab6c604c199fb998341.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 143, + 534, + 465, + 560 + ], + "spans": [], + "index": 27 + } + ] + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 573, + 313, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 314, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 314, + 588 + ], + "score": 1.0, + "content": "3 CONVOLUTIONAL ANALYTIC LISTA", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "We extend the analytic LISTA to the convolutional case in this section, starting from discussing the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "convolutional sparse coding (CSC). Many works studied CSC and proposed efficient algorithms for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "that (Bristow et al., 2013; Heide et al., 2015; Wohlberg, 2014; 2016; Papyan et al., 2017; Garcia-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "Cardona & Wohlberg, 2018; Wang et al., 2018; Liu et al., 2017; 2018). In CSC, the general linear", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 636, + 436, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 436, + 650 + ], + "score": 1.0, + "content": "transform is replaced by convolutions in order to learn spatially invariant features:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 652, + 353, + 686 + ], + "lines": [ + { + "bbox": [ + 256, + 652, + 353, + 686 + ], + "spans": [ + { + "bbox": [ + 256, + 652, + 353, + 686 + ], + "score": 0.95, + "content": "{ \\bf b } = \\sum _ { m = 1 } ^ { M } { \\bf d } _ { m } * { \\bf x } _ { m } ^ { * } + \\varepsilon ,", + "type": "interline_equation", + "image_path": "79e6b6b033d97cf03d6d6bef2bd0bf65e72303fcd2b2a426542a828dfd6a59d3.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 256, + 652, + 353, + 669.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 256, + 669.0, + 353, + 686.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 506, + 713 + ], + "lines": [ + { + "bbox": [ + 103, + 685, + 509, + 708 + ], + "spans": [ + { + "bbox": [ + 103, + 685, + 153, + 708 + ], + "score": 1.0, + "content": "where each", + "type": "text" + }, + { + "bbox": [ + 154, + 691, + 168, + 701 + ], + "score": 0.9, + "content": "\\mathbf { d } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 685, + 297, + 708 + ], + "score": 1.0, + "content": "is a dictionary kernel (or filter).", + "type": "text" + }, + { + "bbox": [ + 297, + 689, + 339, + 702 + ], + "score": 0.92, + "content": "\\lbrace \\mathbf { d } _ { m } \\rbrace _ { m = 1 } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 685, + 444, + 708 + ], + "score": 1.0, + "content": "is the dictionary of filters,", + "type": "text" + }, + { + "bbox": [ + 445, + 691, + 457, + 700 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 685, + 509, + 708 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 103, + 695, + 509, + 718 + ], + "spans": [ + { + "bbox": [ + 103, + 695, + 178, + 718 + ], + "score": 1.0, + "content": "number of filters.", + "type": "text" + }, + { + "bbox": [ + 179, + 700, + 221, + 713 + ], + "score": 0.93, + "content": "\\{ \\mathbf { x } _ { m } ^ { * } \\} _ { m = 1 } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 695, + 509, + 718 + ], + "score": 1.0, + "content": "m=1 is the set of coefficient maps that are assumed to have sparse structure,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 118, + 721, + 416, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 416, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 416, + 734 + ], + "score": 1.0, + "content": "3Some details and a complexity analysis of Stage 1 are discussed in Appendix E.1", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 178, + 97 + ], + "score": 1.0, + "content": "Definition 3. Let", + "type": "text" + }, + { + "bbox": [ + 179, + 81, + 240, + 95 + ], + "score": 0.92, + "content": "\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 79, + 340, + 97 + ], + "score": 1.0, + "content": "be generated by (6) with", + "type": "text" + }, + { + "bbox": [ + 340, + 81, + 409, + 95 + ], + "score": 0.92, + "content": "\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 79, + 428, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 81, + 465, + 93 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 79, + 506, + 97 + ], + "score": 1.0, + "content": ". Then we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 435, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 133, + 108 + ], + "score": 1.0, + "content": "define", + "type": "text" + }, + { + "bbox": [ + 133, + 95, + 142, + 105 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 93, + 410, + 108 + ], + "score": 1.0, + "content": "as the set of parameters that guarantee there is no false positive in", + "type": "text" + }, + { + "bbox": [ + 411, + 94, + 429, + 105 + ], + "score": 0.89, + "content": "\\mathbf { x } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 93, + 435, + 108 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 79, + 506, + 108 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 111, + 478, + 133 + ], + "lines": [ + { + "bbox": [ + 116, + 111, + 478, + 133 + ], + "spans": [ + { + "bbox": [ + 116, + 111, + 478, + 133 + ], + "score": 0.92, + "content": "{ \\mathcal { T } } = \\Big \\{ \\{ \\mathbf { W } ^ { ( k ) } \\in \\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { m i n } ) , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } , \\forall \\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s ) , \\forall k \\Big \\}", + "type": "interline_equation", + "image_path": "4e8753ce984d55a7c2a4c1df63f194e6773e7702b44a866715b252e731ba5d12.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 116, + 111, + 478, + 133 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 265, + 159 + ], + "score": 1.0, + "content": "The conclusion (10) demonstrates that", + "type": "text" + }, + { + "bbox": [ + 266, + 147, + 275, + 156 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 145, + 400, + 159 + ], + "score": 1.0, + "content": "is nonempty because “support", + "type": "text" + }, + { + "bbox": [ + 401, + 145, + 470, + 158 + ], + "score": 0.81, + "content": "( \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) ) \\subset \\mathbb { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 145, + 505, + 159 + ], + "score": 1.0, + "content": "is satis-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 155, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 169, + 173 + ], + "score": 1.0, + "content": "fied as long as", + "type": "text" + }, + { + "bbox": [ + 170, + 157, + 197, + 168 + ], + "score": 0.9, + "content": "\\theta ^ { ( k - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 155, + 300, + 173 + ], + "score": 1.0, + "content": "large enough. Actually,", + "type": "text" + }, + { + "bbox": [ + 300, + 159, + 310, + 169 + ], + "score": 0.82, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 155, + 506, + 173 + ], + "score": 1.0, + "content": "contains almost all “good” parameters because", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 169, + 441, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 357, + 182 + ], + "score": 1.0, + "content": "considerable false positives lead to large recovery errors. With", + "type": "text" + }, + { + "bbox": [ + 357, + 170, + 367, + 180 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 169, + 441, + 182 + ], + "score": 1.0, + "content": "defined, we have:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 145, + 506, + 182 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 345, + 200 + ], + "score": 1.0, + "content": "Theorem 2 (Recovery error lower bound). Let the sequence", + "type": "text" + }, + { + "bbox": [ + 345, + 184, + 407, + 198 + ], + "score": 0.91, + "content": "\\{ \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) \\} _ { k = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 183, + 506, + 200 + ], + "score": 1.0, + "content": "be generated by (6) with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 191, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 107, + 196, + 176, + 210 + ], + "score": 0.92, + "content": "\\{ \\mathbf { W } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 191, + 186, + 222 + ], + "score": 1.0, + "content": "aall", + "type": "text" + }, + { + "bbox": [ + 195, + 197, + 231, + 208 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 191, + 397, + 222 + ], + "score": 1.0, + "content": ". Under Assumption 2, for all parameters ave", + "type": "text" + }, + { + "bbox": [ + 397, + 197, + 486, + 211 + ], + "score": 0.93, + "content": "\\{ \\mathbf { \\bar { W } } ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k = 0 } ^ { \\infty } \\in \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 196, + 507, + 212 + ], + "score": 1.0, + "content": ")}∞k=0 ∈ T and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 186, + 210, + 209, + 219 + ], + "spans": [ + { + "bbox": [ + 186, + 210, + 209, + 219 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 183, + 507, + 222 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 225, + 386, + 241 + ], + "lines": [ + { + "bbox": [ + 223, + 225, + 386, + 241 + ], + "spans": [ + { + "bbox": [ + 223, + 225, + 386, + 241 + ], + "score": 0.91, + "content": "\\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 2 } \\geq \\epsilon \\| \\mathbf { x } ^ { * } \\| _ { 2 } \\exp ( - \\bar { c } k ) ,", + "type": "interline_equation", + "image_path": "510fb2fcf239f765a9cc93a25510fdd8b826ca7b2eb36522b47a02d04bc0aa05.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 223, + 225, + 386, + 241 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 405, + 261 + ], + "lines": [ + { + "bbox": [ + 104, + 243, + 403, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 243, + 205, + 264 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 205, + 246, + 271, + 260 + ], + "score": 0.91, + "content": "( 1 - \\epsilon s ^ { 3 / 2 } - \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 243, + 302, + 264 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 247, + 403, + 261 + ], + "score": 0.87, + "content": "\\bar { c } = s \\log ( 3 ) - \\log ( \\bar { \\sigma } _ { m i n } ) .", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 243, + 403, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 506, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "This theorem illustrates that, with high probability, the convergence rate of LISTA cannot be faster", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 278, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 482, + 293 + ], + "score": 1.0, + "content": "than a linear rate. Thus, the parameters given in (9), that leads to the linear convergence if", + "type": "text" + }, + { + "bbox": [ + 482, + 279, + 494, + 291 + ], + "score": 0.89, + "content": "\\gamma ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 278, + 506, + 293 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 290, + 498, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 498, + 304 + ], + "score": 1.0, + "content": "bounded within an interval near 1, are optimal with respect to the order of convergence of LISTA.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 268, + 506, + 304 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 311, + 339, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 342, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 342, + 324 + ], + "score": 1.0, + "content": "2.3 ANALYTIC LISTA: LESS PARAMETERS TO LEARN", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 109, + 325, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 261, + 341 + ], + "score": 1.0, + "content": "Following Theorems 1 and 2, we set", + "type": "text" + }, + { + "bbox": [ + 262, + 325, + 331, + 338 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 322, + 364, + 341 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 365, + 325, + 383, + 338 + ], + "score": 0.89, + "content": "\\gamma ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 322, + 506, + 341 + ], + "score": 1.0, + "content": "is a scalar, and propose Tied", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 322, + 506, + 341 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 339, + 401, + 360 + ], + "lines": [ + { + "bbox": [ + 208, + 339, + 401, + 360 + ], + "spans": [ + { + "bbox": [ + 208, + 339, + 401, + 360 + ], + "score": 0.86, + "content": "\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - \\gamma ^ { ( k ) } \\mathbf { W } ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } ) \\Big ) ,", + "type": "interline_equation", + "image_path": "10f52ec24b40d3fd861aecf9edbcb46182777d7bccc51a3fc8408236e2936f9f.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 208, + 339, + 401, + 360 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 133, + 379 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 362, + 250, + 377 + ], + "score": 0.93, + "content": "\\boldsymbol { \\Theta } = \\left\\{ \\{ \\gamma ^ { ( k ) } \\} _ { k } , \\{ \\theta ^ { ( k ) } \\} _ { k } , \\mathbf { W } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 362, + 390, + 379 + ], + "score": 1.0, + "content": "are parameters to train. The matrix", + "type": "text" + }, + { + "bbox": [ + 391, + 364, + 404, + 374 + ], + "score": 0.34, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 362, + 506, + 379 + ], + "score": 1.0, + "content": "is tied over all the layers.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 300, + 389 + ], + "score": 1.0, + "content": "Further, we notice that the selection of W from", + "type": "text" + }, + { + "bbox": [ + 301, + 376, + 329, + 388 + ], + "score": 0.9, + "content": "\\mathcal { W } ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 375, + 378, + 389 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 379, + 376, + 389, + 386 + ], + "score": 0.58, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "only. Hence we propose the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 401, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 401, + 400 + ], + "score": 1.0, + "content": "analytic LISTA (ALISTA) that decomposes tied-LISTA into two stages:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 362, + 506, + 400 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 403, + 402, + 425 + ], + "lines": [ + { + "bbox": [ + 208, + 403, + 402, + 425 + ], + "spans": [ + { + "bbox": [ + 208, + 403, + 402, + 425 + ], + "score": 0.92, + "content": "\\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - \\gamma ^ { ( k ) } \\tilde { \\mathbf { W } } ^ { T } ( \\mathbf { D } \\mathbf { x } ^ { ( k ) } - \\mathbf { b } ) \\Big ) ,", + "type": "interline_equation", + "image_path": "d0d53ca6ce78306068072b685183b5aa3ec02b0a7a2c79f65dd6e2ce27423f09.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 208, + 403, + 402, + 425 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 432, + 396, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 398, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 133, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 431, + 146, + 442 + ], + "score": 0.83, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 430, + 398, + 447 + ], + "score": 1.0, + "content": "is pre-computed by solving the following problem (Stage 1)3:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 430, + 398, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 448, + 457, + 473 + ], + "lines": [ + { + "bbox": [ + 153, + 448, + 457, + 473 + ], + "spans": [ + { + "bbox": [ + 153, + 448, + 457, + 473 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { { { \\mathbf { W } } } } \\in \\underset { { \\mathbf { W } } \\in \\mathbb { R } ^ { N \\times M } } { \\arg \\operatorname* { m i n } } \\left\\| \\mathbf { W } ^ { T } { \\mathbf { D } } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( { \\mathbf { W } } _ { : , m } \\right) ^ { T } { \\mathbf { D } } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M , } \\end{array}", + "type": "interline_equation", + "image_path": "e243cb1fff86ec715b2ec44172d032059e8dac517e180f6e60abe7efcf3244d6.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 153, + 448, + 457, + 473 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 504, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 150, + 495 + ], + "score": 1.0, + "content": "Then with", + "type": "text" + }, + { + "bbox": [ + 150, + 479, + 164, + 491 + ], + "score": 0.74, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 478, + 190, + 495 + ], + "score": 1.0, + "content": "fixed,", + "type": "text" + }, + { + "bbox": [ + 191, + 479, + 243, + 493 + ], + "score": 0.93, + "content": "\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 478, + 506, + 495 + ], + "score": 1.0, + "content": "in (15) are learned from end to end (Stage 2). (16) reformulates", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 269, + 505 + ], + "score": 1.0, + "content": "(8) to minimizing the Frobenius norm of", + "type": "text" + }, + { + "bbox": [ + 269, + 491, + 297, + 502 + ], + "score": 0.9, + "content": "{ \\bf W } ^ { T } { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "(a quadratic objective), over linear constraints. This", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 503, + 465, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 465, + 515 + ], + "score": 1.0, + "content": "is a standard convex quadratic program, which is easier to solve than to solve (8) directly.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 478, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 150, + 516, + 461, + 527 + ], + "lines": [ + { + "bbox": [ + 149, + 514, + 461, + 530 + ], + "spans": [ + { + "bbox": [ + 149, + 514, + 461, + 530 + ], + "score": 1.0, + "content": "Table 1: Summary: variants of LISTA and the number of parameters to learn.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 149, + 514, + 461, + 530 + ] + }, + { + "type": "table", + "bbox": [ + 143, + 534, + 465, + 560 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 143, + 534, + 465, + 560 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 534, + 465, + 560 + ], + "spans": [ + { + "bbox": [ + 143, + 534, + 465, + 560 + ], + "score": 0.962, + "html": "
Vanilla LISTA (4)LISTA-CPSS (6)TiLISTA (14)ALISTA (15)
O(KM²+K+MN)O(KNM+K)O(NM+ K)O(K)
", + "type": "table", + "image_path": "ca71257d3e26a0167489a41309483e948a3981082b4ccab6c604c199fb998341.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 143, + 534, + 465, + 560 + ], + "spans": [], + "index": 27 + } + ] + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 573, + 313, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 314, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 314, + 588 + ], + "score": 1.0, + "content": "3 CONVOLUTIONAL ANALYTIC LISTA", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "We extend the analytic LISTA to the convolutional case in this section, starting from discussing the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "convolutional sparse coding (CSC). Many works studied CSC and proposed efficient algorithms for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "that (Bristow et al., 2013; Heide et al., 2015; Wohlberg, 2014; 2016; Papyan et al., 2017; Garcia-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "Cardona & Wohlberg, 2018; Wang et al., 2018; Liu et al., 2017; 2018). 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\\left( { \\bf w } _ { m } ^ { ( k ) } \\right) ^ { \\prime } \\ast \\Big ( \\sum _ { \\bar { m } = 1 } ^ { M } { \\bf d } _ { \\bar { m } } \\ast { \\bf x } _ { \\bar { m } } ^ { ( k ) } - { \\bf b } \\Big ) \\right) , \\quad m = 1 , 2 , \\cdots , M ,", + "type": "interline_equation", + "image_path": "6a41f5b8ec4e924d64e16be5089ef290d15a52a1ee59134fa9f323ec3227516f.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 142, + 253, + 468, + 264.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 142, + 264.6666666666667, + 468, + 276.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 142, + 276.33333333333337, + 468, + 288.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 508, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 188, + 310 + ], + "score": 1.0, + "content": "where {w(k)m }Mm=1", + "type": "text" + }, + { + "bbox": [ + 182, + 290, + 290, + 312 + ], + "score": 1.0, + "content": "share the same sizes with", + "type": "text" + }, + { + "bbox": [ + 291, + 294, + 333, + 307 + ], + "score": 0.93, + "content": "\\lbrace \\mathbf { d } _ { m } \\rbrace _ { m = 1 } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 290, + 353, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 353, + 294, + 367, + 307 + ], + "score": 0.86, + "content": "( \\cdot ) ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 290, + 508, + 312 + ], + "score": 1.0, + "content": "means a 180 rotation of the filter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 351, + 321 + ], + "score": 1.0, + "content": "(Chalasani et al., 2013). 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Since convolution can be written as a matrix form (19), (20) is equivalent to", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 336, + 506, + 368 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 366, + 435, + 388 + ], + "lines": [ + { + "bbox": [ + 175, + 366, + 435, + 388 + ], + "spans": [ + { + "bbox": [ + 175, + 366, + 435, + 388 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbf { x } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( \\mathbf { x } ^ { ( k ) } - ( \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { ( k ) } ) ) ^ { T } \\big ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) \\mathbf { x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "be0b87973f2b484a025c5e178ff1e406d836e65fe850fb9fb80abe8294b281f8.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 175, + 366, + 435, + 388 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 503, + 416 + ], + "lines": [ + { + "bbox": [ + 103, + 389, + 508, + 417 + ], + "spans": [ + { + "bbox": [ + 103, + 389, + 211, + 417 + ], + "score": 1.0, + "content": "Then by just substituting can be applied to the conv", + "type": "text" + }, + { + "bbox": [ + 211, + 392, + 249, + 405 + ], + "score": 0.89, + "content": "\\mathbf { D } , \\mathbf { W } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 389, + 272, + 417 + ], + "score": 1.0, + "content": "with LIST", + "type": "text" + }, + { + "bbox": [ + 272, + 392, + 373, + 406 + ], + "score": 0.91, + "content": "\\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) , \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { ( k ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 389, + 508, + 417 + ], + "score": 1.0, + "content": "respectively, Theorems 1 and 2", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 103, + 389, + 508, + 417 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 102, + 415, + 508, + 439 + ], + "spans": [ + { + "bbox": [ + 102, + 415, + 189, + 439 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 189, + 419, + 255, + 433 + ], + "score": 0.92, + "content": "\\mathbf { D } = \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 415, + 277, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 277, + 419, + 374, + 433 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { ( k ) } = \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\big ( \\mathbf { w } ^ { ( k ) } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 415, + 508, + 439 + ], + "score": 1.0, + "content": ". With Assumption 1 and other", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 274, + 444 + ], + "score": 1.0, + "content": "settings the same with those in Theorem", + "type": "text" + }, + { + "bbox": [ + 274, + 433, + 280, + 442 + ], + "score": 0.36, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 432, + 285, + 444 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 285, + 432, + 303, + 443 + ], + "score": 0.37, + "content": "( I O )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "holds. With Assumption 2 and other settings the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 275, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 275, + 454 + ], + "score": 1.0, + "content": "same with those in Theorem 2, (13) holds.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 102, + 415, + 508, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 464, + 503, + 491 + ], + "lines": [ + { + "bbox": [ + 104, + 460, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 472, + 481 + ], + "score": 1.0, + "content": "Similar to the fully connected case (15), based on the results in Proposition 1, we should set", + "type": "text" + }, + { + "bbox": [ + 473, + 462, + 505, + 477 + ], + "score": 0.89, + "content": "\\mathbf { w } _ { m } ^ { ( k ) } =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 402, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 140, + 491 + ], + "score": 0.33, + "content": "\\gamma _ { m } ^ { ( k ) } \\tilde { \\mathbf { w } } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 472, + 146, + 495 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 146, + 478, + 218, + 490 + ], + "score": 0.71, + "content": "m = 1 , 2 , \\cdots , M", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 472, + 249, + 495 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 249, + 478, + 337, + 491 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { w } } = [ \\tilde { \\mathbf { w } } _ { 1 } , \\cdots , \\tilde { \\mathbf { w } } _ { M } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 472, + 402, + 495 + ], + "score": 1.0, + "content": "is chosen from", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 460, + 505, + 495 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 495, + 436, + 532 + ], + "lines": [ + { + "bbox": [ + 175, + 495, + 436, + 532 + ], + "spans": [ + { + "bbox": [ + 175, + 495, + 436, + 532 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { w } } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\underset { \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , 1 \\leq m \\leq M } { \\arg \\operatorname* { m i n } } \\left\\| \\left( \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } ) \\right\\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "69facdcab8d97cc0763450ae507fbf8e8a4df1363bc5de378b427a244649f251.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 175, + 495, + 436, + 507.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 175, + 507.3333333333333, + 436, + 519.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 175, + 519.6666666666666, + 436, + 532.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 506, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 535, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 535, + 385, + 553 + ], + "score": 1.0, + "content": "However, (22) is not as efficient to solve as (16). To see that, matrices", + "type": "text" + }, + { + "bbox": [ + 385, + 537, + 425, + 550 + ], + "score": 0.89, + "content": "\\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 535, + 443, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 537, + 489, + 550 + ], + "score": 0.93, + "content": "\\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 535, + 506, + 553 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 547, + 508, + 568 + ], + "spans": [ + { + "bbox": [ + 103, + 547, + 156, + 568 + ], + "score": 1.0, + "content": "both of size", + "type": "text" + }, + { + "bbox": [ + 156, + 551, + 254, + 564 + ], + "score": 0.9, + "content": "N ^ { 2 } \\times ( N + D - 1 ) ^ { 2 } M", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 547, + 346, + 568 + ], + "score": 1.0, + "content": ", the coherence matrix", + "type": "text" + }, + { + "bbox": [ + 346, + 550, + 446, + 565 + ], + "score": 0.91, + "content": "\\left( \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 547, + 508, + 568 + ], + "score": 1.0, + "content": "is thus of size", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 563, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 251, + 577 + ], + "score": 0.89, + "content": "( N + D - 1 ) ^ { 2 } M \\times ( N + D - 1 ) ^ { 2 } M .", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 563, + 505, + 578 + ], + "score": 1.0, + "content": "In the typical application setting of CSC, b is usually an image", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 341, + 588 + ], + "score": 1.0, + "content": "rather than a small patch. For example, if the image size is", + "type": "text" + }, + { + "bbox": [ + 341, + 576, + 384, + 586 + ], + "score": 0.88, + "content": "1 0 0 \\times 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 576, + 457, + 588 + ], + "score": 1.0, + "content": ", dictionary size is", + "type": "text" + }, + { + "bbox": [ + 457, + 576, + 501, + 586 + ], + "score": 0.9, + "content": "7 \\times 7 \\times 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 576, + 506, + 588 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 585, + 444, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 215, + 598 + ], + "score": 0.91, + "content": "N = 1 0 0 , D = 7 , M = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 585, + 239, + 599 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 239, + 586, + 440, + 599 + ], + "score": 0.88, + "content": "( \\dot { N } + D - 1 ) ^ { 2 } \\bar { M } \\times ( N + D - 1 ) ^ { 2 } M \\approx 5 \\times \\dot { 1 } 0 ^ { 1 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 585, + 444, + 599 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 103, + 535, + 508, + 599 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 606, + 460, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 460, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 460, + 618 + ], + "score": 1.0, + "content": "3.1 CALCULATING CONVOLUTIONAL WEIGHTS ANALYTICALLY AND EFFICIENTLY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "To overcome the computational challenge of solving (22), we exploit the following circular convo-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 253, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 253, + 645 + ], + "score": 1.0, + "content": "lution as an efficient approximation:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 621, + 505, + 645 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 648, + 483, + 684 + ], + "lines": [ + { + "bbox": [ + 109, + 648, + 483, + 684 + ], + "spans": [ + { + "bbox": [ + 109, + 648, + 483, + 684 + ], + "score": 0.93, + "content": "\\mathbf { b } ( i , j ) = \\sum _ { k = 0 } ^ { D - 1 } \\sum _ { l = 0 } ^ { D - 1 } \\sum _ { m = 1 } ^ { M } \\mathbf { d } _ { m } ( k , l ) \\mathbf { x } _ { m } \\big ( ( i + k ) _ { \\mathrm { m o d } N } , ( j + l ) _ { \\mathrm { m o d } N } \\big ) + \\varepsilon ( i , j ) , \\quad 0 \\leq i , j \\leq N - 1 ,", + "type": "interline_equation", + "image_path": "861f01235b24aed94c7892c63dcbfce62a64bee8193a88a53ee79f3b8d455234.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 109, + 648, + 483, + 660.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 109, + 660.0, + 483, + 672.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 109, + 672.0, + 483, + 684.0 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 80, + 475, + 95 + ], + "lines": [ + { + "bbox": [ + 104, + 77, + 479, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 77, + 133, + 98 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 80, + 270, + 94 + ], + "score": 0.89, + "content": " { \\mathbf { b } } \\in \\mathbb { R } ^ { N ^ { 2 } } , \\mathbf { d } _ { m } \\in \\mathbb { R } ^ { D ^ { 2 } } , { \\mathbf { x } } _ { m } \\in \\mathbb { R } ^ { N ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 77, + 479, + 98 + ], + "score": 1.0, + "content": ". Similar to (18), we rewrite (23) in a compact way:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 96, + 401, + 130 + ], + "lines": [ + { + "bbox": [ + 208, + 96, + 401, + 130 + ], + "spans": [ + { + "bbox": [ + 208, + 96, + 401, + 130 + ], + "score": 0.94, + "content": "\\mathbf { b } = \\sum _ { m = 1 } ^ { M } \\mathbf { D } _ { \\mathrm { c i r } , m } ^ { N } ( \\mathbf { d } _ { m } ) \\mathbf { x } _ { m } + \\varepsilon = \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\mathbf { x } + \\varepsilon ,", + "type": "interline_equation", + "image_path": "c53587267e7b1b7d25ead5cbae04b5a46e2556e095687c3e7af8bc3f806ec10f.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 96, + 401, + 113.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 208, + 113.0, + 401, + 130.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 133, + 503, + 157 + ], + "lines": [ + { + "bbox": [ + 104, + 130, + 501, + 148 + ], + "spans": [ + { + "bbox": [ + 104, + 130, + 133, + 148 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 132, + 240, + 146 + ], + "score": 0.92, + "content": "{ \\bf D } _ { \\mathrm { c i r } } ^ { N } ( { \\bf d } ) : \\mathbb { R } ^ { N ^ { 2 } M } \\mathbb { R } ^ { N ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 130, + 406, + 148 + ], + "score": 1.0, + "content": "is a matrix depending on the signal size", + "type": "text" + }, + { + "bbox": [ + 406, + 135, + 416, + 144 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 130, + 493, + 148 + ], + "score": 1.0, + "content": "and the dictionary", + "type": "text" + }, + { + "bbox": [ + 494, + 135, + 501, + 144 + ], + "score": 0.52, + "content": "\\mathbf { d }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 405, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 405, + 158 + ], + "score": 1.0, + "content": "Then the coherence minimization with the circular convolution is given by", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 158, + 414, + 196 + ], + "lines": [ + { + "bbox": [ + 196, + 158, + 414, + 196 + ], + "spans": [ + { + "bbox": [ + 196, + 158, + 414, + 196 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\underset { \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , \\ 1 \\leq m \\leq M } { \\arg \\operatorname* { m i n } } \\left\\| \\left( \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\right\\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f66d39ac9ea7aaf918852cb50a3b65ebad36271f3422b92be7cecb37947ed31a.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 158, + 414, + 177.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 196, + 177.0, + 414, + 196.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 203, + 464, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 465, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 465, + 217 + ], + "score": 1.0, + "content": "The following theorem motivates us to use the solution to (24) to approximate that of (22)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 217, + 423, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 424, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 424, + 231 + ], + "score": 1.0, + "content": "Theorem 3. The solution sets of (22) and (24) satisfy the following properties:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 131, + 235, + 274, + 251 + ], + "lines": [ + { + "bbox": [ + 130, + 235, + 274, + 250 + ], + "spans": [ + { + "bbox": [ + 130, + 235, + 274, + 250 + ], + "score": 0.41, + "content": "I . \\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } , \\forall N \\ge 2 D - 1 .", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 128, + 256, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 128, + 250, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 128, + 255, + 270, + 273 + ], + "score": 1.0, + "content": "2. If at least one of the matrices", + "type": "text" + }, + { + "bbox": [ + 270, + 257, + 365, + 272 + ], + "score": 0.93, + "content": "\\{ \\mathbf { D } _ { \\mathrm { c i r } , 1 } ^ { 2 D - 1 } , \\cdot \\cdot \\cdot , \\mathbf { D } _ { \\mathrm { c i r } , M } ^ { 2 D - 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 250, + 434, + 278 + ], + "score": 1.0, + "content": "is non-singular,", + "type": "text" + }, + { + "bbox": [ + 434, + 257, + 468, + 271 + ], + "score": 0.93, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 250, + 506, + 278 + ], + "score": 1.0, + "content": "involves", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 269, + 292, + 284 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 292, + 284 + ], + "score": 1.0, + "content": "only a unique element. Furthermore,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 283, + 375, + 304 + ], + "lines": [ + { + "bbox": [ + 271, + 283, + 375, + 304 + ], + "spans": [ + { + "bbox": [ + 271, + 283, + 375, + 304 + ], + "score": 0.92, + "content": "\\operatorname * { l i m } _ { N \\infty } \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } .", + "type": "interline_equation", + "image_path": "a11b533d34eab49701b03d1d25838ecfe65d4df8c23d7dab8167f49099e409c3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 271, + 283, + 375, + 304 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 104, + 308, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 174, + 330 + ], + "score": 1.0, + "content": "The solution set", + "type": "text" + }, + { + "bbox": [ + 174, + 312, + 195, + 325 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 308, + 334, + 330 + ], + "score": 1.0, + "content": "is not related with the image size", + "type": "text" + }, + { + "bbox": [ + 335, + 314, + 345, + 323 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 308, + 390, + 330 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 390, + 313, + 447, + 324 + ], + "score": 0.91, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 308, + 505, + 330 + ], + "score": 1.0, + "content": ", thus one can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 284, + 336 + ], + "score": 1.0, + "content": "deal with a much smaller-size problem (let", + "type": "text" + }, + { + "bbox": [ + 284, + 324, + 343, + 335 + ], + "score": 0.89, + "content": "N = 2 D - 1 \\mathrm { \\ Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 324, + 474, + 336 + ], + "score": 1.0, + "content": "). Further, (25) indicates that as", + "type": "text" + }, + { + "bbox": [ + 474, + 325, + 484, + 334 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "gets", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 102, + 333, + 504, + 352 + ], + "spans": [ + { + "bbox": [ + 102, + 333, + 184, + 352 + ], + "score": 1.0, + "content": "(much) larger than", + "type": "text" + }, + { + "bbox": [ + 184, + 337, + 194, + 347 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 333, + 470, + 352 + ], + "score": 1.0, + "content": ", the boundary condition becomes less important. Thus, one can use", + "type": "text" + }, + { + "bbox": [ + 470, + 335, + 504, + 349 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 \\breve { D } - 1 }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 345, + 478, + 367 + ], + "spans": [ + { + "bbox": [ + 102, + 345, + 169, + 367 + ], + "score": 1.0, + "content": "to approximate", + "type": "text" + }, + { + "bbox": [ + 169, + 348, + 197, + 362 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 345, + 478, + 367 + ], + "score": 1.0, + "content": ". In Appendix E.2, we introduce the algorithm details of solving (24).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 424, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 425, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 425, + 380 + ], + "score": 1.0, + "content": "Based on Proposition 1 and Theorem 3, we obtain the convolutional ALISTA:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 380, + 467, + 414 + ], + "lines": [ + { + "bbox": [ + 126, + 380, + 467, + 414 + ], + "spans": [ + { + "bbox": [ + 126, + 380, + 467, + 414 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { m } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\left( \\mathbf { x } _ { m } ^ { ( k ) } - \\gamma _ { m } ^ { ( k ) } \\left( \\tilde { \\mathbf { w } } _ { m } \\right) ^ { \\prime } \\ast \\bigg ( \\sum _ { \\bar { m } = 1 } ^ { M } \\mathbf { d } _ { \\bar { m } } \\ast \\mathbf { x } _ { \\bar { m } } ^ { ( k ) } - \\mathbf { b } \\bigg ) \\right) , \\quad m = 1 , 2 , \\cdots , M ,", + "type": "interline_equation", + "image_path": "948bc3d2d6deb966f18893c596ef4da05e9991b4b0c679e17e070bb04f48b6dd.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 126, + 380, + 467, + 391.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 126, + 391.3333333333333, + 467, + 402.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 126, + 402.66666666666663, + 467, + 413.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 101, + 410, + 510, + 439 + ], + "spans": [ + { + "bbox": [ + 101, + 410, + 133, + 439 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 417, + 267, + 431 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { w } } = [ \\tilde { \\mathbf { w } } _ { 1 } , \\cdots , \\tilde { \\mathbf { w } } _ { M } ] ^ { T } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 410, + 285, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 416, + 394, + 431 + ], + "score": 0.93, + "content": "\\Theta = \\{ \\{ \\gamma _ { m } ^ { ( k ) } \\} _ { m , k } , \\{ \\theta ^ { ( k ) } \\} _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 410, + 510, + 439 + ], + "score": 1.0, + "content": "are the parameters to train.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "(26) is a simplified form, compared to the empirically unfolded CSC model recently proposed in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 441, + 204, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 204, + 453 + ], + "score": 1.0, + "content": "(Sreter & Giryes, 2018)", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 106, + 462, + 358, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 359, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 359, + 477 + ], + "score": 1.0, + "content": "4 ROBUST ALISTA TO MODEL PERTURBATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "Many applications, such as often found in surveillance video scenarios (Zhao et al., 2011; Han", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "et al., 2013), can be formulated as sparse coding models whose dictionaries are subject to small", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 517 + ], + "score": 1.0, + "content": "dynamic perturbations (e.g, slowly varied over time). Specifically, the linear system model (1) may", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 169, + 528 + ], + "score": 1.0, + "content": "have uncertain", + "type": "text" + }, + { + "bbox": [ + 170, + 515, + 180, + 526 + ], + "score": 0.32, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 515, + 187, + 528 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 187, + 514, + 252, + 527 + ], + "score": 0.89, + "content": "\\tilde { \\textbf { D } } = \\textbf { D } + \\varepsilon _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 515, + 286, + 528 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 286, + 517, + 301, + 527 + ], + "score": 0.84, + "content": "\\varepsilon _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "is some small stochastic perturbation. Classical", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "score": 1.0, + "content": "LISTA entangles the learning of all its parameters, and the trained model is tied to one static D.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "The important contribution of ALISTA is to decompose fitting W w.r.t. D, from adapting other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 547, + 273, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 153, + 563 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 153, + 548, + 206, + 561 + ], + "score": 0.94, + "content": "\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 547, + 273, + 563 + ], + "score": 1.0, + "content": "to training data.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 108, + 565, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "In this section, we develop a robust variant of ALISTA that is a fast regressor not only for a given", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 117, + 589 + ], + "score": 0.45, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 579, + 256, + 591 + ], + "score": 1.0, + "content": ", but all its randomly perturbations", + "type": "text" + }, + { + "bbox": [ + 256, + 577, + 266, + 589 + ], + "score": 0.86, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "to some extent. Up to our best knowledge, this approach is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 473, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 473, + 603 + ], + "score": 1.0, + "content": "new. Robust ALISTA can be sketched as the following empirical routine (at each iteration):", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 132, + 604, + 504, + 657 + ], + "lines": [ + { + "bbox": [ + 131, + 603, + 435, + 618 + ], + "spans": [ + { + "bbox": [ + 131, + 604, + 265, + 618 + ], + "score": 1.0, + "content": "• Sample a perturbed dictionary", + "type": "text" + }, + { + "bbox": [ + 265, + 604, + 275, + 616 + ], + "score": 0.78, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 604, + 311, + 618 + ], + "score": 1.0, + "content": ". Sample", + "type": "text" + }, + { + "bbox": [ + 312, + 607, + 320, + 616 + ], + "score": 0.66, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 604, + 337, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 337, + 608, + 343, + 615 + ], + "score": 0.74, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 604, + 422, + 618 + ], + "score": 1.0, + "content": "to generate b w.r.t.", + "type": "text" + }, + { + "bbox": [ + 423, + 603, + 432, + 616 + ], + "score": 0.57, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 604, + 435, + 618 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 132, + 617, + 504, + 631 + ], + "spans": [ + { + "bbox": [ + 132, + 619, + 276, + 631 + ], + "score": 1.0, + "content": "• Apply Stage 1 of ALISTA w.r.t.", + "type": "text" + }, + { + "bbox": [ + 276, + 617, + 287, + 630 + ], + "score": 0.77, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 619, + 333, + 631 + ], + "score": 1.0, + "content": "and obtain", + "type": "text" + }, + { + "bbox": [ + 334, + 618, + 347, + 630 + ], + "score": 0.74, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 619, + 504, + 631 + ], + "score": 1.0, + "content": "; however, instead of an iterative mini-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 140, + 630, + 492, + 644 + ], + "spans": [ + { + "bbox": [ + 140, + 631, + 476, + 644 + ], + "score": 1.0, + "content": "mization algorithm, we use a neural network that unfolds that algorithm to produce", + "type": "text" + }, + { + "bbox": [ + 476, + 630, + 489, + 642 + ], + "score": 0.72, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 631, + 492, + 644 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 130, + 642, + 436, + 660 + ], + "spans": [ + { + "bbox": [ + 130, + 642, + 272, + 660 + ], + "score": 1.0, + "content": "• Apply Stage 2 of ALISTA w.r.t.", + "type": "text" + }, + { + "bbox": [ + 272, + 643, + 285, + 656 + ], + "score": 0.31, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 642, + 303, + 660 + ], + "score": 1.0, + "content": ", D,", + "type": "text" + }, + { + "bbox": [ + 303, + 647, + 311, + 655 + ], + "score": 0.34, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 642, + 378, + 660 + ], + "score": 1.0, + "content": ", and b to obtain", + "type": "text" + }, + { + "bbox": [ + 378, + 644, + 431, + 657 + ], + "score": 0.93, + "content": "\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 642, + 436, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 211, + 674 + ], + "score": 1.0, + "content": "In Robust ALISTA above,", + "type": "text" + }, + { + "bbox": [ + 211, + 660, + 221, + 672 + ], + "score": 0.78, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "becomes a part of the data for training the neural network that generates", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 672, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 120, + 684 + ], + "score": 0.46, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 673, + 506, + 687 + ], + "score": 1.0, + "content": ". This neural network is faster to apply than the minimization algorithm. One might attempt to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 685, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 122, + 700 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 122, + 685, + 132, + 696 + ], + "score": 0.81, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 685, + 241, + 700 + ], + "score": 1.0, + "content": "in the last step, rather than", + "type": "text" + }, + { + "bbox": [ + 242, + 686, + 252, + 696 + ], + "score": 0.56, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 685, + 271, + 700 + ], + "score": 1.0, + "content": ", but", + "type": "text" + }, + { + "bbox": [ + 271, + 686, + 281, + 696 + ], + "score": 0.86, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 685, + 505, + 700 + ], + "score": 1.0, + "content": "makes training less stable, potentially because of larger", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 437, + 711 + ], + "score": 1.0, + "content": "weight variations between training iterations due to the random perturbations in", + "type": "text" + }, + { + "bbox": [ + 438, + 697, + 448, + 709 + ], + "score": 0.78, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". We observe", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 149, + 723 + ], + "score": 1.0, + "content": "that using", + "type": "text" + }, + { + "bbox": [ + 149, + 710, + 159, + 720 + ], + "score": 0.46, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "stabilizes training better and empirically achieves a good prediction. More details of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 311, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 311, + 734 + ], + "score": 1.0, + "content": "training Robust ALISTA are given in Appendix G.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 80, + 475, + 95 + ], + "lines": [ + { + "bbox": [ + 104, + 77, + 479, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 77, + 133, + 98 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 80, + 270, + 94 + ], + "score": 0.89, + "content": " { \\mathbf { b } } \\in \\mathbb { R } ^ { N ^ { 2 } } , \\mathbf { d } _ { m } \\in \\mathbb { R } ^ { D ^ { 2 } } , { \\mathbf { x } } _ { m } \\in \\mathbb { R } ^ { N ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 77, + 479, + 98 + ], + "score": 1.0, + "content": ". Similar to (18), we rewrite (23) in a compact way:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 104, + 77, + 479, + 98 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 96, + 401, + 130 + ], + "lines": [ + { + "bbox": [ + 208, + 96, + 401, + 130 + ], + "spans": [ + { + "bbox": [ + 208, + 96, + 401, + 130 + ], + "score": 0.94, + "content": "\\mathbf { b } = \\sum _ { m = 1 } ^ { M } \\mathbf { D } _ { \\mathrm { c i r } , m } ^ { N } ( \\mathbf { d } _ { m } ) \\mathbf { x } _ { m } + \\varepsilon = \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\mathbf { x } + \\varepsilon ,", + "type": "interline_equation", + "image_path": "c53587267e7b1b7d25ead5cbae04b5a46e2556e095687c3e7af8bc3f806ec10f.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 96, + 401, + 113.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 208, + 113.0, + 401, + 130.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 133, + 503, + 157 + ], + "lines": [ + { + "bbox": [ + 104, + 130, + 501, + 148 + ], + "spans": [ + { + "bbox": [ + 104, + 130, + 133, + 148 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 132, + 240, + 146 + ], + "score": 0.92, + "content": "{ \\bf D } _ { \\mathrm { c i r } } ^ { N } ( { \\bf d } ) : \\mathbb { R } ^ { N ^ { 2 } M } \\mathbb { R } ^ { N ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 130, + 406, + 148 + ], + "score": 1.0, + "content": "is a matrix depending on the signal size", + "type": "text" + }, + { + "bbox": [ + 406, + 135, + 416, + 144 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 130, + 493, + 148 + ], + "score": 1.0, + "content": "and the dictionary", + "type": "text" + }, + { + "bbox": [ + 494, + 135, + 501, + 144 + ], + "score": 0.52, + "content": "\\mathbf { d }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 405, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 405, + 158 + ], + "score": 1.0, + "content": "Then the coherence minimization with the circular convolution is given by", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 130, + 501, + 158 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 158, + 414, + 196 + ], + "lines": [ + { + "bbox": [ + 196, + 158, + 414, + 196 + ], + "spans": [ + { + "bbox": [ + 196, + 158, + 414, + 196 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\underset { \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , \\ 1 \\leq m \\leq M } { \\arg \\operatorname* { m i n } } \\left\\| \\left( \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\right) ^ { T } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\right\\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f66d39ac9ea7aaf918852cb50a3b65ebad36271f3422b92be7cecb37947ed31a.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 158, + 414, + 177.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 196, + 177.0, + 414, + 196.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 203, + 464, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 465, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 465, + 217 + ], + "score": 1.0, + "content": "The following theorem motivates us to use the solution to (24) to approximate that of (22)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 202, + 465, + 217 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 217, + 423, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 424, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 424, + 231 + ], + "score": 1.0, + "content": "Theorem 3. The solution sets of (22) and (24) satisfy the following properties:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 216, + 424, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 235, + 274, + 251 + ], + "lines": [ + { + "bbox": [ + 130, + 235, + 274, + 250 + ], + "spans": [ + { + "bbox": [ + 130, + 235, + 274, + 250 + ], + "score": 0.41, + "content": "I . \\mathcal { W } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } , \\forall N \\ge 2 D - 1 .", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 130, + 235, + 274, + 250 + ] + }, + { + "type": "text", + "bbox": [ + 128, + 256, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 128, + 250, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 128, + 255, + 270, + 273 + ], + "score": 1.0, + "content": "2. If at least one of the matrices", + "type": "text" + }, + { + "bbox": [ + 270, + 257, + 365, + 272 + ], + "score": 0.93, + "content": "\\{ \\mathbf { D } _ { \\mathrm { c i r } , 1 } ^ { 2 D - 1 } , \\cdot \\cdot \\cdot , \\mathbf { D } _ { \\mathrm { c i r } , M } ^ { 2 D - 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 250, + 434, + 278 + ], + "score": 1.0, + "content": "is non-singular,", + "type": "text" + }, + { + "bbox": [ + 434, + 257, + 468, + 271 + ], + "score": 0.93, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 250, + 506, + 278 + ], + "score": 1.0, + "content": "involves", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 269, + 292, + 284 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 292, + 284 + ], + "score": 1.0, + "content": "only a unique element. Furthermore,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 128, + 250, + 506, + 284 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 283, + 375, + 304 + ], + "lines": [ + { + "bbox": [ + 271, + 283, + 375, + 304 + ], + "spans": [ + { + "bbox": [ + 271, + 283, + 375, + 304 + ], + "score": 0.92, + "content": "\\operatorname * { l i m } _ { N \\infty } \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } .", + "type": "interline_equation", + "image_path": "a11b533d34eab49701b03d1d25838ecfe65d4df8c23d7dab8167f49099e409c3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 271, + 283, + 375, + 304 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 104, + 308, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 174, + 330 + ], + "score": 1.0, + "content": "The solution set", + "type": "text" + }, + { + "bbox": [ + 174, + 312, + 195, + 325 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 308, + 334, + 330 + ], + "score": 1.0, + "content": "is not related with the image size", + "type": "text" + }, + { + "bbox": [ + 335, + 314, + 345, + 323 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 308, + 390, + 330 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 390, + 313, + 447, + 324 + ], + "score": 0.91, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 308, + 505, + 330 + ], + "score": 1.0, + "content": ", thus one can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 284, + 336 + ], + "score": 1.0, + "content": "deal with a much smaller-size problem (let", + "type": "text" + }, + { + "bbox": [ + 284, + 324, + 343, + 335 + ], + "score": 0.89, + "content": "N = 2 D - 1 \\mathrm { \\ Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 324, + 474, + 336 + ], + "score": 1.0, + "content": "). 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Thus, one can use", + "type": "text" + }, + { + "bbox": [ + 470, + 335, + 504, + 349 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 \\breve { D } - 1 }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 345, + 478, + 367 + ], + "spans": [ + { + "bbox": [ + 102, + 345, + 169, + 367 + ], + "score": 1.0, + "content": "to approximate", + "type": "text" + }, + { + "bbox": [ + 169, + 348, + 197, + 362 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 345, + 478, + 367 + ], + "score": 1.0, + "content": ". In Appendix E.2, we introduce the algorithm details of solving (24).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 102, + 308, + 505, + 367 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 424, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 425, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 425, + 380 + ], + "score": 1.0, + "content": "Based on Proposition 1 and Theorem 3, we obtain the convolutional ALISTA:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 365, + 425, + 380 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 380, + 467, + 414 + ], + "lines": [ + { + "bbox": [ + 126, + 380, + 467, + 414 + ], + "spans": [ + { + "bbox": [ + 126, + 380, + 467, + 414 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { m } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\left( \\mathbf { x } _ { m } ^ { ( k ) } - \\gamma _ { m } ^ { ( k ) } \\left( \\tilde { \\mathbf { w } } _ { m } \\right) ^ { \\prime } \\ast \\bigg ( \\sum _ { \\bar { m } = 1 } ^ { M } \\mathbf { d } _ { \\bar { m } } \\ast \\mathbf { x } _ { \\bar { m } } ^ { ( k ) } - \\mathbf { b } \\bigg ) \\right) , \\quad m = 1 , 2 , \\cdots , M ,", + "type": "interline_equation", + "image_path": "948bc3d2d6deb966f18893c596ef4da05e9991b4b0c679e17e070bb04f48b6dd.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 126, + 380, + 467, + 391.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 126, + 391.3333333333333, + 467, + 402.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 126, + 402.66666666666663, + 467, + 413.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 101, + 410, + 510, + 439 + ], + "spans": [ + { + "bbox": [ + 101, + 410, + 133, + 439 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 417, + 267, + 431 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { w } } = [ \\tilde { \\mathbf { w } } _ { 1 } , \\cdots , \\tilde { \\mathbf { w } } _ { M } ] ^ { T } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 410, + 285, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 416, + 394, + 431 + ], + "score": 0.93, + "content": "\\Theta = \\{ \\{ \\gamma _ { m } ^ { ( k ) } \\} _ { m , k } , \\{ \\theta ^ { ( k ) } \\} _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 410, + 510, + 439 + ], + "score": 1.0, + "content": "are the parameters to train.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "(26) is a simplified form, compared to the empirically unfolded CSC model recently proposed in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 441, + 204, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 204, + 453 + ], + "score": 1.0, + "content": "(Sreter & Giryes, 2018)", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 101, + 410, + 510, + 453 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 462, + 358, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 359, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 359, + 477 + ], + "score": 1.0, + "content": "4 ROBUST ALISTA TO MODEL PERTURBATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "Many applications, such as often found in surveillance video scenarios (Zhao et al., 2011; Han", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "et al., 2013), can be formulated as sparse coding models whose dictionaries are subject to small", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 517 + ], + "score": 1.0, + "content": "dynamic perturbations (e.g, slowly varied over time). Specifically, the linear system model (1) may", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 169, + 528 + ], + "score": 1.0, + "content": "have uncertain", + "type": "text" + }, + { + "bbox": [ + 170, + 515, + 180, + 526 + ], + "score": 0.32, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 515, + 187, + 528 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 187, + 514, + 252, + 527 + ], + "score": 0.89, + "content": "\\tilde { \\textbf { D } } = \\textbf { D } + \\varepsilon _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 515, + 286, + 528 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 286, + 517, + 301, + 527 + ], + "score": 0.84, + "content": "\\varepsilon _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "is some small stochastic perturbation. Classical", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "score": 1.0, + "content": "LISTA entangles the learning of all its parameters, and the trained model is tied to one static D.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "The important contribution of ALISTA is to decompose fitting W w.r.t. D, from adapting other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 547, + 273, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 153, + 563 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 153, + 548, + 206, + 561 + ], + "score": 0.94, + "content": "\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 547, + 273, + 563 + ], + "score": 1.0, + "content": "to training data.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 481, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 565, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 580 + ], + "score": 1.0, + "content": "In this section, we develop a robust variant of ALISTA that is a fast regressor not only for a given", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 117, + 589 + ], + "score": 0.45, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 579, + 256, + 591 + ], + "score": 1.0, + "content": ", but all its randomly perturbations", + "type": "text" + }, + { + "bbox": [ + 256, + 577, + 266, + 589 + ], + "score": 0.86, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "to some extent. Up to our best knowledge, this approach is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 473, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 473, + 603 + ], + "score": 1.0, + "content": "new. Robust ALISTA can be sketched as the following empirical routine (at each iteration):", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 566, + 505, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 604, + 504, + 657 + ], + "lines": [ + { + "bbox": [ + 131, + 603, + 435, + 618 + ], + "spans": [ + { + "bbox": [ + 131, + 604, + 265, + 618 + ], + "score": 1.0, + "content": "• Sample a perturbed dictionary", + "type": "text" + }, + { + "bbox": [ + 265, + 604, + 275, + 616 + ], + "score": 0.78, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 604, + 311, + 618 + ], + "score": 1.0, + "content": ". Sample", + "type": "text" + }, + { + "bbox": [ + 312, + 607, + 320, + 616 + ], + "score": 0.66, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 604, + 337, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 337, + 608, + 343, + 615 + ], + "score": 0.74, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 604, + 422, + 618 + ], + "score": 1.0, + "content": "to generate b w.r.t.", + "type": "text" + }, + { + "bbox": [ + 423, + 603, + 432, + 616 + ], + "score": 0.57, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 604, + 435, + 618 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 132, + 617, + 504, + 631 + ], + "spans": [ + { + "bbox": [ + 132, + 619, + 276, + 631 + ], + "score": 1.0, + "content": "• Apply Stage 1 of ALISTA w.r.t.", + "type": "text" + }, + { + "bbox": [ + 276, + 617, + 287, + 630 + ], + "score": 0.77, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 619, + 333, + 631 + ], + "score": 1.0, + "content": "and obtain", + "type": "text" + }, + { + "bbox": [ + 334, + 618, + 347, + 630 + ], + "score": 0.74, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 619, + 504, + 631 + ], + "score": 1.0, + "content": "; however, instead of an iterative mini-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 140, + 630, + 492, + 644 + ], + "spans": [ + { + "bbox": [ + 140, + 631, + 476, + 644 + ], + "score": 1.0, + "content": "mization algorithm, we use a neural network that unfolds that algorithm to produce", + "type": "text" + }, + { + "bbox": [ + 476, + 630, + 489, + 642 + ], + "score": 0.72, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 631, + 492, + 644 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 130, + 642, + 436, + 660 + ], + "spans": [ + { + "bbox": [ + 130, + 642, + 272, + 660 + ], + "score": 1.0, + "content": "• Apply Stage 2 of ALISTA w.r.t.", + "type": "text" + }, + { + "bbox": [ + 272, + 643, + 285, + 656 + ], + "score": 0.31, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 642, + 303, + 660 + ], + "score": 1.0, + "content": ", D,", + "type": "text" + }, + { + "bbox": [ + 303, + 647, + 311, + 655 + ], + "score": 0.34, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 642, + 378, + 660 + ], + "score": 1.0, + "content": ", and b to obtain", + "type": "text" + }, + { + "bbox": [ + 378, + 644, + 431, + 657 + ], + "score": 0.93, + "content": "\\{ \\gamma ^ { ( k ) } , \\theta ^ { ( k ) } \\} _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 642, + 436, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 130, + 603, + 504, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 211, + 674 + ], + "score": 1.0, + "content": "In Robust ALISTA above,", + "type": "text" + }, + { + "bbox": [ + 211, + 660, + 221, + 672 + ], + "score": 0.78, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "becomes a part of the data for training the neural network that generates", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 672, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 120, + 684 + ], + "score": 0.46, + "content": "\\tilde { \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 673, + 506, + 687 + ], + "score": 1.0, + "content": ". This neural network is faster to apply than the minimization algorithm. One might attempt to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 685, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 122, + 700 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 122, + 685, + 132, + 696 + ], + "score": 0.81, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 685, + 241, + 700 + ], + "score": 1.0, + "content": "in the last step, rather than", + "type": "text" + }, + { + "bbox": [ + 242, + 686, + 252, + 696 + ], + "score": 0.56, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 685, + 271, + 700 + ], + "score": 1.0, + "content": ", but", + "type": "text" + }, + { + "bbox": [ + 271, + 686, + 281, + 696 + ], + "score": 0.86, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 685, + 505, + 700 + ], + "score": 1.0, + "content": "makes training less stable, potentially because of larger", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 437, + 711 + ], + "score": 1.0, + "content": "weight variations between training iterations due to the random perturbations in", + "type": "text" + }, + { + "bbox": [ + 438, + 697, + 448, + 709 + ], + "score": 0.78, + "content": "\\tilde { \\bf D }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". We observe", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 149, + 723 + ], + "score": 1.0, + "content": "that using", + "type": "text" + }, + { + "bbox": [ + 149, + 710, + 159, + 720 + ], + "score": 0.46, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "stabilizes training better and empirically achieves a good prediction. More details of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 311, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 311, + 734 + ], + "score": 1.0, + "content": "training Robust ALISTA are given in Appendix G.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 660, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 240, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 240, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 240, + 96 + ], + "score": 1.0, + "content": "5 NUMERICAL RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 97, + 503, + 109 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 504, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 504, + 110 + ], + "score": 1.0, + "content": "In this section, we conduct extensive experiments on both synthesized and real data to demonstrate:6", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 128, + 114, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 132, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 132, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "• We experimentally validate Theorems 1 and 2, and show that ALISTA is as effective as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 142, + 126, + 488, + 137 + ], + "spans": [ + { + "bbox": [ + 142, + 126, + 488, + 137 + ], + "score": 1.0, + "content": "classical LISTA (Gregor & LeCun, 2010; Chen et al., 2018)but is much easier to train.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 132, + 140, + 415, + 151 + ], + "spans": [ + { + "bbox": [ + 132, + 140, + 415, + 151 + ], + "score": 1.0, + "content": "• Similar conclusions can be drawn for convolutional analytic LISTA.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 133, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 133, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "• The robust analytic LISTA further shows remarkable robustness in sparse code prediction,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 165, + 366, + 178 + ], + "spans": [ + { + "bbox": [ + 141, + 166, + 184, + 178 + ], + "score": 1.0, + "content": "given that", + "type": "text" + }, + { + "bbox": [ + 184, + 165, + 194, + 176 + ], + "score": 0.3, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 166, + 366, + 178 + ], + "score": 1.0, + "content": "is randomly perturbed within some extent.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 504, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 504, + 198 + ], + "score": 1.0, + "content": "Notation For brevity, we let LISTA denote the vanilla LISTA model (4) in (Gregor & LeCun,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "score": 1.0, + "content": "2010); LISTA-CPSS refers to the lately-proposed fast LISTA variant (Chen et al., 2018) with weight", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 221 + ], + "score": 1.0, + "content": "coupling and support selection; TiLISTA is the tied LISTA (14); and ALISTA is our proposed Analytic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 217, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 506, + 233 + ], + "score": 1.0, + "content": "LISTA (15). If the model is for convolutional case, then we add “Conv” as the prefix for model name,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 401, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 401, + 242 + ], + "score": 1.0, + "content": "such as “Conv ALISTA” that represents the convolutional analytic LISTA.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 256, + 375, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 375, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 375, + 270 + ], + "score": 1.0, + "content": "5.1 VALIDATION OF THEOREMS 1 AND 2 (ANALYTIC LISTA)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 189, + 285 + ], + "score": 1.0, + "content": "We follow the same", + "type": "text" + }, + { + "bbox": [ + 189, + 273, + 229, + 284 + ], + "score": 0.56, + "content": "N = 2 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 273, + 232, + 285 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 232, + 273, + 274, + 284 + ], + "score": 0.64, + "content": "M = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "setting as (Chen et al., 2018) by default. 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127 + ], + "score": 1.0, + "content": "• We experimentally validate Theorems 1 and 2, and show that ALISTA is as effective as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 142, + 126, + 488, + 137 + ], + "spans": [ + { + "bbox": [ + 142, + 126, + 488, + 137 + ], + "score": 1.0, + "content": "classical LISTA (Gregor & LeCun, 2010; Chen et al., 2018)but is much easier to train.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 132, + 140, + 415, + 151 + ], + "spans": [ + { + "bbox": [ + 132, + 140, + 415, + 151 + ], + "score": 1.0, + "content": "• Similar conclusions can be drawn for convolutional analytic LISTA.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 133, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 133, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "• The robust analytic LISTA further shows remarkable robustness in sparse code prediction,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 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If the model is for convolutional case, then we add “Conv” as the prefix for model name,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 401, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 401, + 242 + ], + "score": 1.0, + "content": "such as “Conv ALISTA” that represents the convolutional analytic LISTA.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 185, + 506, + 242 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 256, + 375, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 375, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 375, + 270 + ], + "score": 1.0, + "content": "5.1 VALIDATION OF THEOREMS 1 AND 2 (ANALYTIC LISTA)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 189, + 285 + ], + "score": 1.0, + "content": "We follow the same", + "type": "text" + }, + { + "bbox": [ + 189, + 273, + 229, + 284 + ], + "score": 0.56, + "content": "N = 2 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 273, + 232, + 285 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 232, + 273, + 274, + 284 + ], + "score": 0.64, + "content": "M = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "setting as (Chen et al., 2018) by default. 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A test set", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "of 1000 samples generated in the above manner is fixed for all tests in our simulations. 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As shown in Figure 1, the", + "type": "text" + }, + { + "bbox": [ + 332, + 401, + 338, + 409 + ], + "score": 0.59, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "-axes denotes the indices of layers for the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "networks, or the number of iterations for the iterative algorithms. 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Among the four networks, classical-LISTA is inferior to the other three by an obvious", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "margin. LISTA-CPSS, TiLISTA and ALISTA perform comparably: ALISTA is observed to eventu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "ally achieve the lowest NMSE. Figure 1(a) also supports Theorem 2, that all networks have at most", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "score": 1.0, + "content": "linear convergence, regardless of how freely their parameters can be end-to-end learned.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "Figure 1 (b) - (d) further show that even in the presence of noise, ALISTA can empirically perform", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "comparably with LISTA-CPSS and TiLISTA, and stay clearly better than LISTA and ISTA/FISTA.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 504, + 188 + ], + "score": 1.0, + "content": "Always note that ALISTA the smallest amount of parameters to learn from the end-to-end train-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "ing (Stage 2). The above results endorse that: i) the optimal LISTA layer-wise weights could be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 197, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 162, + 213 + ], + "score": 1.0, + "content": "structured as", + "type": "text" + }, + { + "bbox": [ + 162, + 198, + 232, + 210 + ], + "score": 0.93, + "content": "\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 197, + 506, + 213 + ], + "score": 1.0, + "content": "; and ii) W could be analytically solved rather than learned from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "data, without incurring performance loss. We also observe the significant reduction of training time", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 337, + 233 + ], + "score": 1.0, + "content": "for ALISTA: while LISTA-CPSS of the same depth took", + "type": "text" + }, + { + "bbox": [ + 337, + 221, + 359, + 232 + ], + "score": 0.85, + "content": "{ \\sim } 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "hours to train, ALISTA was trained", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "within only 6 minutes (0.1 hours) to achieve comparable performance, on the same hardware (one", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 185, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 185, + 255 + ], + "score": 1.0, + "content": "1080 Ti on server).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "image", + "bbox": [ + 110, + 261, + 476, + 358 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 261, + 476, + 358 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 261, + 476, + 358 + ], + "spans": [ + { + "bbox": [ + 110, + 261, + 476, + 358 + ], + "score": 0.969, + "type": "image", + "image_path": "b4ad8ecd57838661b04ad8c14453ba10544c6080dccc8fdb5991b9989b099d41.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 110, + 261, + 476, + 293.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 110, + 293.3333333333333, + 476, + 325.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 110, + 325.66666666666663, + 476, + 357.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 362, + 497, + 373 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 359, + 498, + 376 + ], + "spans": [ + { + "bbox": [ + 110, + 359, + 498, + 376 + ], + "score": 1.0, + "content": "Figure 2: Justification of Theorem 1 (noiseless case): parameters obtained by training satisfy (9).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + } + ], + "index": 17.0 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 336, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 336, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 336, + 395 + ], + "score": 1.0, + "content": "We further supply Figures 2 and 3 to justify Theo-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 337, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 337, + 406 + ], + "score": 1.0, + "content": "rem 1 from different perspectives. Figure 2 plots the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 403, + 337, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 189, + 419 + ], + "score": 1.0, + "content": "learned parameters", + "type": "text" + }, + { + "bbox": [ + 189, + 404, + 238, + 417 + ], + "score": 0.93, + "content": "\\{ \\bar { \\gamma ^ { ( k ) } } , \\bar { \\theta ^ { ( k ) } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 403, + 337, + 419 + ], + "score": 1.0, + "content": "in ALISTA (Stage 2),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 416, + 336, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 336, + 429 + ], + "score": 1.0, + "content": "showing that they satisfy the properties proposed in The-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 426, + 338, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 146, + 442 + ], + "score": 1.0, + "content": "orem 1:", + "type": "text" + }, + { + "bbox": [ + 147, + 427, + 165, + 440 + ], + "score": 0.88, + "content": "\\gamma ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 426, + 212, + 442 + ], + "score": 1.0, + "content": "bounded;", + "type": "text" + }, + { + "bbox": [ + 212, + 427, + 229, + 439 + ], + "score": 0.89, + "content": "{ \\theta } ^ { \\bar { ( k ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 426, + 251, + 442 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 427, + 270, + 440 + ], + "score": 0.91, + "content": "\\gamma ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 426, + 338, + 442 + ], + "score": 1.0, + "content": "is proportional", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 438, + 338, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 118, + 454 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 119, + 439, + 259, + 453 + ], + "score": 0.6, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 1 } ~ ( ^ { * } \\mathrm { s u p } _ { \\mathbf { x } ^ { * } } , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 438, + 338, + 454 + ], + "score": 1.0, + "content": "is taken over the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 338, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 338, + 464 + ], + "score": 1.0, + "content": "test set). Figure 3 reports the average magnitude7 of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 338, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 291, + 474 + ], + "score": 1.0, + "content": "the false positives and the true positives in", + "type": "text" + }, + { + "bbox": [ + 292, + 462, + 322, + 474 + ], + "score": 0.92, + "content": "\\mathbf { \\Delta x } ^ { k } ( \\mathbf { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 461, + 338, + 474 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 337, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 337, + 485 + ], + "score": 1.0, + "content": "ALISTA: the “true positives” curve draws the values", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 337, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 119, + 497 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 484, + 229, + 497 + ], + "score": 0.9, + "content": "\\mathbb { E } \\{ \\| \\mathbf { x } _ { \\mathbb { S } } ^ { k } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } / \\| \\mathbf { x } ^ { \\hat { k } } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 483, + 258, + 497 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + }, + { + "bbox": [ + 259, + 485, + 266, + 495 + ], + "score": 0.71, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 483, + 337, + 497 + ], + "score": 1.0, + "content": "(the expectation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 337, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 337, + 507 + ], + "score": 1.0, + "content": "is taken over the test set), while “false positives” for", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24 + }, + { + "type": "image", + "bbox": [ + 351, + 387, + 492, + 454 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 351, + 387, + 492, + 454 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 351, + 387, + 492, + 454 + ], + "spans": [ + { + "bbox": [ + 351, + 387, + 492, + 454 + ], + "score": 0.95, + "type": "image", + "image_path": "113a9556b7deff18282564eac7e07763703a959bbb96cb26d5149e31de0c5122.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 351, + 387, + 492, + 420.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 351, + 420.5, + 492, + 454.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 344, + 463, + 504, + 497 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 343, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 343, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "Figure 3: Justification of Theorem 1", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 343, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 343, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "(noiseless case): Proportion of false", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 343, + 485, + 492, + 497 + ], + "spans": [ + { + "bbox": [ + 343, + 485, + 459, + 497 + ], + "score": 1.0, + "content": "positives vs true positives in", + "type": "text" + }, + { + "bbox": [ + 459, + 485, + 489, + 497 + ], + "score": 0.93, + "content": "\\mathbf { x } ^ { k } ( \\mathbf { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 485, + 492, + 497 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "index": 31.75 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 219, + 519 + ], + "score": 0.92, + "content": "{ \\mathbb { E } } \\{ \\| \\mathbf { x } _ { \\mathbb { S } ^ { c } } ^ { k } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } / \\| \\mathbf { x } ^ { k } ( \\mathbf { x } ^ { * } ) \\| _ { 2 } ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 504, + 506, + 520 + ], + "score": 1.0, + "content": ". False positives take up small proportion over the positives, which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 518, + 361, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 300, + 531 + ], + "score": 1.0, + "content": "supports the Theorem 1 conclusion that support", + "type": "text" + }, + { + "bbox": [ + 300, + 518, + 357, + 531 + ], + "score": 0.84, + "content": "( \\mathbf { x } ^ { k } ( \\mathbf { x } ^ { \\ast } ) ) \\subset \\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 518, + 361, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 419, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 420, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 420, + 555 + ], + "score": 1.0, + "content": "5.2 VALIDATION OF THEOREM 3 (CONVOLUTIONAL ANALYTIC LISTA)", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "score": 1.0, + "content": "For convolutional cases, we use real image data to verify Theorem 3. We train a convolutional", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 149, + 581 + ], + "score": 1.0, + "content": "dictionary", + "type": "text" + }, + { + "bbox": [ + 149, + 569, + 157, + 579 + ], + "score": 0.57, + "content": "\\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 569, + 178, + 581 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 179, + 569, + 245, + 580 + ], + "score": 0.91, + "content": "D = 7 , M = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "on the BSD500 training set (400 images), using the Algorithm 1", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 580, + 495, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 477, + 592 + ], + "score": 1.0, + "content": "in (Liu et al., 2018). We then use it for problems (22) and (24) and solve them with different", + "type": "text" + }, + { + "bbox": [ + 477, + 580, + 491, + 590 + ], + "score": 0.58, + "content": "N s", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 580, + 495, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 102, + 590, + 503, + 617 + ], + "spans": [ + { + "bbox": [ + 102, + 590, + 185, + 617 + ], + "score": 1.0, + "content": "In Table 2, we take", + "type": "text" + }, + { + "bbox": [ + 185, + 596, + 237, + 610 + ], + "score": 0.93, + "content": "\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 590, + 240, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 241, + 596, + 288, + 610 + ], + "score": 0.89, + "content": "\\mathbf { w } ^ { * } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 5 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 590, + 482, + 617 + ], + "score": 1.0, + "content": "(consider 50 as large enough) For this example,", + "type": "text" + }, + { + "bbox": [ + 482, + 596, + 503, + 610 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 101, + 604, + 509, + 628 + ], + "spans": [ + { + "bbox": [ + 101, + 604, + 275, + 628 + ], + "score": 1.0, + "content": "has only one element. Table 2 shows that", + "type": "text" + }, + { + "bbox": [ + 276, + 610, + 321, + 622 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } = \\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 604, + 337, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 338, + 610, + 372, + 621 + ], + "score": 0.91, + "content": "N \\geq 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 604, + 509, + 628 + ], + "score": 1.0, + "content": ", i.e., the solution of the problem", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 199, + 633 + ], + "score": 1.0, + "content": "(24) is independent of", + "type": "text" + }, + { + "bbox": [ + 210, + 620, + 221, + 633 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 222, + 621, + 279, + 632 + ], + "score": 0.91, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 620, + 506, + 633 + ], + "score": 1.0, + "content": ", justifying the first conclusion in Theorem 3. In Table", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 101, + 626, + 509, + 672 + ], + "spans": [ + { + "bbox": [ + 101, + 637, + 133, + 672 + ], + "score": 1.0, + "content": "shows valida", + "type": "text" + }, + { + "bbox": [ + 102, + 626, + 153, + 652 + ], + "score": 1.0, + "content": "3, we take", + "type": "text" + }, + { + "bbox": [ + 134, + 645, + 188, + 657 + ], + "score": 0.91, + "content": "\\mathbf { w } _ { \\mathrm { c o n v } } ^ { N } \\to \\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 632, + 223, + 646 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { w } } _ { \\mathrm { c o n v } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 637, + 343, + 672 + ], + "score": 1.0, + "content": "conv cir co, i.e., the solution of the problem (22) d conclusion of Theorem 3. Visualized", + "type": "text" + }, + { + "bbox": [ + 223, + 626, + 243, + 652 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 632, + 291, + 646 + ], + "score": 0.92, + "content": "\\mathbf { w ^ { * } } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 626, + 324, + 652 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 325, + 632, + 353, + 646 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 626, + 509, + 652 + ], + "score": 1.0, + "content": "also has only one element. Table 3", + "type": "text" + }, + { + "bbox": [ + 388, + 637, + 452, + 672 + ], + "score": 1.0, + "content": "o that of (24) as is displayed in", + "type": "text" + }, + { + "bbox": [ + 462, + 637, + 509, + 672 + ], + "score": 1.0, + "content": "increases,pendix F.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 343, + 656, + 387, + 668 + ], + "spans": [ + { + "bbox": [ + 343, + 656, + 387, + 668 + ], + "score": 0.92, + "content": "\\mathbf { w } ^ { * } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + } + ], + "index": 45 + } + ], + "index": 43 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 681, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 119, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "7The number and proportion of false alarms are a more straightforward performance metric. However, they", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "are sensitive to the threshold. We found that, although using a smaller threshold leads to more false alarms,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "the final recovery quality is better and those false alarms have small magnitudes and are easy to remove by", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "thresholding during post-processing. That’s why we chose to show their magnitudes, implying that we get", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 212, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 212, + 732 + ], + "score": 1.0, + "content": "easy-to-remove false alarms.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "In Figure 1 (a) noise-less case, all four learned models apparently converge much faster than two", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "iterative solvers (ISTA/FISTA curves almost overlap in this y-scale, at the small number of iter-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "ations). Among the four networks, classical-LISTA is inferior to the other three by an obvious", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "margin. LISTA-CPSS, TiLISTA and ALISTA perform comparably: ALISTA is observed to eventu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "ally achieve the lowest NMSE. Figure 1(a) also supports Theorem 2, that all networks have at most", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "score": 1.0, + "content": "linear convergence, regardless of how freely their parameters can be end-to-end learned.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "Figure 1 (b) - (d) further show that even in the presence of noise, ALISTA can empirically perform", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "comparably with LISTA-CPSS and TiLISTA, and stay clearly better than LISTA and ISTA/FISTA.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 504, + 188 + ], + "score": 1.0, + "content": "Always note that ALISTA the smallest amount of parameters to learn from the end-to-end train-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "ing (Stage 2). The above results endorse that: i) the optimal LISTA layer-wise weights could be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 197, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 162, + 213 + ], + "score": 1.0, + "content": "structured as", + "type": "text" + }, + { + "bbox": [ + 162, + 198, + 232, + 210 + ], + "score": 0.93, + "content": "\\mathbf { W } ^ { ( k ) } = \\gamma ^ { ( k ) } \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 197, + 506, + 213 + ], + "score": 1.0, + "content": "; and ii) W could be analytically solved rather than learned from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "data, without incurring performance loss. We also observe the significant reduction of training time", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 337, + 233 + ], + "score": 1.0, + "content": "for ALISTA: while LISTA-CPSS of the same depth took", + "type": "text" + }, + { + "bbox": [ + 337, + 221, + 359, + 232 + ], + "score": 0.85, + "content": "{ \\sim } 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "hours to train, ALISTA was trained", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "within only 6 minutes (0.1 hours) to achieve comparable performance, on the same hardware (one", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 185, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 185, + 255 + ], + "score": 1.0, + "content": "1080 Ti on server).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 155, + 506, + 255 + ] + }, + { + "type": "image", + "bbox": [ + 110, + 261, + 476, + 358 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 261, + 476, + 358 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 261, + 476, + 358 + ], + "spans": [ + { + "bbox": [ + 110, + 261, + 476, + 358 + ], + "score": 0.969, + "type": "image", + "image_path": "b4ad8ecd57838661b04ad8c14453ba10544c6080dccc8fdb5991b9989b099d41.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 110, + 261, + 476, + 293.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 110, + 293.3333333333333, + 476, + 325.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 110, + 325.66666666666663, + 476, + 357.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 362, + 497, + 373 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 359, + 498, + 376 + ], + "spans": [ + { + "bbox": [ + 110, + 359, + 498, + 376 + ], + "score": 1.0, + "content": "Figure 2: Justification of Theorem 1 (noiseless case): parameters obtained by training satisfy (9).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + } + ], + "index": 17.0 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 336, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 336, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 336, + 395 + ], + "score": 1.0, + "content": "We further supply Figures 2 and 3 to justify Theo-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 337, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 337, + 406 + ], + "score": 1.0, + "content": "rem 1 from different perspectives. Figure 2 plots the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 403, + 337, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 189, + 419 + ], + "score": 1.0, + "content": "learned parameters", + "type": "text" + }, + { + "bbox": [ + 189, + 404, + 238, + 417 + ], + "score": 0.93, + "content": "\\{ \\bar { \\gamma ^ { ( k ) } } , \\bar { \\theta ^ { ( k ) } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 403, + 337, + 419 + ], + "score": 1.0, + "content": "in ALISTA (Stage 2),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 416, + 336, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 336, + 429 + ], + "score": 1.0, + "content": "showing that they satisfy the properties proposed in The-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 426, + 338, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 146, + 442 + ], + "score": 1.0, + "content": "orem 1:", + "type": "text" + }, + { + "bbox": [ + 147, + 427, + 165, + 440 + ], + "score": 0.88, + "content": "\\gamma ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 426, + 212, + 442 + ], + "score": 1.0, + "content": "bounded;", + "type": "text" + }, + { + "bbox": [ + 212, + 427, + 229, + 439 + ], + "score": 0.89, + "content": "{ \\theta } ^ { \\bar { ( k ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 426, + 251, + 442 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 427, + 270, + 440 + ], + "score": 0.91, + "content": "\\gamma ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 426, + 338, + 442 + ], + "score": 1.0, + "content": "is proportional", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 438, + 338, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 118, + 454 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 119, + 439, + 259, + 453 + ], + "score": 0.6, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\| \\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } ) - \\mathbf { x } ^ { * } \\| _ { 1 } ~ ( ^ { * } \\mathrm { s u p } _ { \\mathbf { x } ^ { * } } , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 438, + 338, + 454 + ], + "score": 1.0, + "content": "is taken over the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 338, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 338, + 464 + ], + "score": 1.0, + "content": "test set). 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We train a convolutional", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 149, + 581 + ], + "score": 1.0, + "content": "dictionary", + "type": "text" + }, + { + "bbox": [ + 149, + 569, + 157, + 579 + ], + "score": 0.57, + "content": "\\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 569, + 178, + 581 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 179, + 569, + 245, + 580 + ], + "score": 0.91, + "content": "D = 7 , M = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "on the BSD500 training set (400 images), using the Algorithm 1", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 580, + 495, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 477, + 592 + ], + "score": 1.0, + "content": "in (Liu et al., 2018). 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Table 2 shows that", + "type": "text" + }, + { + "bbox": [ + 276, + 610, + 321, + 622 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } = \\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 604, + 337, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 338, + 610, + 372, + 621 + ], + "score": 0.91, + "content": "N \\geq 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 604, + 509, + 628 + ], + "score": 1.0, + "content": ", i.e., the solution of the problem", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 199, + 633 + ], + "score": 1.0, + "content": "(24) is independent of", + "type": "text" + }, + { + "bbox": [ + 210, + 620, + 221, + 633 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 222, + 621, + 279, + 632 + ], + "score": 0.91, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 620, + 506, + 633 + ], + "score": 1.0, + "content": ", justifying the first conclusion in Theorem 3. In Table", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 101, + 626, + 509, + 672 + ], + "spans": [ + { + "bbox": [ + 101, + 637, + 133, + 672 + ], + "score": 1.0, + "content": "shows valida", + "type": "text" + }, + { + "bbox": [ + 102, + 626, + 153, + 652 + ], + "score": 1.0, + "content": "3, we take", + "type": "text" + }, + { + "bbox": [ + 134, + 645, + 188, + 657 + ], + "score": 0.91, + "content": "\\mathbf { w } _ { \\mathrm { c o n v } } ^ { N } \\to \\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 632, + 223, + 646 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { w } } _ { \\mathrm { c o n v } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 637, + 343, + 672 + ], + "score": 1.0, + "content": "conv cir co, i.e., the solution of the problem (22) d conclusion of Theorem 3. Visualized", + "type": "text" + }, + { + "bbox": [ + 223, + 626, + 243, + 652 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 632, + 291, + 646 + ], + "score": 0.92, + "content": "\\mathbf { w ^ { * } } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 626, + 324, + 652 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 325, + 632, + 353, + 646 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 626, + 509, + 652 + ], + "score": 1.0, + "content": "also has only one element. Table 3", + "type": "text" + }, + { + "bbox": [ + 388, + 637, + 452, + 672 + ], + "score": 1.0, + "content": "o that of (24) as is displayed in", + "type": "text" + }, + { + "bbox": [ + 462, + 637, + 509, + 672 + ], + "score": 1.0, + "content": "increases,pendix F.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 343, + 656, + 387, + 668 + ], + "spans": [ + { + "bbox": [ + 343, + 656, + 387, + 668 + ], + "score": 0.92, + "content": "\\mathbf { w } ^ { * } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 101, + 590, + 509, + 672 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 109, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 107, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Besides validating Theorem 3, we also present a real image denoising experiment to verify the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 484, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 471, + 106 + ], + "score": 1.0, + "content": "effectiveness of Conv ALISTA. 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lwir- w*||2/1lw*||2
N=10N= 11N = 12N=13N=15N = 20
2.0×10-29.3×10-33.9 ×10-31.4 ×10-128.8×10-135.9×10-13
", + "type": "table", + "image_path": "3f8cf78e76af573f2fa2e03635213624c37cd3b53ba2d137dd16960276687fb5.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 125, + 125, + 487, + 137.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 125, + 137.66666666666666, + 487, + 150.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 125, + 150.33333333333331, + 487, + 162.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 120, + 166, + 487, + 180 + ], + "lines": [ + { + "bbox": [ + 117, + 160, + 486, + 186 + ], + "spans": [ + { + "bbox": [ + 117, + 160, + 326, + 186 + ], + "score": 1.0, + "content": "Table 3: Validation of Conclusion 2 in Theorem 3.", + "type": "text" + }, + { + "bbox": [ + 326, + 167, + 354, + 177 + ], + "score": 0.78, + "content": "D = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 160, + 358, + 186 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 358, + 166, + 423, + 180 + ], + "score": 0.86, + "content": "\\mathbf { w } _ { \\mathrm { c o n v } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 160, + 442, + 186 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 442, + 166, + 486, + 180 + ], + "score": 0.87, + "content": "\\mathbf { w } ^ { * } \\in \\mathbf { w } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table", + "bbox": [ + 196, + 182, + 412, + 219 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 196, + 182, + 412, + 219 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 196, + 182, + 412, + 219 + ], + "spans": [ + { + "bbox": [ + 196, + 182, + 412, + 219 + ], + "score": 0.955, + "html": "
|/wconv - w*||2/lw*|2
N=3N=5N= 10N=15N = 20
0.18920.08500.02840.01610.0113
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We choose", + "type": "text" + }, + { + "bbox": [ + 434, + 380, + 487, + 391 + ], + "score": 0.91, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 379, + 507, + 397 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "score": 1.0, + "content": "0.03. Other simulation settings are by default the same as in Section 5.1. We then build the Robust", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "ALISTA model, following the strategy in Section 4 and using a 4-layer encoder for approximating", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "its second step (see Appendix G for details). Correspondingly, we compare Robust ALISTA with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "TiLISTA and ALISTA with specific data augmentation: we straightforwardly augment their training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 434, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 359, + 449 + ], + "score": 1.0, + "content": "sets, by including all data generated with randomly perturbed", + "type": "text" + }, + { + "bbox": [ + 360, + 434, + 374, + 447 + ], + "score": 0.53, + "content": "\\breve { \\tilde { \\mathbf { D } } } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "when training Robust ALISTA.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 447, + 379, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 379, + 459 + ], + "score": 1.0, + "content": "We also include the data-free FISTA algorithm into the comparison.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "Figure 4 plots the results when the trained models are applied on the testing data, generated with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 258, + 488 + ], + "score": 1.0, + "content": "the same dictionary and perturbed by", + "type": "text" + }, + { + "bbox": [ + 258, + 475, + 295, + 487 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , \\sigma _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 474, + 336, + 488 + ], + "score": 1.0, + "content": ". 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Not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 484, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 493, + 501 + ], + "score": 1.0, + "content": "surprisingly, FISTA is unaffected, while the other three data-driven models all slight degrade as", + "type": "text" + }, + { + "bbox": [ + 493, + 488, + 504, + 497 + ], + "score": 0.81, + "content": "\\sigma _ { t }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "increases. Compared to the augmented TiLISTA and ALISTA whose performance are both inferior", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "to FISTA, the proposed Robust ALISTA appears to be much more favorable in improving robustness", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 231, + 533 + ], + "score": 1.0, + "content": "to model perturbations. In both", + "type": "text" + }, + { + "bbox": [ + 232, + 520, + 255, + 530 + ], + "score": 0.89, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "cases, it consistently achieves much lower NMSE than FISTA,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 152, + 543 + ], + "score": 1.0, + "content": "even when", + "type": "text" + }, + { + "bbox": [ + 153, + 532, + 163, + 541 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 529, + 256, + 543 + ], + "score": 1.0, + "content": "has slightly surpassed", + "type": "text" + }, + { + "bbox": [ + 257, + 532, + 280, + 541 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ". Although the NMSE of ALISTA may decrease faster", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 115, + 555 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 116, + 543, + 127, + 552 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 540, + 432, + 555 + ], + "score": 1.0, + "content": "continues growing larger, such decrease could be alleviated by improving", + "type": "text" + }, + { + "bbox": [ + 432, + 542, + 456, + 552 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "in training,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 183, + 565 + ], + "score": 1.0, + "content": "e.g., by comparing", + "type": "text" + }, + { + "bbox": [ + 183, + 552, + 234, + 563 + ], + "score": 0.9, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "and 0.03. Robust ALISTA demonstrates remarkable robustness and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 562, + 408, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 340, + 576 + ], + "score": 1.0, + "content": "maintains the best NMSE performance, within at least the", + "type": "text" + }, + { + "bbox": [ + 340, + 563, + 380, + 574 + ], + "score": 0.88, + "content": "[ 0 , \\sigma _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 562, + 408, + 576 + ], + "score": 1.0, + "content": "range.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 581, + 308, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 310, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 310, + 596 + ], + "score": 1.0, + "content": "6 CONCLUSIONS AND FUTURE WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Based on the recent theoretical advances of LISTA, we have made further steps to reduce the train-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "ing complexity and improve the robustness of LISTA. Specifically, we no longer train any matrix for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "LISTA but directly use the solution to an analytic minimization problem to solve for its layer-wise", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 504, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 504, + 646 + ], + "score": 1.0, + "content": "weights. Therefore, only two scalar sequences (stepsizes and thresholds) still need to be trained.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Excluding the matrix from training is backed by our theoretical upper and lower bounds. The re-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "sulting method, Analytic LISTA or ALISTA, is not only faster to train but performs as well as the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "score": 1.0, + "content": "state-of-the-art variant of LISTA by (Chen et al., 2018). This discovery motivates us to further re-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "place the minimization algorithm by its unfolding neural network, and train this neural network to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "more quickly produce the weight matrix. The resulting algorithm is used to handle perturbations", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "in the model dictionary — we only train once for a dictionary with all its small perturbations. Our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "future work will investigate the theoretical sensitivity of ALISTA (and its convolutional version) to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 192, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 192, + 734 + ], + "score": 1.0, + "content": "noisy measurements.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 109, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 107, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Besides validating Theorem 3, we also present a real image denoising experiment to verify the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 484, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 471, + 106 + ], + "score": 1.0, + "content": "effectiveness of Conv ALISTA. The detailed settings and results are presented in Appendix", + "type": "text" + }, + { + "bbox": [ + 472, + 94, + 480, + 104 + ], + "score": 0.27, + "content": "_ \\mathrm { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 93, + 484, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 107, + 82, + 505, + 106 + ] + }, + { + "type": "table", + "bbox": [ + 125, + 125, + 487, + 163 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 127, + 109, + 483, + 123 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 105, + 480, + 127 + ], + "spans": [ + { + "bbox": [ + 124, + 105, + 331, + 127 + ], + "score": 1.0, + "content": "Table 2: Validation of Conclusion 1 in Theorem 3.", + "type": "text" + }, + { + "bbox": [ + 332, + 110, + 360, + 120 + ], + "score": 0.84, + "content": "D = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 105, + 363, + 127 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 364, + 108, + 415, + 123 + ], + "score": 0.91, + "content": "\\mathbf { w } _ { \\mathrm { c i r } } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 105, + 433, + 127 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 433, + 109, + 480, + 122 + ], + "score": 0.9, + "content": "\\mathbf { w } ^ { * } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 5 0 }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 125, + 125, + 487, + 163 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 125, + 487, + 163 + ], + "spans": [ + { + "bbox": [ + 125, + 125, + 487, + 163 + ], + "score": 0.955, + "html": "
lwir- w*||2/1lw*||2
N=10N= 11N = 12N=13N=15N = 20
2.0×10-29.3×10-33.9 ×10-31.4 ×10-128.8×10-135.9×10-13
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|/wconv - w*||2/lw*|2
N=3N=5N= 10N=15N = 20
0.18920.08500.02840.01610.0113
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We choose", + "type": "text" + }, + { + "bbox": [ + 434, + 380, + 487, + 391 + ], + "score": 0.91, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 379, + 507, + 397 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "score": 1.0, + "content": "0.03. Other simulation settings are by default the same as in Section 5.1. We then build the Robust", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "ALISTA model, following the strategy in Section 4 and using a 4-layer encoder for approximating", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "its second step (see Appendix G for details). Correspondingly, we compare Robust ALISTA with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "TiLISTA and ALISTA with specific data augmentation: we straightforwardly augment their training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 434, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 359, + 449 + ], + "score": 1.0, + "content": "sets, by including all data generated with randomly perturbed", + "type": "text" + }, + { + "bbox": [ + 360, + 434, + 374, + 447 + ], + "score": 0.53, + "content": "\\breve { \\tilde { \\mathbf { D } } } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "when training Robust ALISTA.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 447, + 379, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 379, + 459 + ], + "score": 1.0, + "content": "We also include the data-free FISTA algorithm into the comparison.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 368, + 507, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "Figure 4 plots the results when the trained models are applied on the testing data, generated with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 258, + 488 + ], + "score": 1.0, + "content": "the same dictionary and perturbed by", + "type": "text" + }, + { + "bbox": [ + 258, + 475, + 295, + 487 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , \\sigma _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 474, + 336, + 488 + ], + "score": 1.0, + "content": ". We vary", + "type": "text" + }, + { + "bbox": [ + 336, + 477, + 347, + 486 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 474, + 459, + 488 + ], + "score": 1.0, + "content": "from zero to slightly above", + "type": "text" + }, + { + "bbox": [ + 459, + 477, + 482, + 487 + ], + "score": 0.89, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 474, + 506, + 488 + ], + "score": 1.0, + "content": ". Not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 484, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 493, + 501 + ], + "score": 1.0, + "content": "surprisingly, FISTA is unaffected, while the other three data-driven models all slight degrade as", + "type": "text" + }, + { + "bbox": [ + 493, + 488, + 504, + 497 + ], + "score": 0.81, + "content": "\\sigma _ { t }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "increases. Compared to the augmented TiLISTA and ALISTA whose performance are both inferior", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "to FISTA, the proposed Robust ALISTA appears to be much more favorable in improving robustness", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 231, + 533 + ], + "score": 1.0, + "content": "to model perturbations. In both", + "type": "text" + }, + { + "bbox": [ + 232, + 520, + 255, + 530 + ], + "score": 0.89, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "cases, it consistently achieves much lower NMSE than FISTA,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 152, + 543 + ], + "score": 1.0, + "content": "even when", + "type": "text" + }, + { + "bbox": [ + 153, + 532, + 163, + 541 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 529, + 256, + 543 + ], + "score": 1.0, + "content": "has slightly surpassed", + "type": "text" + }, + { + "bbox": [ + 257, + 532, + 280, + 541 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ". Although the NMSE of ALISTA may decrease faster", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 115, + 555 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 116, + 543, + 127, + 552 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 540, + 432, + 555 + ], + "score": 1.0, + "content": "continues growing larger, such decrease could be alleviated by improving", + "type": "text" + }, + { + "bbox": [ + 432, + 542, + 456, + 552 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "in training,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 183, + 565 + ], + "score": 1.0, + "content": "e.g., by comparing", + "type": "text" + }, + { + "bbox": [ + 183, + 552, + 234, + 563 + ], + "score": 0.9, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "and 0.03. Robust ALISTA demonstrates remarkable robustness and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 562, + 408, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 340, + 576 + ], + "score": 1.0, + "content": "maintains the best NMSE performance, within at least the", + "type": "text" + }, + { + "bbox": [ + 340, + 563, + 380, + 574 + ], + "score": 0.88, + "content": "[ 0 , \\sigma _ { m a x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 562, + 408, + 576 + ], + "score": 1.0, + "content": "range.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 464, + 506, + 576 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 581, + 308, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 310, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 310, + 596 + ], + "score": 1.0, + "content": "6 CONCLUSIONS AND FUTURE WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Based on the recent theoretical advances of LISTA, we have made further steps to reduce the train-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "ing complexity and improve the robustness of LISTA. Specifically, we no longer train any matrix for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "LISTA but directly use the solution to an analytic minimization problem to solve for its layer-wise", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 504, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 504, + 646 + ], + "score": 1.0, + "content": "weights. Therefore, only two scalar sequences (stepsizes and thresholds) still need to be trained.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Excluding the matrix from training is backed by our theoretical upper and lower bounds. The re-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "sulting method, Analytic LISTA or ALISTA, is not only faster to train but performs as well as the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "score": 1.0, + "content": "state-of-the-art variant of LISTA by (Chen et al., 2018). This discovery motivates us to further re-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "place the minimization algorithm by its unfolding neural network, and train this neural network to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "more quickly produce the weight matrix. The resulting algorithm is used to handle perturbations", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "in the model dictionary — we only train once for a dictionary with all its small perturbations. Our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "future work will investigate the theoretical sensitivity of ALISTA (and its convolutional version) to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 192, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 192, + 734 + ], + "score": 1.0, + "content": "noisy measurements.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 600, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 98, + 504, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 504, + 114 + ], + "score": 1.0, + "content": "Thomas Blumensath and Mike E. 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In AAAI", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 116, + 174, + 292, + 188 + ], + "spans": [ + { + "bbox": [ + 116, + 174, + 292, + 188 + ], + "score": 1.0, + "content": "Conference on Artificial Intelligence, 2018.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 152, + 506, + 188 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 245, + 96 + ], + "score": 1.0, + "content": "A PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 104, + 103, + 504, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 232, + 120 + ], + "score": 1.0, + "content": "In this proof, we use the notion", + "type": "text" + }, + { + "bbox": [ + 232, + 105, + 250, + 117 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 103, + 293, + 120 + ], + "score": 1.0, + "content": "to replace", + "type": "text" + }, + { + "bbox": [ + 293, + 105, + 329, + 118 + ], + "score": 0.93, + "content": "\\mathbf { x } ^ { ( k ) } ( \\mathbf { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 103, + 417, + 120 + ], + "score": 1.0, + "content": "for simplicity. We fix", + "type": "text" + }, + { + "bbox": [ + 417, + 106, + 428, + 117 + ], + "score": 0.65, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 103, + 480, + 120 + ], + "score": 1.0, + "content": "in the proof,", + "type": "text" + }, + { + "bbox": [ + 480, + 106, + 504, + 118 + ], + "score": 0.9, + "content": "\\tilde { \\mu } ( \\mathbf { D } )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 217, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 205, + 130 + ], + "score": 1.0, + "content": "can be simply written as", + "type": "text" + }, + { + "bbox": [ + 206, + 118, + 213, + 129 + ], + "score": 0.84, + "content": "\\tilde { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 117, + 217, + 130 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 344, + 146 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 344, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 344, + 148 + ], + "score": 1.0, + "content": "Before proving Theorem 1, we present and prove a lemma.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 104, + 148, + 409, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 409, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 364, + 162 + ], + "score": 1.0, + "content": "Lemma 1. With all the settings the same with those in Theorem", + "type": "text" + }, + { + "bbox": [ + 365, + 150, + 370, + 159 + ], + "score": 0.53, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 147, + 409, + 162 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 165, + 360, + 181 + ], + "lines": [ + { + "bbox": [ + 251, + 165, + 360, + 181 + ], + "spans": [ + { + "bbox": [ + 251, + 165, + 360, + 181 + ], + "score": 0.91, + "content": "\\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ) \\subset \\mathbb { S } , \\quad \\forall k .", + "type": "interline_equation", + "image_path": "19a107bb58d0b3b9dd480ba6f4e99e06604e287aa0d68d63bd819bbcdf1bd597.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 251, + 165, + 360, + 181 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 405, + 201 + ], + "lines": [ + { + "bbox": [ + 101, + 184, + 402, + 207 + ], + "spans": [ + { + "bbox": [ + 101, + 184, + 295, + 207 + ], + "score": 1.0, + "content": "In another word, there are no false positives in", + "type": "text" + }, + { + "bbox": [ + 295, + 186, + 313, + 199 + ], + "score": 0.75, + "content": "\\mathbf { x } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 184, + 318, + 207 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 319, + 185, + 402, + 201 + ], + "score": 0.74, + "content": "x _ { i } ^ { ( k ) } = 0 , \\forall i \\notin \\mathbb { S } , \\forall k", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 195, + 225 + ], + "score": 1.0, + "content": "Proof. Take arbitrary", + "type": "text" + }, + { + "bbox": [ + 195, + 212, + 253, + 225 + ], + "score": 0.93, + "content": "\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 212, + 411, + 225 + ], + "score": 1.0, + "content": ". We prove Lemma 1 by induction. As", + "type": "text" + }, + { + "bbox": [ + 411, + 213, + 437, + 223 + ], + "score": 0.88, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 212, + 505, + 225 + ], + "score": 1.0, + "content": ", (27) is satisfied", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 223, + 385, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 129, + 238 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 224, + 167, + 235 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { ( 0 ) } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 223, + 199, + 238 + ], + "score": 1.0, + "content": ". Fixing", + "type": "text" + }, + { + "bbox": [ + 199, + 226, + 206, + 235 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 223, + 301, + 238 + ], + "score": 1.0, + "content": ", and assuming support", + "type": "text" + }, + { + "bbox": [ + 301, + 224, + 345, + 237 + ], + "score": 0.85, + "content": "( \\mathbf { x } ^ { ( k ) } ) \\subset \\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 223, + 385, + 238 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 241, + 437, + 293 + ], + "lines": [ + { + "bbox": [ + 174, + 241, + 437, + 293 + ], + "spans": [ + { + "bbox": [ + 174, + 241, + 437, + 293 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { x _ { i } ^ { ( k + 1 ) } = \\eta _ { \\theta ^ { ( k ) } } \\Big ( x _ { i } ^ { ( k ) } - \\gamma ^ { ( k ) } ( \\mathbf { W } _ { : , i } ) ^ { T } \\big ( \\mathbf { D x } ^ { ( k ) } - \\mathbf { b } \\big ) \\Big ) } \\\\ & { \\quad \\quad \\quad = \\eta _ { \\theta ^ { ( k ) } } \\Big ( - \\gamma ^ { ( k ) } \\displaystyle \\sum _ { j \\in \\mathbb { S } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } \\big ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } \\big ) \\Big ) , \\quad \\forall i \\notin \\mathbb { S } . } \\end{array}", + "type": "interline_equation", + "image_path": "ba67cda30d2837a8a8e87de6941218c4d28e1e10fd9211440af8a7870c6e7ad2.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 174, + 241, + 437, + 258.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 174, + 258.3333333333333, + 437, + 275.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 275.66666666666663, + 437, + 292.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 297, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 104, + 296, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 248, + 313 + ], + "score": 1.0, + "content": "By (9), the thresholds are taken as", + "type": "text" + }, + { + "bbox": [ + 248, + 298, + 394, + 312 + ], + "score": 0.94, + "content": "\\theta ^ { ( k ) } = \\tilde { \\mu } \\gamma ^ { ( k ) } \\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 296, + 447, + 313 + ], + "score": 1.0, + "content": ". 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x _ { j } ^ { * } ) + ( 1 - \\gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "b0e019ad95b75de9faab7bcef7891705e00259182818d2c5e506887b486a5e2c.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 164, + 611, + 447, + 636.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 164, + 636.3333333333334, + 447, + 661.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 164, + 661.6666666666667, + 447, + 687.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 690, + 335, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 335, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 309, + 703 + ], + "score": 1.0, + "content": "Then the following inclusion formula holds for all", + "type": "text" + }, + { + "bbox": [ + 309, + 691, + 331, + 701 + ], + "score": 0.86, + "content": "i \\in \\mathbb S", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 690, + 335, + 703 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 707, + 505, + 736 + ], + "lines": [ + { + "bbox": [ + 111, + 707, + 505, + 736 + ], + "spans": [ + { + "bbox": [ + 111, + 707, + 505, + 736 + ], + "score": 0.89, + "content": "\\mathbf { \\Phi } _ { i } ^ { ( k + 1 ) } - 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We fix", + "type": "text" + }, + { + "bbox": [ + 417, + 106, + 428, + 117 + ], + "score": 0.65, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 103, + 480, + 120 + ], + "score": 1.0, + "content": "in the proof,", + "type": "text" + }, + { + "bbox": [ + 480, + 106, + 504, + 118 + ], + "score": 0.9, + "content": "\\tilde { \\mu } ( \\mathbf { D } )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 217, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 205, + 130 + ], + "score": 1.0, + "content": "can be simply written as", + "type": "text" + }, + { + "bbox": [ + 206, + 118, + 213, + 129 + ], + "score": 0.84, + "content": "\\tilde { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 117, + 217, + 130 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 103, + 504, + 130 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 344, + 146 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 344, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 344, + 148 + ], + "score": 1.0, + "content": "Before proving Theorem 1, we present and prove a lemma.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 132, + 344, + 148 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 148, + 409, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 409, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 364, + 162 + ], + "score": 1.0, + "content": "Lemma 1. With all the settings the same with those in Theorem", + "type": "text" + }, + { + "bbox": [ + 365, + 150, + 370, + 159 + ], + "score": 0.53, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 147, + 409, + 162 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 147, + 409, + 162 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 165, + 360, + 181 + ], + "lines": [ + { + "bbox": [ + 251, + 165, + 360, + 181 + ], + "spans": [ + { + "bbox": [ + 251, + 165, + 360, + 181 + ], + "score": 0.91, + "content": "\\mathrm { s u p p o r t } ( \\mathbf { x } ^ { ( k ) } ) \\subset \\mathbb { S } , \\quad \\forall k .", + "type": "interline_equation", + "image_path": "19a107bb58d0b3b9dd480ba6f4e99e06604e287aa0d68d63bd819bbcdf1bd597.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 251, + 165, + 360, + 181 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 405, + 201 + ], + "lines": [ + { + "bbox": [ + 101, + 184, + 402, + 207 + ], + "spans": [ + { + "bbox": [ + 101, + 184, + 295, + 207 + ], + "score": 1.0, + "content": "In another word, there are no false positives in", + "type": "text" + }, + { + "bbox": [ + 295, + 186, + 313, + 199 + ], + "score": 0.75, + "content": "\\mathbf { x } ^ { ( k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 184, + 318, + 207 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 319, + 185, + 402, + 201 + ], + "score": 0.74, + "content": "x _ { i } ^ { ( k ) } = 0 , \\forall i \\notin \\mathbb { S } , \\forall k", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 101, + 184, + 402, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 195, + 225 + ], + "score": 1.0, + "content": "Proof. 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x _ { j } ^ { * } ) + ( 1 - \\gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "b0e019ad95b75de9faab7bcef7891705e00259182818d2c5e506887b486a5e2c.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 164, + 611, + 447, + 636.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 164, + 636.3333333333334, + 447, + 661.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 164, + 661.6666666666667, + 447, + 687.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 690, + 335, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 335, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 309, + 703 + ], + "score": 1.0, + "content": "Then the following inclusion formula holds for all", + "type": "text" + }, + { + "bbox": [ + 309, + 691, + 331, + 701 + ], + "score": 0.86, + "content": "i \\in \\mathbb S", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 690, + 335, + 703 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 690, + 335, + 703 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 707, + 505, + 736 + ], + "lines": [ + { + "bbox": [ + 111, + 707, + 505, + 736 + ], + "spans": [ + { + "bbox": [ + 111, + 707, + 505, + 736 + ], + "score": 0.89, + "content": "\\mathbf { \\Phi } _ { i } ^ { ( k + 1 ) } - \\mathbf { \\Phi } _ { x _ { i } ^ { * } } ^ { * } \\in - \\gamma ^ { ( k ) } \\sum _ { \\substack { j \\in \\mathbb { S } , j \\neq i } } ( \\mathbf { W } _ { : , i } ) ^ { T } \\mathbf { D } _ { : , j } ( x _ { j } ^ { ( k ) } - x _ { j } ^ { * } ) - \\theta ^ { ( k ) } \\partial \\ell _ { 1 } ( x _ { i } ^ { ( k + 1 ) } ) + ( 1 - \\gamma ^ { ( k ) } ) ( x _ { i } ^ { ( k ) } - x _ { i } ^ { * } ) .", + "type": "interline_equation", + "image_path": "29a256724b5c73fbb96859fbf4f89c3c0616d291f8d88156fa969348c9710b55.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 111, + 707, + 505, + 736 + ], + "spans": [], + "index": 32 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 187, + 95 + ], + "score": 1.0, + "content": "By the definition of", + "type": "text" + }, + { + "bbox": [ + 188, + 83, + 203, + 93 + ], + "score": 0.89, + "content": "\\partial \\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 81, + 277, + 95 + ], + "score": 1.0, + "content": ", every element in", + "type": "text" + }, + { + "bbox": [ + 277, + 82, + 342, + 95 + ], + "score": 0.91, + "content": "\\partial \\ell _ { 1 } ( x ) , \\forall x \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "has a magnitude less than or equal to 1.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 185, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 158, + 105 + ], + "score": 1.0, + "content": "Thus, for all", + "type": "text" + }, + { + "bbox": [ + 158, + 94, + 180, + 104 + ], + "score": 0.88, + "content": "i \\in \\mathbb S", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 93, + 185, + 105 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 106, + 479, + 166 + ], + "lines": [ + { + "bbox": [ + 132, + 106, + 479, + 166 + ], + "spans": [ + { + "bbox": [ + 132, + 106, + 479, + 166 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { | x _ { i } ^ { ( k + 1 ) } - x _ { i } ^ { * } | \\le \\displaystyle \\sum _ { j \\in \\mathbb S , j \\ne i } \\gamma ^ { ( k ) } \\Big | ( \\mathbf W _ { : , i } ) ^ { T } \\mathbf D _ { : , j } \\Big | | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } | + \\theta ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } | } \\\\ & { \\qquad \\le \\widetilde { \\mu } \\gamma ^ { ( k ) } \\displaystyle \\sum _ { j \\in \\mathbb S , j \\ne i } | x _ { j } ^ { ( k ) } - x _ { j } ^ { * } | + \\theta ^ { ( k ) } + | 1 - \\gamma ^ { ( k ) } | | x _ { i } ^ { ( k ) } - x _ { i } ^ { * } | . } \\end{array}", + "type": "interline_equation", + "image_path": "9d43a100754ec2c8718aa9e66be81b0cd09d44be92bd3db0f78fe7be26b31b72.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 132, + 106, + 479, + 126.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 132, + 126.0, + 479, + 146.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 132, + 146.0, + 479, + 166.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 381, + 185 + ], + "lines": [ + { + "bbox": [ + 104, + 168, + 384, + 187 + ], + "spans": [ + { + "bbox": [ + 104, + 168, + 196, + 187 + ], + "score": 1.0, + "content": "Equation (27) implies", + "type": "text" + }, + { + "bbox": [ + 196, + 169, + 319, + 185 + ], + "score": 0.94, + "content": "\\| \\mathbf { x } ^ { ( k ) } - 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\\gamma ^ { ( k ) } \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | _ { 1 } } \\\\ & { = \\tilde { \\mu } \\gamma ^ { ( k ) } ( \\big | \\mathbb { S } \\big | - 1 ) \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | _ { 1 } + \\theta ^ { ( k ) } \\big | \\mathbb { S } \\big | + \\big | 1 - \\gamma ^ { ( k ) } \\big | \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | . } \\end{array}", + "type": "interline_equation", + "image_path": "d094400df99d7946ab901830153a0a9d64e91fd8cc4e305777691852d0a61bab.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 132, + 189, + 479, + 223.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 223.0, + 479, + 257.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 257.0, + 479, + 291.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 403, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 403, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 294, + 313 + ], + "score": 1.0, + "content": "Taking supremum of the above inequality over", + "type": "text" + }, + { + "bbox": [ + 295, + 299, + 352, + 311 + ], + "score": 0.92, + "content": "\\mathbf { x } ^ { * } \\in \\mathcal { X } ( B , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 297, + 369, + 313 + ], + "score": 1.0, + "content": ", by", + "type": "text" + }, + { + "bbox": [ + 369, + 299, + 398, + 311 + ], + "score": 0.88, + "content": "| \\mathbb { S } | \\le s", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 297, + 403, + 313 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 314, + 474, + 338 + ], + "lines": [ + { + "bbox": [ + 136, + 314, + 474, + 338 + ], + "spans": [ + { + "bbox": [ + 136, + 314, + 474, + 338 + ], + "score": 0.9, + "content": "\\operatorname* { s u p } _ { \\mathbf { x } ^ { * } } \\{ \\| \\mathbf { x } ^ { ( k + 1 ) } - 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We fix", + "type": "text" + }, + { + "bbox": [ + 220, + 595, + 230, + 605 + ], + "score": 0.57, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 592, + 285, + 609 + ], + "score": 1.0, + "content": "and sample a", + "type": "text" + }, + { + "bbox": [ + 286, + 595, + 324, + 606 + ], + "score": 0.91, + "content": "\\mathbf { x } ^ { * } \\sim P _ { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 592, + 329, + 609 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 592, + 329, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 171, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 171, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 171, + 625 + ], + "score": 1.0, + "content": "If we can prove", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 610, + 171, + 625 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 621, + 415, + 643 + ], + "lines": [ + { + "bbox": [ + 194, + 621, + 415, + 643 + ], + "spans": [ + { + "bbox": [ + 194, + 621, + 415, + 643 + ], + "score": 0.9, + "content": "P \\Big ( ( 1 3 ) \\mathrm { d o e s ~ n o t ~ h o l d } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) \\leq \\epsilon | \\mathbb { S } | + \\epsilon ^ { | \\mathbb { S } | } ,", + "type": "interline_equation", + "image_path": "fb23775c1cd50131b85d7b8feca23f22b133206a84682e331ccca223c326bf86.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 194, + 621, + 415, + 643 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 320, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 320, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 320, + 658 + ], + "score": 1.0, + "content": "then the lower bound (13) in Theorem 2 is proved by", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 641, + 320, + 658 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 658, + 469, + 736 + ], + "lines": [ + { + "bbox": [ + 140, + 658, + 469, + 736 + ], + "spans": [ + { + "bbox": [ + 140, + 658, + 469, + 736 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { P \\Big ( ( 1 3 ) \\mathrm { h o l d s } \\Big ) = \\displaystyle \\sum _ { \\mathbb { S } , 2 \\leq | \\mathbb { S } | \\leq s } P \\Big ( ( 1 3 ) \\mathrm { h o l d s } \\Big | \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) P \\Big ( \\mathrm { s u p p o r t } ( \\mathbf { x } ^ { * } ) = \\mathbb { S } \\Big ) } \\\\ & { \\qquad \\geq ( 1 - 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Let", + "type": "text" + }, + { + "bbox": [ + 155, + 481, + 246, + 496 + ], + "score": 0.89, + "content": "\\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } ) = \\mathbf { s } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 476, + 262, + 500 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 262, + 484, + 325, + 495 + ], + "score": 0.9, + "content": "t = 1 , 2 , \\cdots , j", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 476, + 354, + 500 + ], + "score": 1.0, + "content": ". 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1 , 1 \\} , \\forall i = 1 , \\cdots , n { \\Big \\} } .", + "type": "interline_equation", + "image_path": "89ad8d5ad0636a39a33d3e911125f654c1f015e92ba29e5688e5e59bb1375a4c.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 182, + 434, + 427, + 456 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 104, + 462, + 508, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 132, + 484 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 132, + 466, + 236, + 481 + ], + "score": 0.93, + "content": "\\mathrm { s i g n } \\big ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } \\big ) \\ \\in \\ \\mathrm { S i } ( | \\mathbb { S } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 462, + 267, + 484 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 267, + 468, + 330, + 480 + ], + "score": 0.76, + "content": "t = 1 , 2 , \\cdots , j", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 462, + 335, + 484 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 335, + 466, + 420, + 481 + ], + "score": 0.91, + "content": "\\{ \\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - 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Let", + "type": "text" + }, + { + "bbox": [ + 155, + 481, + 246, + 496 + ], + "score": 0.89, + "content": "\\mathrm { s i g n } ( \\mathbf { x } _ { \\mathbb { S } } ^ { ( k - j + t ) } ) = \\mathbf { s } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 476, + 262, + 500 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 262, + 484, + 325, + 495 + ], + "score": 0.9, + "content": "t = 1 , 2 , \\cdots , j", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 476, + 354, + 500 + ], + "score": 1.0, + "content": ". 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1 } \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } ) ^ { k - j } \\mathbb { 3 } ^ { ( j - k ) \\vert \\mathbb { S } \\vert } + \\epsilon ^ { \\vert \\mathbb { S } \\vert } = \\sum _ { j = 1 } ^ { k } \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } ( \\bar { \\sigma } _ { \\operatorname* { m i n } } ) ^ { j } 3 ^ { - j \\vert \\mathbb { S } \\vert } + \\epsilon ^ { \\vert \\mathbb { S } \\vert } } \\ \\\\ { { \\displaystyle = \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } \\frac { \\bar { \\sigma } _ { \\operatorname* { m i n } } 3 ^ { - \\vert \\mathbb { S } \\vert } } { 1 - \\bar { \\sigma } _ { \\operatorname* { m i n } } 3 ^ { - \\vert \\mathbb { S } \\vert } } \\Big ( 1 - ( \\bar { \\sigma } _ { \\operatorname* { m i n } } 3 ^ { - \\vert \\mathbb { S } \\vert } ) ^ { k } \\Big ) + \\epsilon ^ { \\vert \\mathbb { S } \\vert } \\leq \\epsilon \\vert \\mathbb { S } \\vert ^ { 3 / 2 } + \\epsilon ^ { \\vert \\mathbb { S } \\vert } } . } \\end{array}", + "type": "interline_equation", + "image_path": "e47d7755e253012a629d6e016ae9dccf4770f99512774a01f790b19220532a5d.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 158, + 221, + 451, + 248.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 158, + 248.66666666666666, + 451, + 276.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 158, + 276.3333333333333, + 451, + 304.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 189, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 190, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 190, + 317 + ], + "score": 1.0, + "content": "Then (28) is proved.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 331, + 244, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 245, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 245, + 346 + ], + "score": 1.0, + "content": "C PROOF OF THEOREM 3", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 355, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "score": 1.0, + "content": "There are two conclusions in Theorem 3. 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k _ { 2 } , l _ { 2 } , m _ { 2 } ) = \\displaystyle \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { j = 0 } ^ { N - 1 } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k _ { 1 } , l _ { 1 } , m _ { 1 } ) \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( i , j ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) } \\\\ & { \\qquad = \\displaystyle \\sum _ { i \\in \\mathbb { Z } ( k _ { 1 } , k _ { 2 } ) } \\sum _ { j \\in \\mathcal { I } ( l _ { 1 } , l _ { 2 } ) } \\mathbf { d } _ { m _ { 1 } } \\big ( I ( i , k _ { 1 } ) , I ( j , l _ { 1 } ) \\big ) \\mathbf { w } _ { m _ { 2 } } \\big ( I ( i , k _ { 2 } ) , I ( j , l _ { 2 } ) \\big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "428b664be9bfa30165a2f2480078f55455952cf22507003364ab42bf4ff64d77.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 111, + 669, + 500, + 691.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 111, + 691.3333333333334, + 500, + 713.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 111, + 713.6666666666667, + 500, + 736.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 133, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 135, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 135, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 96, + 465, + 126 + ], + "lines": [ + { + "bbox": [ + 144, + 96, + 465, + 126 + ], + "spans": [ + { + "bbox": [ + 144, + 96, + 465, + 126 + ], + "score": 0.86, + "content": "\\begin{array} { r l } & { \\mathcal { T } ( k _ { 1 } , k _ { 2 } ) = \\{ i | 0 \\leq i \\leq N - 1 , 0 \\leq I ( i , k _ { 1 } ) \\leq D - 1 , 0 \\leq I ( i , k _ { 2 } ) \\leq D - 1 \\} , } \\\\ & { \\mathcal { I } ( l _ { 1 } , l _ { 2 } ) = \\{ j | 0 \\leq j \\leq N - 1 , 0 \\leq I ( j , l _ { 1 } ) \\leq D - 1 , 0 \\leq I ( j , l _ { 2 } ) \\leq D - 1 \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "f6d967411bec56ca235aee7847e9c417607e1338fcb565c44cc9449345fb3e4e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 144, + 96, + 465, + 126 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 129, + 291, + 141 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 291, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 180, + 142 + ], + "score": 1.0, + "content": "Lemma 2. Given", + "type": "text" + }, + { + "bbox": [ + 181, + 130, + 235, + 141 + ], + "score": 0.9, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 128, + 291, + 142 + ], + "score": 1.0, + "content": ", it holds that:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 149, + 506, + 203 + ], + "lines": [ + { + "bbox": [ + 104, + 148, + 501, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 123, + 163 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 123, + 150, + 183, + 162 + ], + "score": 0.92, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 148, + 242, + 163 + ], + "score": 1.0, + "content": "if and only if “", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 501, + 163 + ], + "score": 0.76, + "content": "0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime \\prime } o r ^ { \\ast } 0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 120, + 160, + 151, + 174 + ], + "spans": [ + { + "bbox": [ + 120, + 160, + 151, + 174 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 178, + 500, + 193 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 123, + 193 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 123, + 180, + 181, + 192 + ], + "score": 0.91, + "content": "\\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 178, + 241, + 193 + ], + "score": 1.0, + "content": "if and only if “", + "type": "text" + }, + { + "bbox": [ + 242, + 179, + 500, + 192 + ], + "score": 0.62, + "content": "0 \\leq ( l _ { 1 } - l _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime \\prime } o r ^ { \\textit { \\infty } } 0 < ( l _ { 2 } - l _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 121, + 190, + 151, + 204 + ], + "spans": [ + { + "bbox": [ + 121, + 190, + 151, + 204 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 105, + 214, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 363, + 229 + ], + "score": 1.0, + "content": "Proof. Now we prove Conclusion (a). Firstly, we prove “if.” If", + "type": "text" + }, + { + "bbox": [ + 363, + 215, + 487, + 227 + ], + "score": 0.91, + "content": "0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 213, + 506, + 229 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 224, + 200, + 239 + ], + "spans": [ + { + "bbox": [ + 107, + 226, + 160, + 237 + ], + "score": 0.91, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 224, + 200, + 239 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 241, + 455, + 257 + ], + "lines": [ + { + "bbox": [ + 156, + 241, + 455, + 257 + ], + "spans": [ + { + "bbox": [ + 156, + 241, + 455, + 257 + ], + "score": 0.86, + "content": "\\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \\big \\{ ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\big | \\delta \\in \\mathbb { Z } , ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq \\delta \\leq D - 1 \\big \\} \\neq \\emptyset .", + "type": "interline_equation", + "image_path": "725c17b996fd12d4a91b289e1f1f9c49e273dc5c163cf0343b0d91035ac7946b.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 156, + 241, + 455, + 257 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 261, + 349, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 348, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 115, + 276 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 261, + 236, + 274 + ], + "score": 0.91, + "content": "0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 259, + 254, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 262, + 308, + 273 + ], + "score": 0.9, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 259, + 348, + 276 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 278, + 455, + 294 + ], + "lines": [ + { + "bbox": [ + 156, + 278, + 455, + 294 + ], + "spans": [ + { + "bbox": [ + 156, + 278, + 455, + 294 + ], + "score": 0.87, + "content": "\\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \\big \\{ ( k _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\big | \\delta \\in \\mathbb { Z } , ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq \\delta \\leq D - 1 \\big \\} \\neq \\emptyset .", + "type": "interline_equation", + "image_path": "2f5a3e62ce951c6abf961cf6e7caeec393dfaf2b6efdc0b75efe8e0d934d82bf.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 156, + 278, + 455, + 294 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 240, + 312 + ], + "score": 1.0, + "content": "Secondly, we prove “only if.” If", + "type": "text" + }, + { + "bbox": [ + 240, + 299, + 302, + 311 + ], + "score": 0.93, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 297, + 376, + 312 + ], + "score": 1.0, + "content": ", we can select an", + "type": "text" + }, + { + "bbox": [ + 376, + 299, + 432, + 311 + ], + "score": 0.94, + "content": "i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 297, + 454, + 312 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 455, + 299, + 505, + 311 + ], + "score": 0.89, + "content": "r _ { 1 } = ( k _ { 1 } -", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 107, + 310, + 137, + 322 + ], + "score": 0.9, + "content": "i ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 309, + 156, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 310, + 236, + 322 + ], + "score": 0.92, + "content": "r _ { 2 } = ( k _ { 2 } - i ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 309, + 321, + 322 + ], + "score": 1.0, + "content": ". By the definition of", + "type": "text" + }, + { + "bbox": [ + 321, + 309, + 360, + 321 + ], + "score": 0.93, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 309, + 399, + 322 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 399, + 311, + 480, + 321 + ], + "score": 0.9, + "content": "0 \\leq r _ { 1 } , r _ { 2 } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 309, + 506, + 322 + ], + "score": 1.0, + "content": ". Two", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 273, + 333 + ], + "score": 1.0, + "content": "cases should be considered here. Case 1:", + "type": "text" + }, + { + "bbox": [ + 274, + 321, + 307, + 332 + ], + "score": 0.86, + "content": "r _ { 1 } \\geq r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 320, + 337, + 333 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 338, + 321, + 468, + 332 + ], + "score": 0.92, + "content": "0 \\le r _ { 1 } - r _ { 2 } \\le D - 1 \\le N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 320, + 505, + 333 + ], + "score": 1.0, + "content": ", it holds", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 331, + 258, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 124, + 344 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 331, + 229, + 344 + ], + "score": 0.93, + "content": "r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 331, + 258, + 344 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 347, + 442, + 397 + ], + "lines": [ + { + "bbox": [ + 168, + 347, + 442, + 397 + ], + "spans": [ + { + "bbox": [ + 168, + 347, + 442, + 397 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \\mathrm { m o d } N } = \\left( ( k _ { 1 } - i ) _ { \\mathrm { m o d } N } - ( k _ { 2 } - i ) _ { \\mathrm { m o d } N } \\right) _ { \\mathrm { m o d } N } } \\\\ & { ~ = \\left( ( k _ { 1 } - i ) - ( k _ { 2 } - i ) \\right) _ { \\mathrm { m o d } N } } \\\\ & { ~ = ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } . } \\end{array}", + "type": "interline_equation", + "image_path": "ca3617944362923737e8fb9e8c72895ab921e7c71fb76c317d388bb5dfd69f77.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 168, + 347, + 442, + 363.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 168, + 363.6666666666667, + 442, + 380.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 168, + 380.33333333333337, + 442, + 397.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 400, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 162, + 414 + ], + "score": 1.0, + "content": "The equality", + "type": "text" + }, + { + "bbox": [ + 163, + 401, + 259, + 412 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbf { \\dot { \\mathbf { \\varphi } } } 0 \\leq r _ { 1 } - r _ { 2 } \\leq D - { \\mathbf { \\varphi } } 1 ^ { , } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 399, + 359, + 414 + ], + "score": 1.0, + "content": "leads to the conclusion", + "type": "text" + }, + { + "bbox": [ + 360, + 401, + 488, + 413 + ], + "score": 0.93, + "content": "0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\mathfrak { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 399, + 506, + 414 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 411, + 488, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 160, + 424 + ], + "score": 1.0, + "content": "case 2 where", + "type": "text" + }, + { + "bbox": [ + 160, + 413, + 192, + 423 + ], + "score": 0.91, + "content": "r _ { 1 } < r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 411, + 254, + 424 + ], + "score": 1.0, + "content": ", we can obtain", + "type": "text" + }, + { + "bbox": [ + 254, + 412, + 375, + 424 + ], + "score": 0.92, + "content": "0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 411, + 488, + 424 + ], + "score": 1.0, + "content": "with the similar arguments.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 485, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 485, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 485, + 442 + ], + "score": 1.0, + "content": "Conclusion (b) can be proved by the same argument with the proof of (a). Lemma 2 is proved.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 155, + 465 + ], + "score": 1.0, + "content": "Now we fix", + "type": "text" + }, + { + "bbox": [ + 155, + 452, + 178, + 464 + ], + "score": 0.92, + "content": "k _ { 1 } , l _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 452, + 292, + 465 + ], + "score": 1.0, + "content": "and consider what values of", + "type": "text" + }, + { + "bbox": [ + 293, + 453, + 315, + 464 + ], + "score": 0.92, + "content": "k _ { 2 } , l _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 452, + 335, + 465 + ], + "score": 1.0, + "content": "give", + "type": "text" + }, + { + "bbox": [ + 336, + 452, + 396, + 464 + ], + "score": 0.93, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 452, + 414, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 452, + 471, + 464 + ], + "score": 0.93, + "content": "\\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 452, + 505, + 465 + ], + "score": 1.0, + "content": ". Define", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 463, + 277, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 191, + 475 + ], + "score": 1.0, + "content": "four index sets given", + "type": "text" + }, + { + "bbox": [ + 192, + 464, + 272, + 475 + ], + "score": 0.92, + "content": "0 \\leq k _ { 1 } , l _ { 1 } \\leq N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 463, + 277, + 475 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 479, + 392, + 510 + ], + "lines": [ + { + "bbox": [ + 219, + 479, + 392, + 510 + ], + "spans": [ + { + "bbox": [ + 219, + 479, + 392, + 510 + ], + "score": 0.82, + "content": "\\begin{array} { l c l } { { } } & { { } } & { { \\displaystyle \\mathcal { K } ( k _ { 1 } ) = \\{ k | 0 \\leq ( k _ { 1 } - k ) _ { \\mathrm { m o d } N } \\leq D - 1 \\} } } \\\\ { { } } & { { } } & { { \\displaystyle \\bar { \\mathcal { K } } ( k _ { 1 } ) = \\{ k | 0 < ( k - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 \\} } } \\end{array}", + "type": "interline_equation", + "image_path": "c72902ed5871cffecbcfd321dbe10bf14477e7a4292f7acf44ecfed536128f59.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 479, + 392, + 494.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 219, + 494.5, + 392, + 510.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 513, + 387, + 527 + ], + "lines": [ + { + "bbox": [ + 223, + 513, + 387, + 527 + ], + "spans": [ + { + "bbox": [ + 223, + 513, + 387, + 527 + ], + "score": 0.5, + "content": "\\mathcal { L } ( l _ { 1 } ) = \\{ l | 0 \\leq ( l _ { 1 } - l ) _ { \\mathrm { m o d } N } \\leq D - 1 \\}", + "type": "interline_equation", + "image_path": "81f3971f568ba0e7152da654d00d547cbcfa760d752c6ce81ed2ae3436f622e8.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 223, + 513, + 387, + 527 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 529, + 386, + 543 + ], + "lines": [ + { + "bbox": [ + 224, + 529, + 386, + 543 + ], + "spans": [ + { + "bbox": [ + 224, + 529, + 386, + 543 + ], + "score": 0.38, + "content": "\\bar { \\mathcal { L } } ( l _ { 1 } ) = \\{ l | 0 < ( l - l _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 \\}", + "type": "interline_equation", + "image_path": "58a4d217877b7a695f662731f6db1cf9935c8e898601de1f57d41d8682872e04.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 224, + 529, + 386, + 543 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 259, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 259, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 163, + 558 + ], + "score": 1.0, + "content": "Lemma 3. If", + "type": "text" + }, + { + "bbox": [ + 163, + 545, + 216, + 556 + ], + "score": 0.89, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 543, + 259, + 558 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 392, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 391, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 196, + 579 + ], + "score": 1.0, + "content": "(a) The cardinality of", + "type": "text" + }, + { + "bbox": [ + 196, + 564, + 391, + 578 + ], + "score": 0.74, + "content": "\\begin{array} { r } { \\mathcal { K } ( k _ { 1 } ) , \\bar { \\mathcal { K } } ( k _ { 1 } ) \\colon | \\mathcal { K } ( k _ { 1 } ) | = D , | \\bar { \\mathcal { K } } ( k _ { 1 } ) | = D - 1 . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 209, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 210, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 123, + 599 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 123, + 583, + 207, + 597 + ], + "score": 0.89, + "content": "\\mathcal { K } ( k _ { 1 } ) \\cap \\bar { \\mathcal { K } } ( k _ { 1 } ) = \\emptyset .", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 583, + 210, + 599 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 380, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 379, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 197, + 617 + ], + "score": 1.0, + "content": "(c) The cardinality of", + "type": "text" + }, + { + "bbox": [ + 197, + 602, + 379, + 616 + ], + "score": 0.83, + "content": "\\mathcal { L } ( l _ { 1 } ) , \\bar { \\mathcal { L } } ( l _ { 1 } ) \\colon | \\mathcal { L } ( l _ { 1 } ) | = D , | \\bar { \\mathcal { L } } ( l _ { 1 } ) | = D - 1 .", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 203, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 200, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 123, + 636 + ], + "score": 1.0, + "content": "(d)", + "type": "text" + }, + { + "bbox": [ + 123, + 621, + 200, + 634 + ], + "score": 0.89, + "content": "\\mathcal { L } ( l _ { 1 } ) \\cap \\bar { \\mathcal { L } } ( l _ { 1 } ) = \\emptyset", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 441, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 442, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 293, + 660 + ], + "score": 1.0, + "content": "Proof. Now we prove Conclusion (a). The set", + "type": "text" + }, + { + "bbox": [ + 293, + 646, + 319, + 658 + ], + "score": 0.92, + "content": "\\kappa ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 644, + 442, + 660 + ], + "score": 1.0, + "content": "can be equivalently written as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 662, + 407, + 676 + ], + "lines": [ + { + "bbox": [ + 204, + 662, + 407, + 676 + ], + "spans": [ + { + "bbox": [ + 204, + 662, + 407, + 676 + ], + "score": 0.87, + "content": "\\mathcal { K } ( k _ { 1 } ) = \\{ ( k _ { 1 } - r _ { k } ) _ { \\mathrm { m o d } N } | r _ { k } = 0 , 1 , \\cdots , D - 1 \\}", + "type": "interline_equation", + "image_path": "6d6ea5e3fa7ac38d1f3cc7982a792cd70490824ae24093f8d455dcf814977df7.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 204, + 662, + 407, + 676 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 680, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 104, + 679, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 122, + 695 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 681, + 221, + 693 + ], + "score": 0.93, + "content": "k ( r _ { k } ) = ( k _ { 1 } - r _ { k } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 679, + 312, + 695 + ], + "score": 1.0, + "content": ". We want to show that", + "type": "text" + }, + { + "bbox": [ + 313, + 680, + 372, + 694 + ], + "score": 0.93, + "content": "k ( r _ { k } ^ { 1 } ) \\neq k ( r _ { k } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 679, + 415, + 695 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 415, + 680, + 448, + 694 + ], + "score": 0.92, + "content": "r _ { k } ^ { 1 } \\neq r _ { k } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 679, + 505, + 695 + ], + "score": 1.0, + "content": ". Without loss", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 691, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 207, + 706 + ], + "score": 1.0, + "content": "of generality, we assume", + "type": "text" + }, + { + "bbox": [ + 208, + 691, + 298, + 704 + ], + "score": 0.92, + "content": "0 \\leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 691, + 506, + 706 + ], + "score": 1.0, + "content": ". 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Given", + "type": "text" + }, + { + "bbox": [ + 181, + 130, + 235, + 141 + ], + "score": 0.9, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 128, + 291, + 142 + ], + "score": 1.0, + "content": ", it holds that:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 128, + 291, + 142 + ] + }, + { + "type": "list", + "bbox": [ + 105, + 149, + 506, + 203 + ], + "lines": [ + { + "bbox": [ + 104, + 148, + 501, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 123, + 163 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 123, + 150, + 183, + 162 + ], + "score": 0.92, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 148, + 242, + 163 + ], + "score": 1.0, + "content": "if and only if “", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 501, + 163 + ], + "score": 0.76, + "content": "0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime \\prime } o r ^ { \\ast } 0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 120, + 160, + 151, + 174 + ], + "spans": [ + { + "bbox": [ + 120, + 160, + 151, + 174 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 178, + 500, + 193 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 123, + 193 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 123, + 180, + 181, + 192 + ], + "score": 0.91, + "content": "\\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 178, + 241, + 193 + ], + "score": 1.0, + "content": "if and only if “", + "type": "text" + }, + { + "bbox": [ + 242, + 179, + 500, + 192 + ], + "score": 0.62, + "content": "0 \\leq ( l _ { 1 } - l _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime \\prime } o r ^ { \\textit { \\infty } } 0 < ( l _ { 2 } - l _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 121, + 190, + 151, + 204 + ], + "spans": [ + { + "bbox": [ + 121, + 190, + 151, + 204 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 6, + "is_list_end_line": true + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 148, + 501, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 214, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 363, + 229 + ], + "score": 1.0, + "content": "Proof. Now we prove Conclusion (a). Firstly, we prove “if.” If", + "type": "text" + }, + { + "bbox": [ + 363, + 215, + 487, + 227 + ], + "score": 0.91, + "content": "0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 213, + 506, + 229 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 224, + 200, + 239 + ], + "spans": [ + { + "bbox": [ + 107, + 226, + 160, + 237 + ], + "score": 0.91, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 224, + 200, + 239 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 213, + 506, + 239 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 241, + 455, + 257 + ], + "lines": [ + { + "bbox": [ + 156, + 241, + 455, + 257 + ], + "spans": [ + { + "bbox": [ + 156, + 241, + 455, + 257 + ], + "score": 0.86, + "content": "\\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \\big \\{ ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\big | \\delta \\in \\mathbb { Z } , ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq \\delta \\leq D - 1 \\big \\} \\neq \\emptyset .", + "type": "interline_equation", + "image_path": "725c17b996fd12d4a91b289e1f1f9c49e273dc5c163cf0343b0d91035ac7946b.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 156, + 241, + 455, + 257 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 261, + 349, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 348, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 115, + 276 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 261, + 236, + 274 + ], + "score": 0.91, + "content": "0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 259, + 254, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 262, + 308, + 273 + ], + "score": 0.9, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 259, + 348, + 276 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 259, + 348, + 276 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 278, + 455, + 294 + ], + "lines": [ + { + "bbox": [ + 156, + 278, + 455, + 294 + ], + "spans": [ + { + "bbox": [ + 156, + 278, + 455, + 294 + ], + "score": 0.87, + "content": "\\mathcal { Z } ( k _ { 1 } , k _ { 2 } ) = \\big \\{ ( k _ { 2 } - \\delta ) _ { \\mathrm { m o d } N } \\big | \\delta \\in \\mathbb { Z } , ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq \\delta \\leq D - 1 \\big \\} \\neq \\emptyset .", + "type": "interline_equation", + "image_path": "2f5a3e62ce951c6abf961cf6e7caeec393dfaf2b6efdc0b75efe8e0d934d82bf.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 156, + 278, + 455, + 294 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 240, + 312 + ], + "score": 1.0, + "content": "Secondly, we prove “only if.” If", + "type": "text" + }, + { + "bbox": [ + 240, + 299, + 302, + 311 + ], + "score": 0.93, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 297, + 376, + 312 + ], + "score": 1.0, + "content": ", we can select an", + "type": "text" + }, + { + "bbox": [ + 376, + 299, + 432, + 311 + ], + "score": 0.94, + "content": "i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 297, + 454, + 312 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 455, + 299, + 505, + 311 + ], + "score": 0.89, + "content": "r _ { 1 } = ( k _ { 1 } -", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 107, + 310, + 137, + 322 + ], + "score": 0.9, + "content": "i ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 309, + 156, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 310, + 236, + 322 + ], + "score": 0.92, + "content": "r _ { 2 } = ( k _ { 2 } - i ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 309, + 321, + 322 + ], + "score": 1.0, + "content": ". By the definition of", + "type": "text" + }, + { + "bbox": [ + 321, + 309, + 360, + 321 + ], + "score": 0.93, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 309, + 399, + 322 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 399, + 311, + 480, + 321 + ], + "score": 0.9, + "content": "0 \\leq r _ { 1 } , r _ { 2 } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 309, + 506, + 322 + ], + "score": 1.0, + "content": ". Two", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 273, + 333 + ], + "score": 1.0, + "content": "cases should be considered here. Case 1:", + "type": "text" + }, + { + "bbox": [ + 274, + 321, + 307, + 332 + ], + "score": 0.86, + "content": "r _ { 1 } \\geq r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 320, + 337, + 333 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 338, + 321, + 468, + 332 + ], + "score": 0.92, + "content": "0 \\le r _ { 1 } - r _ { 2 } \\le D - 1 \\le N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 320, + 505, + 333 + ], + "score": 1.0, + "content": ", it holds", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 331, + 258, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 124, + 344 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 331, + 229, + 344 + ], + "score": 0.93, + "content": "r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 331, + 258, + 344 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 297, + 506, + 344 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 347, + 442, + 397 + ], + "lines": [ + { + "bbox": [ + 168, + 347, + 442, + 397 + ], + "spans": [ + { + "bbox": [ + 168, + 347, + 442, + 397 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { r _ { 1 } - r _ { 2 } = ( r _ { 1 } - r _ { 2 } ) _ { \\mathrm { m o d } N } = \\left( ( k _ { 1 } - i ) _ { \\mathrm { m o d } N } - ( k _ { 2 } - i ) _ { \\mathrm { m o d } N } \\right) _ { \\mathrm { m o d } N } } \\\\ & { ~ = \\left( ( k _ { 1 } - i ) - ( k _ { 2 } - i ) \\right) _ { \\mathrm { m o d } N } } \\\\ & { ~ = ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } . } \\end{array}", + "type": "interline_equation", + "image_path": "ca3617944362923737e8fb9e8c72895ab921e7c71fb76c317d388bb5dfd69f77.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 168, + 347, + 442, + 363.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 168, + 363.6666666666667, + 442, + 380.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 168, + 380.33333333333337, + 442, + 397.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 400, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 162, + 414 + ], + "score": 1.0, + "content": "The equality", + "type": "text" + }, + { + "bbox": [ + 163, + 401, + 259, + 412 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbf { \\dot { \\mathbf { \\varphi } } } 0 \\leq r _ { 1 } - r _ { 2 } \\leq D - { \\mathbf { \\varphi } } 1 ^ { , } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 399, + 359, + 414 + ], + "score": 1.0, + "content": "leads to the conclusion", + "type": "text" + }, + { + "bbox": [ + 360, + 401, + 488, + 413 + ], + "score": 0.93, + "content": "0 \\leq ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N } \\leq D - 1 ^ { \\mathfrak { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 399, + 506, + 414 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 411, + 488, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 160, + 424 + ], + "score": 1.0, + "content": "case 2 where", + "type": "text" + }, + { + "bbox": [ + 160, + 413, + 192, + 423 + ], + "score": 0.91, + "content": "r _ { 1 } < r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 411, + 254, + 424 + ], + "score": 1.0, + "content": ", we can obtain", + "type": "text" + }, + { + "bbox": [ + 254, + 412, + 375, + 424 + ], + "score": 0.92, + "content": "0 < ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 411, + 488, + 424 + ], + "score": 1.0, + "content": "with the similar arguments.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 399, + 506, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 485, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 485, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 485, + 442 + ], + "score": 1.0, + "content": "Conclusion (b) can be proved by the same argument with the proof of (a). Lemma 2 is proved.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 427, + 485, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 155, + 465 + ], + "score": 1.0, + "content": "Now we fix", + "type": "text" + }, + { + "bbox": [ + 155, + 452, + 178, + 464 + ], + "score": 0.92, + "content": "k _ { 1 } , l _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 452, + 292, + 465 + ], + "score": 1.0, + "content": "and consider what values of", + "type": "text" + }, + { + "bbox": [ + 293, + 453, + 315, + 464 + ], + "score": 0.92, + "content": "k _ { 2 } , l _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 452, + 335, + 465 + ], + "score": 1.0, + "content": "give", + "type": "text" + }, + { + "bbox": [ + 336, + 452, + 396, + 464 + ], + "score": 0.93, + "content": "\\mathcal { T } ( k _ { 1 } , k _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 452, + 414, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 452, + 471, + 464 + ], + "score": 0.93, + "content": "\\mathcal { I } ( l _ { 1 } , l _ { 2 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 452, + 505, + 465 + ], + "score": 1.0, + "content": ". Define", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 463, + 277, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 191, + 475 + ], + "score": 1.0, + "content": "four index sets given", + "type": "text" + }, + { + "bbox": [ + 192, + 464, + 272, + 475 + ], + "score": 0.92, + "content": "0 \\leq k _ { 1 } , l _ { 1 } \\leq N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 463, + 277, + 475 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 452, + 505, + 475 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 479, + 392, + 510 + ], + "lines": [ + { + "bbox": [ + 219, + 479, + 392, + 510 + ], + "spans": [ + { + "bbox": [ + 219, + 479, + 392, + 510 + ], + "score": 0.82, + "content": "\\begin{array} { l c l } { { } } & { { } } & { { \\displaystyle \\mathcal { K } ( k _ { 1 } ) = \\{ k | 0 \\leq ( k _ { 1 } - k ) _ { \\mathrm { m o d } N } \\leq D - 1 \\} } } \\\\ { { } } & { { } } & { { \\displaystyle \\bar { \\mathcal { K } } ( k _ { 1 } ) = \\{ k | 0 < ( k - k _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 \\} } } \\end{array}", + "type": "interline_equation", + "image_path": "c72902ed5871cffecbcfd321dbe10bf14477e7a4292f7acf44ecfed536128f59.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 479, + 392, + 494.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 219, + 494.5, + 392, + 510.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 513, + 387, + 527 + ], + "lines": [ + { + "bbox": [ + 223, + 513, + 387, + 527 + ], + "spans": [ + { + "bbox": [ + 223, + 513, + 387, + 527 + ], + "score": 0.5, + "content": "\\mathcal { L } ( l _ { 1 } ) = \\{ l | 0 \\leq ( l _ { 1 } - l ) _ { \\mathrm { m o d } N } \\leq D - 1 \\}", + "type": "interline_equation", + "image_path": "81f3971f568ba0e7152da654d00d547cbcfa760d752c6ce81ed2ae3436f622e8.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 223, + 513, + 387, + 527 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 529, + 386, + 543 + ], + "lines": [ + { + "bbox": [ + 224, + 529, + 386, + 543 + ], + "spans": [ + { + "bbox": [ + 224, + 529, + 386, + 543 + ], + "score": 0.38, + "content": "\\bar { \\mathcal { L } } ( l _ { 1 } ) = \\{ l | 0 < ( l - l _ { 1 } ) _ { \\mathrm { m o d } N } \\leq D - 1 \\}", + "type": "interline_equation", + "image_path": "58a4d217877b7a695f662731f6db1cf9935c8e898601de1f57d41d8682872e04.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 224, + 529, + 386, + 543 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 259, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 259, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 163, + 558 + ], + "score": 1.0, + "content": "Lemma 3. If", + "type": "text" + }, + { + "bbox": [ + 163, + 545, + 216, + 556 + ], + "score": 0.89, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 543, + 259, + 558 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 543, + 259, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 392, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 391, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 196, + 579 + ], + "score": 1.0, + "content": "(a) The cardinality of", + "type": "text" + }, + { + "bbox": [ + 196, + 564, + 391, + 578 + ], + "score": 0.74, + "content": "\\begin{array} { r } { \\mathcal { K } ( k _ { 1 } ) , \\bar { \\mathcal { K } } ( k _ { 1 } ) \\colon | \\mathcal { K } ( k _ { 1 } ) | = D , | \\bar { \\mathcal { K } } ( k _ { 1 } ) | = D - 1 . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 564, + 391, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 209, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 210, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 123, + 599 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 123, + 583, + 207, + 597 + ], + "score": 0.89, + "content": "\\mathcal { K } ( k _ { 1 } ) \\cap \\bar { \\mathcal { K } } ( k _ { 1 } ) = \\emptyset .", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 583, + 210, + 599 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 583, + 210, + 599 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 380, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 379, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 197, + 617 + ], + "score": 1.0, + "content": "(c) The cardinality of", + "type": "text" + }, + { + "bbox": [ + 197, + 602, + 379, + 616 + ], + "score": 0.83, + "content": "\\mathcal { L } ( l _ { 1 } ) , \\bar { \\mathcal { L } } ( l _ { 1 } ) \\colon | \\mathcal { L } ( l _ { 1 } ) | = D , | \\bar { \\mathcal { L } } ( l _ { 1 } ) | = D - 1 .", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 601, + 379, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 203, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 200, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 123, + 636 + ], + "score": 1.0, + "content": "(d)", + "type": "text" + }, + { + "bbox": [ + 123, + 621, + 200, + 634 + ], + "score": 0.89, + "content": "\\mathcal { L } ( l _ { 1 } ) \\cap \\bar { \\mathcal { L } } ( l _ { 1 } ) = \\emptyset", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 620, + 200, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 441, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 442, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 293, + 660 + ], + "score": 1.0, + "content": "Proof. Now we prove Conclusion (a). The set", + "type": "text" + }, + { + "bbox": [ + 293, + 646, + 319, + 658 + ], + "score": 0.92, + "content": "\\kappa ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 644, + 442, + 660 + ], + "score": 1.0, + "content": "can be equivalently written as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 644, + 442, + 660 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 662, + 407, + 676 + ], + "lines": [ + { + "bbox": [ + 204, + 662, + 407, + 676 + ], + "spans": [ + { + "bbox": [ + 204, + 662, + 407, + 676 + ], + "score": 0.87, + "content": "\\mathcal { K } ( k _ { 1 } ) = \\{ ( k _ { 1 } - r _ { k } ) _ { \\mathrm { m o d } N } | r _ { k } = 0 , 1 , \\cdots , D - 1 \\}", + "type": "interline_equation", + "image_path": "6d6ea5e3fa7ac38d1f3cc7982a792cd70490824ae24093f8d455dcf814977df7.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 204, + 662, + 407, + 676 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 680, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 104, + 679, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 122, + 695 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 681, + 221, + 693 + ], + "score": 0.93, + "content": "k ( r _ { k } ) = ( k _ { 1 } - r _ { k } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 679, + 312, + 695 + ], + "score": 1.0, + "content": ". We want to show that", + "type": "text" + }, + { + "bbox": [ + 313, + 680, + 372, + 694 + ], + "score": 0.93, + "content": "k ( r _ { k } ^ { 1 } ) \\neq k ( r _ { k } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 679, + 415, + 695 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + }, + { + "bbox": [ + 415, + 680, + 448, + 694 + ], + "score": 0.92, + "content": "r _ { k } ^ { 1 } \\neq r _ { k } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 679, + 505, + 695 + ], + "score": 1.0, + "content": ". Without loss", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 691, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 207, + 706 + ], + "score": 1.0, + "content": "of generality, we assume", + "type": "text" + }, + { + "bbox": [ + 208, + 691, + 298, + 704 + ], + "score": 0.92, + "content": "0 \\leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 691, + 506, + 706 + ], + "score": 1.0, + "content": ". By the definition of modulo operation, There exist", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 703, + 216, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 158, + 716 + ], + "score": 1.0, + "content": "two integers", + "type": "text" + }, + { + "bbox": [ + 158, + 703, + 176, + 715 + ], + "score": 0.91, + "content": "q , q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 703, + 216, + 716 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 679, + 506, + 716 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 718, + 410, + 733 + ], + "lines": [ + { + "bbox": [ + 200, + 718, + 410, + 733 + ], + "spans": [ + { + "bbox": [ + 200, + 718, + 410, + 733 + ], + "score": 0.9, + "content": "k ( r _ { k } ^ { 1 } ) = q N + k _ { 1 } - r _ { k } ^ { 1 } , \\quad k ( r _ { k } ^ { 2 } ) = q ^ { \\prime } N + k _ { 1 } - r _ { k } ^ { 2 } .", + "type": "interline_equation", + "image_path": "886496ab9907ee51c6b95ef828e89ce3441d725612365c70494604f39d272243.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 200, + 718, + 410, + 733 + ], + "spans": [], + "index": 38 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 103, + 78, + 505, + 100 + ], + "spans": [ + { + "bbox": [ + 103, + 78, + 143, + 100 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 144, + 82, + 205, + 95 + ], + "score": 0.93, + "content": "k ( r _ { k } ^ { 1 } ) = k ( r _ { k } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 78, + 482, + 100 + ], + "score": 1.0, + "content": ". Taking the difference between the above two equations, we obtain", + "type": "text" + }, + { + "bbox": [ + 483, + 82, + 505, + 94 + ], + "score": 0.83, + "content": "r _ { k } ^ { 2 } \\mathrm { ~ - ~ }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 173, + 105 + ], + "score": 0.86, + "content": "r _ { k } ^ { 1 } = ( q ^ { \\prime } - \\overset { \\cdot } { q } ) N", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 92, + 192, + 109 + ], + "score": 1.0, + "content": ", i.e,", + "type": "text" + }, + { + "bbox": [ + 192, + 95, + 203, + 104 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 92, + 235, + 109 + ], + "score": 1.0, + "content": "divides", + "type": "text" + }, + { + "bbox": [ + 235, + 93, + 268, + 106 + ], + "score": 0.93, + "content": "\\overline { { r _ { k } ^ { 2 } } } - r _ { k } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 92, + 313, + 109 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + }, + { + "bbox": [ + 314, + 93, + 407, + 105 + ], + "score": 0.91, + "content": "0 \\leq r _ { k } ^ { 1 } < r _ { k } ^ { 2 } \\leq D - \\mathrm { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 92, + 441, + 109 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 441, + 93, + 505, + 105 + ], + "score": 0.84, + "content": "1 \\leq r _ { k } ^ { 2 } - r _ { k } ^ { 1 } \\leq", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 99, + 508, + 121 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 174, + 115 + ], + "score": 0.87, + "content": "D - 1 \\leq N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 99, + 275, + 121 + ], + "score": 1.0, + "content": "k k, which contradicts with", + "type": "text" + }, + { + "bbox": [ + 275, + 105, + 287, + 114 + ], + "score": 0.68, + "content": "^ { \\circ } N", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 99, + 324, + 121 + ], + "score": 1.0, + "content": "dividing", + "type": "text" + }, + { + "bbox": [ + 324, + 105, + 356, + 117 + ], + "score": 0.9, + "content": "r _ { k } ^ { 2 } - r _ { k } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 99, + 441, + 121 + ], + "score": 1.0, + "content": "k .” Thus, it holds that", + "type": "text" + }, + { + "bbox": [ + 441, + 105, + 501, + 117 + ], + "score": 0.9, + "content": "k ( r _ { k } ^ { 1 } ) \\stackrel { \\sim } { = } k ( \\stackrel { \\sim } { r } _ { k } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 99, + 508, + 121 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 224, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 165, + 128 + ], + "score": 1.0, + "content": "Then we have", + "type": "text" + }, + { + "bbox": [ + 165, + 115, + 219, + 127 + ], + "score": 0.93, + "content": "| \\kappa ( k _ { 1 } ) | = D", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 114, + 224, + 128 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 210, + 144 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 211, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 211, + 145 + ], + "score": 1.0, + "content": "In the same way, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 147, + 407, + 161 + ], + "lines": [ + { + "bbox": [ + 204, + 147, + 407, + 161 + ], + "spans": [ + { + "bbox": [ + 204, + 147, + 407, + 161 + ], + "score": 0.88, + "content": "\\bar { \\mathcal { K } } ( k _ { 1 } ) = \\{ ( k _ { 1 } + r _ { k } ) _ { \\mathrm { m o d } N } | r _ { k } = 1 , 2 , \\cdot \\cdot \\cdot , D - 1 \\}", + "type": "interline_equation", + "image_path": "72ca797a8a77a6ae0def61c6ca53cc7674bce437925b8a81131f08fed5d10454.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 204, + 147, + 407, + 161 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 163, + 303, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 303, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 123, + 177 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 163, + 196, + 176 + ], + "score": 0.93, + "content": "| \\bar { \\kappa } ( k _ { 1 } ) | = D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 163, + 303, + 177 + ], + "score": 1.0, + "content": ". Conclusion (a) is proved.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 273, + 193 + ], + "score": 1.0, + "content": "Now we prove Conclusion (b). Suppose", + "type": "text" + }, + { + "bbox": [ + 273, + 180, + 361, + 193 + ], + "score": 0.93, + "content": "{ \\mathcal { K } } ( k _ { 1 } ) \\cap { \\bar { \\mathcal { K } } } ( k _ { 1 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 180, + 395, + 193 + ], + "score": 1.0, + "content": ". Pick a", + "type": "text" + }, + { + "bbox": [ + 395, + 180, + 483, + 193 + ], + "score": 0.92, + "content": "k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } ) \\cap \\bar { \\mathcal { K } } ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 180, + 506, + 193 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 190, + 204 + ], + "score": 0.92, + "content": "r _ { 3 } = ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 191, + 207, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 192, + 291, + 204 + ], + "score": 0.92, + "content": "r _ { 4 } = ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 191, + 353, + 205 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + }, + { + "bbox": [ + 353, + 192, + 418, + 203 + ], + "score": 0.91, + "content": "0 \\leq r _ { 3 } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 191, + 436, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 193, + 501, + 203 + ], + "score": 0.9, + "content": "0 < r _ { 4 } \\le D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 191, + 505, + 205 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 419, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 360, + 216 + ], + "score": 1.0, + "content": "By the definition of modulo operation, There exist two integers", + "type": "text" + }, + { + "bbox": [ + 361, + 204, + 378, + 215 + ], + "score": 0.9, + "content": "q , q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 202, + 419, + 216 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 217, + 396, + 231 + ], + "lines": [ + { + "bbox": [ + 215, + 217, + 396, + 231 + ], + "spans": [ + { + "bbox": [ + 215, + 217, + 396, + 231 + ], + "score": 0.92, + "content": "k _ { 1 } - k _ { 2 } = q N + r _ { 3 } , \\quad k _ { 2 } - k _ { 1 } = q ^ { \\prime } N + r _ { 4 }", + "type": "interline_equation", + "image_path": "5919f9aca37c17a9e9eab41af79e40f681389b4bd3a82e6886da4282d208e8e4.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 215, + 217, + 396, + 231 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 158, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 159, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 159, + 250 + ], + "score": 1.0, + "content": "which imply", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 244, + 358, + 258 + ], + "lines": [ + { + "bbox": [ + 252, + 244, + 358, + 258 + ], + "spans": [ + { + "bbox": [ + 252, + 244, + 358, + 258 + ], + "score": 0.9, + "content": "r _ { 3 } + r _ { 4 } + ( q + q ^ { \\prime } ) N = 0 .", + "type": "interline_equation", + "image_path": "8ab8cb05aaba5299179504d6a3ba7ff9265e3768b5a9e510e585560028c1e5aa.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 252, + 244, + 358, + 258 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 259, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 146, + 272 + ], + "score": 1.0, + "content": "However,", + "type": "text" + }, + { + "bbox": [ + 147, + 260, + 235, + 271 + ], + "score": 0.92, + "content": "0 < r _ { 3 } + r _ { 4 } \\le 2 D - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 257, + 305, + 272 + ], + "score": 1.0, + "content": "contradicts with", + "type": "text" + }, + { + "bbox": [ + 305, + 259, + 449, + 271 + ], + "score": 0.91, + "content": "\\dot { \\boldsymbol { q } } \\in \\mathbb { Z } , \\boldsymbol { q } ^ { \\prime } \\in \\mathbb { Z } , N \\in \\mathbb { Z } , N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 257, + 505, + 272 + ], + "score": 1.0, + "content": ".” Conclusion", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 162, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 162, + 283 + ], + "score": 1.0, + "content": "(b) is proved.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "score": 1.0, + "content": "Conclusions (c) and (d) are actually the same with Conclusions (a) and (b) respectively. Thus, it", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 150, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 150, + 310 + ], + "score": 1.0, + "content": "holds that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 312, + 401, + 343 + ], + "lines": [ + { + "bbox": [ + 209, + 312, + 401, + 343 + ], + "spans": [ + { + "bbox": [ + 209, + 312, + 401, + 343 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathcal { L } ( l _ { 1 } ) = \\{ ( l _ { 1 } - r _ { l } ) _ { \\mathrm { m o d } N } | r _ { l } = 0 , 1 , \\cdots , D - 1 \\} } \\\\ & { \\bar { \\mathcal { L } } ( l _ { 1 } ) = \\{ ( l _ { 1 } + r _ { l } ) _ { \\mathrm { m o d } N } | r _ { l } = 1 , 2 , \\cdots , D - 1 \\} } \\end{array}", + "type": "interline_equation", + "image_path": "aa6be485f5d65c78ef5004c13ca4b07409ac8080df3bb351b13a27c498613be5.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 312, + 401, + 327.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 209, + 327.5, + 401, + 343.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 334, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 334, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 123, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 346, + 249, + 359 + ], + "score": 0.93, + "content": "| \\mathcal { L } ( l _ { 1 } ) | = D , | \\bar { \\mathcal { L } } ( l _ { 1 } ) | = D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 345, + 334, + 360 + ], + "score": 1.0, + "content": ". Lemma 3 is proved.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 385, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 385, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 385, + 383 + ], + "score": 1.0, + "content": "With the preparations, we can prove Conclusion 1 of Theorem 3 now.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 237, + 407 + ], + "score": 1.0, + "content": "Proof of Theorem 3, Conclusion", + "type": "text" + }, + { + "bbox": [ + 237, + 395, + 243, + 404 + ], + "score": 0.45, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 393, + 304, + 407 + ], + "score": 1.0, + "content": ". Firstly we fix", + "type": "text" + }, + { + "bbox": [ + 304, + 394, + 399, + 406 + ], + "score": 0.91, + "content": "k _ { 1 } \\in \\{ 0 , 1 , \\cdots , N - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 393, + 453, + 407 + ], + "score": 1.0, + "content": "and consider", + "type": "text" + }, + { + "bbox": [ + 453, + 394, + 501, + 406 + ], + "score": 0.92, + "content": "k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 393, + 505, + 407 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 122, + 419 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 405, + 210, + 417 + ], + "score": 0.91, + "content": "r _ { k } \\stackrel { \\cdot } { = } ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 403, + 381, + 419 + ], + "score": 1.0, + "content": ". Then equation (37) implies that, for any", + "type": "text" + }, + { + "bbox": [ + 382, + 405, + 437, + 417 + ], + "score": 0.94, + "content": "i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 403, + 497, + 419 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + }, + { + "bbox": [ + 498, + 406, + 504, + 415 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 108, + 415, + 221, + 428 + ], + "spans": [ + { + "bbox": [ + 108, + 417, + 179, + 428 + ], + "score": 0.9, + "content": "( r _ { k } \\le \\delta \\le D - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 415, + 221, + 428 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 429, + 439, + 464 + ], + "lines": [ + { + "bbox": [ + 171, + 429, + 439, + 464 + ], + "spans": [ + { + "bbox": [ + 171, + 429, + 439, + 464 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { I ( i , k _ { 1 } ) = \\bigr ( k _ { 1 } - ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta , } \\\\ & { I ( i , k _ { 2 } ) = \\bigr ( k _ { 2 } - ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { k } ) _ { \\mathrm { m o d } N } = \\delta - r _ { k } . } \\end{array}", + "type": "interline_equation", + "image_path": "d4398c39472066166050e7744f8bcde3f7ef3bb9d313ea89fcaf36b2df577446.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 171, + 429, + 439, + 440.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 171, + 440.6666666666667, + 439, + 452.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 171, + 452.33333333333337, + 439, + 464.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 466, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 248, + 480 + ], + "score": 1.0, + "content": "Now we consider another case for", + "type": "text" + }, + { + "bbox": [ + 248, + 467, + 259, + 478 + ], + "score": 0.84, + "content": "k _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 465, + 264, + 480 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 265, + 466, + 315, + 478 + ], + "score": 0.87, + "content": "k _ { 2 } \\in \\bar { \\mathcal { K } } ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 465, + 319, + 480 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 320, + 466, + 408, + 479 + ], + "score": 0.9, + "content": "r _ { k } = ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 465, + 505, + 480 + ], + "score": 1.0, + "content": ". 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For any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 552, + 339, + 565 + ], + "spans": [ + { + "bbox": [ + 107, + 553, + 160, + 565 + ], + "score": 0.93, + "content": "j \\in \\mathcal { I } ( l _ { 1 } , l _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 552, + 218, + 565 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + }, + { + "bbox": [ + 219, + 554, + 225, + 563 + ], + "score": 0.57, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 553, + 297, + 564 + ], + "score": 0.79, + "content": "( r _ { l } \\le \\delta \\le D - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 552, + 339, + 565 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 567, + 435, + 601 + ], + "lines": [ + { + "bbox": [ + 176, + 567, + 435, + 601 + ], + "spans": [ + { + "bbox": [ + 176, + 567, + 435, + 601 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { I ( j , l _ { 1 } ) = \\bigr ( l _ { 1 } - ( l _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta , } \\\\ & { I ( j , l _ { 2 } ) = \\bigr ( l _ { 2 } - ( l _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { l } ) _ { \\mathrm { m o d } N } = \\delta - r _ { l } . } \\end{array}", + "type": "interline_equation", + "image_path": "feb573099fd8477293e71c33fad7006cbf042a24fff449d4e806e64fcb190979.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 176, + 567, + 435, + 578.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 176, + 578.3333333333334, + 435, + 589.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 176, + 589.6666666666667, + 435, + 601.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 178, + 618 + ], + "score": 1.0, + "content": "Another case for", + "type": "text" + }, + { + "bbox": [ + 179, + 604, + 187, + 615 + ], + "score": 0.69, + "content": "l _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 602, + 194, + 618 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 195, + 603, + 242, + 616 + ], + "score": 0.79, + "content": "l _ { 2 } ~ \\in ~ \\bar { \\mathcal { L } } ( l _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 602, + 247, + 618 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 248, + 604, + 333, + 616 + ], + "score": 0.88, + "content": "r _ { l } = ( l _ { 2 } - l _ { 1 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 602, + 376, + 618 + ], + "score": 1.0, + "content": ". 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Conclusion (a) is proved.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 163, + 303, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 273, + 193 + ], + "score": 1.0, + "content": "Now we prove Conclusion (b). Suppose", + "type": "text" + }, + { + "bbox": [ + 273, + 180, + 361, + 193 + ], + "score": 0.93, + "content": "{ \\mathcal { K } } ( k _ { 1 } ) \\cap { \\bar { \\mathcal { K } } } ( k _ { 1 } ) \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 180, + 395, + 193 + ], + "score": 1.0, + "content": ". Pick a", + "type": "text" + }, + { + "bbox": [ + 395, + 180, + 483, + 193 + ], + "score": 0.92, + "content": "k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } ) \\cap \\bar { \\mathcal { K } } ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 180, + 506, + 193 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 190, + 204 + ], + "score": 0.92, + "content": "r _ { 3 } = ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 191, + 207, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 192, + 291, + 204 + ], + "score": 0.92, + "content": "r _ { 4 } = ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 191, + 353, + 205 + ], + "score": 1.0, + "content": ". Then we have", + "type": "text" + }, + { + "bbox": [ + 353, + 192, + 418, + 203 + ], + "score": 0.91, + "content": "0 \\leq r _ { 3 } \\leq D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 191, + 436, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 193, + 501, + 203 + ], + "score": 0.9, + "content": "0 < r _ { 4 } \\le D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 191, + 505, + 205 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 419, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 360, + 216 + ], + "score": 1.0, + "content": "By the definition of modulo operation, There exist two integers", + "type": "text" + }, + { + "bbox": [ + 361, + 204, + 378, + 215 + ], + "score": 0.9, + "content": "q , q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 202, + 419, + 216 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 180, + 506, + 216 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 217, + 396, + 231 + ], + "lines": [ + { + "bbox": [ + 215, + 217, + 396, + 231 + ], + "spans": [ + { + "bbox": [ + 215, + 217, + 396, + 231 + ], + "score": 0.92, + "content": "k _ { 1 } - k _ { 2 } = q N + r _ { 3 } , \\quad k _ { 2 } - k _ { 1 } = q ^ { \\prime } N + r _ { 4 }", + "type": "interline_equation", + "image_path": "5919f9aca37c17a9e9eab41af79e40f681389b4bd3a82e6886da4282d208e8e4.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 215, + 217, + 396, + 231 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 158, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 159, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 159, + 250 + ], + "score": 1.0, + "content": "which imply", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 232, + 159, + 250 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 244, + 358, + 258 + ], + "lines": [ + { + "bbox": [ + 252, + 244, + 358, + 258 + ], + "spans": [ + { + "bbox": [ + 252, + 244, + 358, + 258 + ], + "score": 0.9, + "content": "r _ { 3 } + r _ { 4 } + ( q + q ^ { \\prime } ) N = 0 .", + "type": "interline_equation", + "image_path": "8ab8cb05aaba5299179504d6a3ba7ff9265e3768b5a9e510e585560028c1e5aa.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 252, + 244, + 358, + 258 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 259, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 146, + 272 + ], + "score": 1.0, + "content": "However,", + "type": "text" + }, + { + "bbox": [ + 147, + 260, + 235, + 271 + ], + "score": 0.92, + "content": "0 < r _ { 3 } + r _ { 4 } \\le 2 D - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 257, + 305, + 272 + ], + "score": 1.0, + "content": "contradicts with", + "type": "text" + }, + { + "bbox": [ + 305, + 259, + 449, + 271 + ], + "score": 0.91, + "content": "\\dot { \\boldsymbol { q } } \\in \\mathbb { Z } , \\boldsymbol { q } ^ { \\prime } \\in \\mathbb { Z } , N \\in \\mathbb { Z } , N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 257, + 505, + 272 + ], + "score": 1.0, + "content": ".” Conclusion", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 162, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 162, + 283 + ], + "score": 1.0, + "content": "(b) is proved.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 257, + 505, + 283 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "score": 1.0, + "content": "Conclusions (c) and (d) are actually the same with Conclusions (a) and (b) respectively. Thus, it", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 150, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 150, + 310 + ], + "score": 1.0, + "content": "holds that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 286, + 505, + 310 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 312, + 401, + 343 + ], + "lines": [ + { + "bbox": [ + 209, + 312, + 401, + 343 + ], + "spans": [ + { + "bbox": [ + 209, + 312, + 401, + 343 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathcal { L } ( l _ { 1 } ) = \\{ ( l _ { 1 } - r _ { l } ) _ { \\mathrm { m o d } N } | r _ { l } = 0 , 1 , \\cdots , D - 1 \\} } \\\\ & { \\bar { \\mathcal { L } } ( l _ { 1 } ) = \\{ ( l _ { 1 } + r _ { l } ) _ { \\mathrm { m o d } N } | r _ { l } = 1 , 2 , \\cdots , D - 1 \\} } \\end{array}", + "type": "interline_equation", + "image_path": "aa6be485f5d65c78ef5004c13ca4b07409ac8080df3bb351b13a27c498613be5.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 312, + 401, + 327.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 209, + 327.5, + 401, + 343.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 334, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 334, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 123, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 346, + 249, + 359 + ], + "score": 0.93, + "content": "| \\mathcal { L } ( l _ { 1 } ) | = D , | \\bar { \\mathcal { L } } ( l _ { 1 } ) | = D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 345, + 334, + 360 + ], + "score": 1.0, + "content": ". Lemma 3 is proved.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 345, + 334, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 385, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 385, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 385, + 383 + ], + "score": 1.0, + "content": "With the preparations, we can prove Conclusion 1 of Theorem 3 now.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 370, + 385, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 237, + 407 + ], + "score": 1.0, + "content": "Proof of Theorem 3, Conclusion", + "type": "text" + }, + { + "bbox": [ + 237, + 395, + 243, + 404 + ], + "score": 0.45, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 393, + 304, + 407 + ], + "score": 1.0, + "content": ". Firstly we fix", + "type": "text" + }, + { + "bbox": [ + 304, + 394, + 399, + 406 + ], + "score": 0.91, + "content": "k _ { 1 } \\in \\{ 0 , 1 , \\cdots , N - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 393, + 453, + 407 + ], + "score": 1.0, + "content": "and consider", + "type": "text" + }, + { + "bbox": [ + 453, + 394, + 501, + 406 + ], + "score": 0.92, + "content": "k _ { 2 } \\in \\mathcal { K } ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 393, + 505, + 407 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 122, + 419 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 405, + 210, + 417 + ], + "score": 0.91, + "content": "r _ { k } \\stackrel { \\cdot } { = } ( k _ { 1 } - k _ { 2 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 403, + 381, + 419 + ], + "score": 1.0, + "content": ". Then equation (37) implies that, for any", + "type": "text" + }, + { + "bbox": [ + 382, + 405, + 437, + 417 + ], + "score": 0.94, + "content": "i \\in \\mathcal { T } ( k _ { 1 } , k _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 403, + 497, + 419 + ], + "score": 1.0, + "content": ", there exists a", + "type": "text" + }, + { + "bbox": [ + 498, + 406, + 504, + 415 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 108, + 415, + 221, + 428 + ], + "spans": [ + { + "bbox": [ + 108, + 417, + 179, + 428 + ], + "score": 0.9, + "content": "( r _ { k } \\le \\delta \\le D - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 415, + 221, + 428 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 393, + 505, + 428 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 429, + 439, + 464 + ], + "lines": [ + { + "bbox": [ + 171, + 429, + 439, + 464 + ], + "spans": [ + { + "bbox": [ + 171, + 429, + 439, + 464 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { I ( i , k _ { 1 } ) = \\bigr ( k _ { 1 } - ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta ) _ { \\mathrm { m o d } N } = \\delta , } \\\\ & { I ( i , k _ { 2 } ) = \\bigr ( k _ { 2 } - ( k _ { 1 } - \\delta ) _ { \\mathrm { m o d } N } \\bigr ) _ { \\mathrm { m o d } N } = ( \\delta - r _ { k } ) _ { \\mathrm { m o d } N } = \\delta - r _ { k } . } \\end{array}", + "type": "interline_equation", + "image_path": "d4398c39472066166050e7744f8bcde3f7ef3bb9d313ea89fcaf36b2df577446.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 171, + 429, + 439, + 440.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 171, + 440.6666666666667, + 439, + 452.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 171, + 452.33333333333337, + 439, + 464.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 466, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 248, + 480 + ], + "score": 1.0, + "content": "Now we consider another case for", + "type": "text" + }, + { + "bbox": [ + 248, + 467, + 259, + 478 + ], + "score": 0.84, + "content": "k _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 465, + 264, + 480 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 265, + 466, + 315, + 478 + ], + "score": 0.87, + "content": "k _ { 2 } \\in \\bar { \\mathcal { K } } ( k _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 465, + 319, + 480 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 320, + 466, + 408, + 479 + ], + "score": 0.9, + "content": "r _ { k } = ( k _ { 2 } - k _ { 1 } ) _ { \\mathrm { m o d } N }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 465, + 505, + 480 + ], + "score": 1.0, + "content": ". 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1 } \\sum _ { r _ { l } = 0 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } , \\delta _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } - r _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "790001e0586d81602cc7cd2a390b494bc22037cbc5f249112b6e40786cc8bcd3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 173, + 245, + 438, + 257.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 257.3333333333333, + 438, + 269.66666666666663 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 173, + 269.66666666666663, + 438, + 281.99999999999994 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 295, + 300 + ], + "lines": [ + { + "bbox": [ + 107, + 288, + 296, + 300 + ], + "spans": [ + { + "bbox": [ + 107, + 288, + 296, + 300 + ], + "score": 1.0, + "content": "Combining (40), (41), (44) and (45), we obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 306, + 437, + 342 + ], + "lines": [ + { + "bbox": [ + 173, + 306, + 437, + 342 + ], + "spans": [ + { + "bbox": [ + 173, + 306, + 437, + 342 + ], + "score": 0.93, + "content": "f _ { 2 } = \\sum _ { r _ { k } = 1 } ^ { D - 1 } \\sum _ { r _ { l } = 0 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } , \\delta _ { l } - r _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "8fc6e447a2891b3a4d47c636c360b84a030a171d4b12b94efea3a75951f50c73.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 173, + 306, + 437, + 318.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 173, + 318.0, + 437, + 330.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 173, + 330.0, + 437, + 342.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 295, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 296, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 296, + 361 + ], + "score": 1.0, + "content": "Combining (39), (42), (43) and (46), we obtain", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 365, + 437, + 402 + ], + "lines": [ + { + "bbox": [ + 173, + 365, + 437, + 402 + ], + "spans": [ + { + "bbox": [ + 173, + 365, + 437, + 402 + ], + "score": 0.94, + "content": "f _ { 3 } = \\sum _ { r _ { k } = 0 } ^ { D - 1 } \\sum _ { r _ { l } = 1 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } , \\delta _ { l } - r _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "fae52290b43843a26fe744429cc44cae4ea1f79a73222d68806b9cab153ce9bf.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 173, + 365, + 437, + 377.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 173, + 377.3333333333333, + 437, + 389.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 173, + 389.66666666666663, + 437, + 401.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 295, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 296, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 296, + 421 + ], + "score": 1.0, + "content": "Combining (40), (42), (44) and (46), we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 425, + 438, + 462 + ], + "lines": [ + { + "bbox": [ + 173, + 425, + 438, + 462 + ], + "spans": [ + { + "bbox": [ + 173, + 425, + 438, + 462 + ], + "score": 0.93, + "content": "f _ { 4 } = \\sum _ { r _ { k } = 1 } ^ { D - 1 } \\sum _ { r _ { l } = 1 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } - r _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } , \\delta _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "c371ba2864143c47dc29238f9f079c304663375ed10edb60886803e159d666c2.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 173, + 425, + 438, + 437.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 173, + 437.3333333333333, + 438, + 449.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 173, + 449.66666666666663, + 438, + 461.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 250, + 487 + ], + "score": 1.0, + "content": "By the above explicit formulas of", + "type": "text" + }, + { + "bbox": [ + 251, + 475, + 314, + 486 + ], + "score": 0.9, + "content": "f _ { i } , 1 \\le i \\le 4", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 473, + 357, + 487 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 357, + 475, + 409, + 486 + ], + "score": 0.92, + "content": "f _ { 1 } , f _ { 2 } , f _ { 3 } , f _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "are all independent of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 129, + 496 + ], + "score": 0.91, + "content": "k _ { 1 } , l _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 484, + 148, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 149, + 486, + 159, + 495 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 484, + 275, + 498 + ], + "score": 1.0, + "content": ". They are only related with", + "type": "text" + }, + { + "bbox": [ + 275, + 487, + 307, + 497 + ], + "score": 0.88, + "content": "m _ { 1 } , m _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 484, + 346, + 498 + ], + "score": 1.0, + "content": "for fixed", + "type": "text" + }, + { + "bbox": [ + 346, + 486, + 354, + 496 + ], + "score": 0.65, + "content": "\\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 484, + 373, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 373, + 487, + 384, + 495 + ], + "score": 0.53, + "content": "\\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 484, + 506, + 498 + ], + "score": 1.0, + "content": ". Thus, we are able to denote", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 496, + 356, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 178, + 509 + ], + "score": 0.91, + "content": "f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 496, + 191, + 510 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 191, + 497, + 236, + 509 + ], + "score": 0.93, + "content": "f ( m _ { 1 } , m _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 496, + 356, + 510 + ], + "score": 1.0, + "content": "for simplicity. Consequently,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 514, + 488, + 660 + ], + "lines": [ + { + "bbox": [ + 121, + 514, + 488, + 660 + ], + "spans": [ + { + "bbox": [ + 121, + 514, + 488, + 660 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\displaystyle \\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c r i } } ^ { N } \\| _ { F } ^ { 2 } = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { m = 1 } ^ { M } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { N - 1 } \\sum _ { m = 1 } ^ { M } \\sum _ { m = 2 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { M } \\sum _ { \\iota _ { 1 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\cdot N ^ { 2 } \\cdot \\displaystyle \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) = \\sum _ { m _ { 1 } = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( m _ { 1 } , m _ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "caac35b84b1bd105e0982f9c3b9fae3b13d665df447b4dcf9be820bddfd4233c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 121, + 514, + 488, + 562.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 121, + 562.6666666666666, + 488, + 611.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 121, + 611.3333333333333, + 488, + 659.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 294, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 296, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 132, + 681 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + }, + { + "bbox": [ + 132, + 665, + 216, + 680 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 664, + 281, + 681 + ], + "score": 1.0, + "content": "is dependent of", + "type": "text" + }, + { + "bbox": [ + 281, + 667, + 291, + 677 + ], + "score": 0.85, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 664, + 296, + 681 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 685, + 463, + 712 + ], + "lines": [ + { + "bbox": [ + 146, + 685, + 463, + 712 + ], + "spans": [ + { + "bbox": [ + 146, + 685, + 463, + 712 + ], + "score": 0.9, + "content": "\\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } = \\frac { 1 } { ( 2 D - 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1 } \\sum _ { r _ { l } = 0 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } , \\delta _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } - r _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "790001e0586d81602cc7cd2a390b494bc22037cbc5f249112b6e40786cc8bcd3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 173, + 245, + 438, + 257.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 257.3333333333333, + 438, + 269.66666666666663 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 173, + 269.66666666666663, + 438, + 281.99999999999994 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 295, + 300 + ], + "lines": [ + { + "bbox": [ + 107, + 288, + 296, + 300 + ], + "spans": [ + { + "bbox": [ + 107, + 288, + 296, + 300 + ], + "score": 1.0, + "content": "Combining (40), (41), (44) and (45), we obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 107, + 288, + 296, + 300 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 306, + 437, + 342 + ], + "lines": [ + { + "bbox": [ + 173, + 306, + 437, + 342 + ], + "spans": [ + { + "bbox": [ + 173, + 306, + 437, + 342 + ], + "score": 0.93, + "content": "f _ { 2 } = \\sum _ { r _ { k } = 1 } ^ { D - 1 } \\sum _ { r _ { l } = 0 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } , \\delta _ { l } - r _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "8fc6e447a2891b3a4d47c636c360b84a030a171d4b12b94efea3a75951f50c73.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 173, + 306, + 437, + 318.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 173, + 318.0, + 437, + 330.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 173, + 330.0, + 437, + 342.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 295, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 296, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 296, + 361 + ], + "score": 1.0, + "content": "Combining (39), (42), (43) and (46), we obtain", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 347, + 296, + 361 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 365, + 437, + 402 + ], + "lines": [ + { + "bbox": [ + 173, + 365, + 437, + 402 + ], + "spans": [ + { + "bbox": [ + 173, + 365, + 437, + 402 + ], + "score": 0.94, + "content": "f _ { 3 } = \\sum _ { r _ { k } = 0 } ^ { D - 1 } \\sum _ { r _ { l } = 1 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } , \\delta _ { l } - r _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "fae52290b43843a26fe744429cc44cae4ea1f79a73222d68806b9cab153ce9bf.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 173, + 365, + 437, + 377.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 173, + 377.3333333333333, + 437, + 389.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 173, + 389.66666666666663, + 437, + 401.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 295, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 296, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 296, + 421 + ], + "score": 1.0, + "content": "Combining (40), (42), (44) and (46), we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 407, + 296, + 421 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 425, + 438, + 462 + ], + "lines": [ + { + "bbox": [ + 173, + 425, + 438, + 462 + ], + "spans": [ + { + "bbox": [ + 173, + 425, + 438, + 462 + ], + "score": 0.93, + "content": "f _ { 4 } = \\sum _ { r _ { k } = 1 } ^ { D - 1 } \\sum _ { r _ { l } = 1 } ^ { D - 1 } \\sum _ { \\delta _ { k } = r _ { k } } ^ { D - 1 } \\sum _ { \\delta _ { l } = r _ { l } } ^ { D - 1 } \\Big ( \\mathbf { d } _ { m _ { 1 } } ( \\delta _ { k } - r _ { k } , \\delta _ { l } - r _ { l } ) \\mathbf { w } _ { m _ { 2 } } ( \\delta _ { k } , \\delta _ { l } ) \\Big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "c371ba2864143c47dc29238f9f079c304663375ed10edb60886803e159d666c2.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 173, + 425, + 438, + 437.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 173, + 437.3333333333333, + 438, + 449.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 173, + 449.66666666666663, + 438, + 461.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 250, + 487 + ], + "score": 1.0, + "content": "By the above explicit formulas of", + "type": "text" + }, + { + "bbox": [ + 251, + 475, + 314, + 486 + ], + "score": 0.9, + "content": "f _ { i } , 1 \\le i \\le 4", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 473, + 357, + 487 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 357, + 475, + 409, + 486 + ], + "score": 0.92, + "content": "f _ { 1 } , f _ { 2 } , f _ { 3 } , f _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "are all independent of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 129, + 496 + ], + "score": 0.91, + "content": "k _ { 1 } , l _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 484, + 148, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 149, + 486, + 159, + 495 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 484, + 275, + 498 + ], + "score": 1.0, + "content": ". They are only related with", + "type": "text" + }, + { + "bbox": [ + 275, + 487, + 307, + 497 + ], + "score": 0.88, + "content": "m _ { 1 } , m _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 484, + 346, + 498 + ], + "score": 1.0, + "content": "for fixed", + "type": "text" + }, + { + "bbox": [ + 346, + 486, + 354, + 496 + ], + "score": 0.65, + "content": "\\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 484, + 373, + 498 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 373, + 487, + 384, + 495 + ], + "score": 0.53, + "content": "\\mathbf { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 484, + 506, + 498 + ], + "score": 1.0, + "content": ". 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Consequently,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 473, + 506, + 510 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 514, + 488, + 660 + ], + "lines": [ + { + "bbox": [ + 121, + 514, + 488, + 660 + ], + "spans": [ + { + "bbox": [ + 121, + 514, + 488, + 660 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\displaystyle \\frac { 1 } { N ^ { 2 } } \\| ( \\mathbf { D } _ { \\mathrm { c r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c r i } } ^ { N } \\| _ { F } ^ { 2 } = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { m = 1 } ^ { M } \\Big ( \\mathbf { B } _ { \\mathrm { c o h } } ( k _ { 1 } , l _ { 1 } , m _ { 1 } ; k _ { 2 } , l _ { 2 } , m _ { 2 } ) \\Big ) ^ { 2 } } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 1 } \\sum _ { \\iota _ { 2 } = 0 } ^ { M } \\sum _ { m = 1 } ^ { M } \\sum _ { \\iota _ { 2 } = 1 } ^ { M } f ( k _ { 1 } , l _ { 1 } , m _ { 1 } , m _ { 2 } ) } & { } \\\\ { = \\frac { 1 } { N ^ { 2 } } \\sum _ { { \\boldsymbol k } _ { 1 } = 0 } ^ { N - 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1 } = \\left[ \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\right] , \\quad \\Delta _ { \\mathbf { W } } ^ { N } = \\left[ \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot ; \\cdot ) \\right] , \\quad ( i , j ) \\in \\mathcal { T } _ { \\Delta } .", + "type": "interline_equation", + "image_path": "e2ea999af7366ed6a17ad9df51b1ffdc13071896cebe8b19440de9bbf5177aa1.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 154, + 388, + 455, + 415 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 131, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 133, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 133, + 434 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 432, + 426, + 448 + ], + "lines": [ + { + "bbox": [ + 184, + 432, + 426, + 448 + ], + "spans": [ + { + "bbox": [ + 184, + 432, + 426, + 448 + ], + "score": 0.89, + "content": "( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } = ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } + ( \\boldsymbol { \\Delta } _ { \\mathbf { D } } ^ { N } ) ^ { T } \\boldsymbol { \\Delta } _ { \\mathbf { W } } ^ { N } .", + "type": "interline_equation", + "image_path": "d3cd9be27e671c38cf93d05e92e587fb28e6ae249c969dabe7c2b6edd1882864.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 184, + 432, + 426, + 448 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 271, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 271, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 188, + 465 + ], + "score": 1.0, + "content": "Lemma 4. For any", + "type": "text" + }, + { + "bbox": [ + 188, + 452, + 234, + 464 + ], + "score": 0.93, + "content": "( i , j ) \\in \\mathcal { I } _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 452, + 271, + 465 + ], + "score": 1.0, + "content": ", one has", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 469, + 374, + 504 + ], + "lines": [ + { + "bbox": [ + 237, + 469, + 374, + 504 + ], + "spans": [ + { + "bbox": [ + 237, + 469, + 374, + 504 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , : ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } , } \\\\ { \\| \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , : ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "40e97ad7ae049f3b9db903876b87b8c949f1f3ed8fc61e7e134a4b9ef17adde1.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 469, + 374, + 486.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 237, + 486.5, + 374, + 504.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 364, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 365, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 260, + 532 + ], + "score": 1.0, + "content": "Proof. 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1 } = \\left[ \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\right] , \\quad \\Delta _ { \\mathbf { W } } ^ { N } = \\left[ \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot ; \\cdot ) \\right] , \\quad ( i , j ) \\in \\mathcal { T } _ { \\Delta } .", + "type": "interline_equation", + "image_path": "e2ea999af7366ed6a17ad9df51b1ffdc13071896cebe8b19440de9bbf5177aa1.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 154, + 388, + 455, + 415 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 131, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 133, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 133, + 434 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 421, + 133, + 434 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 432, + 426, + 448 + ], + "lines": [ + { + "bbox": [ + 184, + 432, + 426, + 448 + ], + "spans": [ + { + "bbox": [ + 184, + 432, + 426, + 448 + ], + "score": 0.89, + "content": "( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } = ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } + ( \\boldsymbol { \\Delta } _ { \\mathbf { D } } ^ { N } ) ^ { T } \\boldsymbol { \\Delta } _ { \\mathbf { W } } ^ { N } .", + "type": "interline_equation", + "image_path": "d3cd9be27e671c38cf93d05e92e587fb28e6ae249c969dabe7c2b6edd1882864.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 184, + 432, + 426, + 448 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 271, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 271, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 188, + 465 + ], + "score": 1.0, + "content": "Lemma 4. For any", + "type": "text" + }, + { + "bbox": [ + 188, + 452, + 234, + 464 + ], + "score": 0.93, + "content": "( i , j ) \\in \\mathcal { I } _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 452, + 271, + 465 + ], + "score": 1.0, + "content": ", one has", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 452, + 271, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 469, + 374, + 504 + ], + "lines": [ + { + "bbox": [ + 237, + 469, + 374, + 504 + ], + "spans": [ + { + "bbox": [ + 237, + 469, + 374, + 504 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , : ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } , } \\\\ { \\| \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , : ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "40e97ad7ae049f3b9db903876b87b8c949f1f3ed8fc61e7e134a4b9ef17adde1.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 469, + 374, + 486.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 237, + 486.5, + 374, + 504.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 364, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 365, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 260, + 532 + ], + "score": 1.0, + "content": "Proof. Equation (35) implies that, for", + "type": "text" + }, + { + "bbox": [ + 260, + 518, + 361, + 531 + ], + "score": 0.79, + "content": "( i , j ) \\in \\mathcal { T } _ { 1 } , 1 \\le m \\le M", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 517, + 365, + 532 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 517, + 365, + 532 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 537, + 513, + 579 + ], + "lines": [ + { + "bbox": [ + 111, + 537, + 513, + 579 + ], + "spans": [ + { + "bbox": [ + 111, + 537, + 513, + 579 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathsf { \\Pi } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) = \\left\\{ \\begin{array} { l l } { \\mathbf { d } _ { m } ( k - i , l - j ) , } & { i \\le k \\le N + D - 2 , j \\le l \\le j + D - 1 } \\\\ { \\mathbf { d } _ { m } ( k - i + N + D - 1 , l - j ) , } & { 0 \\le k \\le i - N , j \\le l \\le j + D - 1 } \\\\ { 0 , } & { k , l \\mathrm { ~ t a k e n ~ a s ~ o t h e r s } } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "1055f36a68b10e22f3c6c811e8c82c55a76702b97734d56352642ec04308f4a4.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 537, + 513, + 551.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 551.0, + 513, + 565.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 565.0, + 513, + 579.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 210, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 211, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 163, + 601 + ], + "score": 1.0, + "content": "Thus, for any", + "type": "text" + }, + { + "bbox": [ + 163, + 587, + 206, + 599 + ], + "score": 0.92, + "content": "( i , j ) \\in \\mathcal { T } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 585, + 211, + 601 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 585, + 211, + 601 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 605, + 471, + 640 + ], + "lines": [ + { + "bbox": [ + 141, + 605, + 471, + 640 + ], + "spans": [ + { + "bbox": [ + 141, + 605, + 471, + 640 + ], + "score": 0.93, + "content": "\\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , \\cdot ) \\| _ { 2 } ^ { 2 } = \\sum _ { k = 0 } ^ { N + D - 2 } \\sum _ { l = 0 } ^ { N + D - 2 } \\sum _ { m = 1 } ^ { M } \\left| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; k , l , m ) \\right| ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "e562a64c2e6428a594f9c99cb9df9ceedf3d426f922b428a61ce02e9def55d15.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 141, + 605, + 471, + 616.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 141, + 616.6666666666666, + 471, + 628.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 141, + 628.3333333333333, + 471, + 639.9999999999999 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 646, + 147, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 149, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 149, + 660 + ], + "score": 1.0, + "content": "Similarly,", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 644, + 149, + 660 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 658, + 413, + 674 + ], + "lines": [ + { + "bbox": [ + 198, + 658, + 413, + 674 + ], + "spans": [ + { + "bbox": [ + 198, + 658, + 413, + 674 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\| \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\cdot , \\cdot ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } , \\quad ( i , j ) \\in \\mathcal { T } _ { 2 } \\cup \\mathcal { T } _ { 3 } . } \\end{array}", + "type": "interline_equation", + "image_path": "d612a43357729aca2e9e7edf60fd828e809d80e82f3d8cbf62a3890adfea825a.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 198, + 658, + 413, + 674 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 423, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 424, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 424, + 693 + ], + "score": 1.0, + "content": "Equation (50) is proved. With the same argument, equation (51) is also proved.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 677, + 424, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 255, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 256, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 163, + 714 + ], + "score": 1.0, + "content": "Lemma 5. If", + "type": "text" + }, + { + "bbox": [ + 163, + 701, + 217, + 712 + ], + "score": 0.88, + "content": "N \\geq 2 D - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 700, + 256, + 714 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 700, + 256, + 714 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 718, + 443, + 734 + ], + "lines": [ + { + "bbox": [ + 167, + 718, + 443, + 734 + ], + "spans": [ + { + "bbox": [ + 167, + 718, + 443, + 734 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\| _ { F } ^ { 2 } \\leq \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) ( 2 D - 1 ) ^ { 2 } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "791fd665e4ca093a49aa43f01d01f0226ea1e31fa4f0242f77dbcbd1bc747c3a.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 167, + 718, + 443, + 734 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 308, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 309, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 309, + 95 + ], + "score": 1.0, + "content": "Proof. For simplicity, we denote two row vectors:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 100, + 402, + 136 + ], + "lines": [ + { + "bbox": [ + 207, + 100, + 402, + 136 + ], + "spans": [ + { + "bbox": [ + 207, + 100, + 402, + 136 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathbf { d } _ { i , j } \\triangleq { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\colon , \\colon ) \\in \\mathbb { R } ^ { 1 \\times ( N + D - 1 ) ^ { 2 } M } } \\\\ & { \\mathbf { w } _ { i , j } \\triangleq \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\colon , \\colon ) \\in \\mathbb { R } ^ { 1 \\times ( N + D - 1 ) ^ { 2 } M } } \\end{array}", + "type": "interline_equation", + "image_path": "0ed4d47e034e4e9fb972e11d615dbd40a34bdb797ab77a04de86bcf87addc9d6.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 100, + 402, 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Z } _ { \\Delta } } \\sum _ { ( i _ { 2 } , j _ { 2 } ) \\in \\mathcal { Z } _ { \\Delta } } \\left. \\mathbf { d } _ { i _ { 1 } , j _ { 1 } } ^ { T } \\mathbf { w } _ { i _ { 1 } , j _ { 1 } } , \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\mathbf { w } _ { i _ { 2 } , j _ { 2 } } \\right. _ { F } ,", + "type": "interline_equation", + "image_path": "d7107d113f061c453b1576a574967f28fc42d44d26311d86a98fc2b2d648db9e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 113, + 159, + 496, + 170.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 170.66666666666666, + 496, + 182.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 182.33333333333331, + 496, + 193.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 133, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 135, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 135, + 213 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 218, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 111, + 218, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 111, + 218, + 505, + 239 + ], + "score": 0.88, + "content": "\\mathbf { \\mathop { ' } d } _ { i _ { 1 } , j _ { 1 } } ^ { T } \\mathbf { \\boldsymbol { w } } _ { i _ { 1 } , j _ { 1 } } , \\mathbf { \\boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\mathbf { \\boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } \\bigg \\rangle _ { F } = \\operatorname { t r a c e } \\Bigl ( \\mathbf { w } _ { i _ { 1 } , j _ { 1 } } ^ { T } \\mathbf { \\boldsymbol { d } } _ { i _ { 1 } , j _ { 1 } } \\mathbf { \\boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\mathbf { \\boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } \\Bigr ) = \\bigl ( \\mathbf { \\boldsymbol { d } } _ { i _ { 1 } , j _ { 1 } } \\mathbf { \\boldsymbol { d } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\bigr ) \\cdot \\bigl ( \\mathbf { \\boldsymbol { w } } _ { i _ { 1 } , j _ { 1 } } \\mathbf { \\boldsymbol { w } } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\bigr ) .", + "type": "interline_equation", + "image_path": "e38464ee4cfe60460468271fb54d7da1b1f5240ee9f64102bfd4e41215b37fe3.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 111, + 218, + 505, + 239 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 131, + 256 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 132, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 132, + 258 + ], + "score": 1.0, + "content": "Since", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 261, + 470, + 297 + ], + "lines": [ + { + "bbox": [ + 140, + 261, + 470, + 297 + ], + "spans": [ + { + "bbox": [ + 140, + 261, + 470, + 297 + ], + "score": 0.93, + "content": "\\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } = \\sum _ { k = 0 } ^ { N - 1 } \\sum _ { l = 0 } ^ { N - 1 } \\sum _ { m = 1 } ^ { M } = \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i _ { 1 } , j _ { 1 } ; k , l , m ) \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i _ { 2 } , j _ { 2 } ; k , l , m ) ,", + "type": "interline_equation", + "image_path": "36e13485b92043bd67fec88933d31cafc4ca6ab65654fe714aa0f9a6a641d04a.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 140, + 261, + 470, + 273.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 140, + 273.0, + 470, + 285.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 140, + 285.0, + 470, + 297.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 394, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 396, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 296, + 320 + ], + "score": 1.0, + "content": "with the same argument in Lemma 2, we have:", + "type": "text" + }, + { + "bbox": [ + 296, + 303, + 362, + 318 + ], + "score": 0.93, + "content": "\\mathbf { d } _ { i _ { 1 } , j _ { 1 } } \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 301, + 396, + 320 + ], + "score": 1.0, + "content": "implies", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 323, + 488, + 366 + ], + "lines": [ + { + "bbox": [ + 122, + 323, + 486, + 366 + ], + "spans": [ + { + "bbox": [ + 122, + 323, + 486, + 366 + ], + "score": 0.26, + "content": "\\begin{array} { r l } & { i _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } \\triangleq \\{ i | 0 \\le ( i _ { 1 } - i ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\mathrm { o r } 0 \\le ( i - i _ { 1 } ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\} } \\\\ & { j _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } \\triangleq \\{ j | 0 \\le ( j _ { 1 } - j ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\mathrm { o r } 0 \\le ( j - j _ { 1 } ) _ { \\mathrm { m o d } ( N + D - 1 ) } \\le D - 1 \\} } \\end{array}", + "type": "interline_equation", + "image_path": "82306b2596bd1a1fe5be753ecbb54253acf905dde5cf2abcbebfe25d5960a295.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 121, + 323, + 488, + 337.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 121, + 337.3333333333333, + 488, + 351.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 121, + 351.66666666666663, + 488, + 365.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 129, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 131, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 131, + 381 + ], + "score": 1.0, + "content": "Then", + "type": "text" + 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\\sum _ { \\Delta } \\sum _ { i _ { 2 } \\in \\mathbb { Z } _ { \\Delta } ^ { \\prime } } \\displaystyle \\sum _ { j _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } } \\\\ { } & { = | \\mathcal { Z } _ { \\Delta } | \\cdot | \\mathcal { I } _ { \\Delta } ^ { \\prime } | \\cdot | \\mathcal { I } _ { \\Delta } ^ { \\prime } | \\cdot \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } } \\\\ { } & { = \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) ( 2 D - 1 ) ^ { 2 } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "d92bb80359f5e8631b8a6315f945a8187284d1b7d90a99022adf8dda9444807c.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 159, + 378, + 451, + 409.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 159, + 409.0, + 451, + 440.0 + ], + 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Inequality (52) is proved.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 385, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 503, + 387, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 387, + 517 + ], + "score": 1.0, + "content": "With these preparations, we can prove Theorem 3, Conclusion 2 now.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 292, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 293, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 293, + 546 + ], + "score": 1.0, + "content": "Proof of Theorem 3, Conclusion 2. Define set", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 551, + 428, + 573 + ], + "lines": [ + { + "bbox": [ + 182, + 551, + 428, + 573 + ], + "spans": [ + { + "bbox": [ + 182, + 551, + 428, + 573 + ], + "score": 0.91, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } } = \\Big \\{ \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } \\Big | \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , \\forall m = 1 , \\cdots , M \\Big \\} .", + "type": "interline_equation", + "image_path": "1c7e9261265d7f20fbf76185a230fef6e2087f505c3a22a44a7e5df76270c7ea.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 182, + 551, + 428, + 573 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 272, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 273, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 131, + 594 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 579, + 186, + 590 + ], + "score": 0.93, + "content": "\\mathbf { d } \\in \\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 576, + 273, + 594 + ], + "score": 1.0, + "content": ", the set is nonempty:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 597, + 336, + 611 + ], + "lines": [ + { + "bbox": [ + 274, + 597, + 336, + 611 + ], + "spans": [ + { + "bbox": [ + 274, + 597, + 336, + 611 + ], + "score": 0.87, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } } \\neq \\emptyset .", + "type": "interline_equation", + "image_path": "b8bfd965b18f857304ea2f27899df09bb7aa1b7065c5f92938d88d1483aa80b4.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 274, + 597, + 336, + 611 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 338, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 336, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 175, + 634 + ], + "score": 1.0, + "content": "Define functions", + "type": "text" + }, + { + "bbox": [ + 175, + 617, + 336, + 632 + ], + "score": 0.91, + "content": "F _ { \\mathrm { c o n v } } ^ { N } : \\mathbb { R } ^ { D ^ { 2 } M } \\to \\mathbb { R } , F _ { \\mathrm { c i r } } ^ { N } : \\mathbb { R } ^ { D ^ { 2 } M } \\to \\mathbb { R } .", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 638, + 447, + 689 + ], + "lines": [ + { + "bbox": [ + 163, + 638, + 447, + 689 + ], + "spans": [ + { + "bbox": [ + 163, + 638, + 447, + 689 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) = \\displaystyle \\frac { 1 } { N + D - 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For simplicity, we denote two row vectors:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 309, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 100, + 402, + 136 + ], + "lines": [ + { + "bbox": [ + 207, + 100, + 402, + 136 + ], + "spans": [ + { + "bbox": [ + 207, + 100, + 402, + 136 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathbf { d } _ { i , j } \\triangleq { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\colon , \\colon ) \\in \\mathbb { R } ^ { 1 \\times ( N + D - 1 ) ^ { 2 } M } } \\\\ & { \\mathbf { w } _ { i , j } \\triangleq \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } ( i , j ; \\cdot , \\colon , \\colon ) \\in \\mathbb { R } ^ { 1 \\times ( N + D - 1 ) ^ { 2 } M } } \\end{array}", + "type": "interline_equation", + "image_path": "0ed4d47e034e4e9fb972e11d615dbd40a34bdb797ab77a04de86bcf87addc9d6.jpg" + } + ] + } + ], + "index": 1.5, + 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_ { 1 } , j _ { 1 } } \\mathbf { d } _ { i _ { 2 } , j _ { 2 } } ^ { T } ) \\cdot ( \\mathbf { w } _ { i _ { 1 } , j _ { 1 } } \\mathbf { w } _ { i _ { 2 } , j _ { 2 } } ^ { T } ) } \\\\ { \\leq } & { \\displaystyle \\sum _ { ( i _ { 1 } , j _ { 1 } ) \\in \\mathbb { Z } } \\displaystyle \\sum _ { \\Delta } \\sum _ { i _ { 2 } \\in \\mathbb { Z } _ { \\Delta } ^ { \\prime } } \\displaystyle \\sum _ { j _ { 2 } \\in \\mathcal { I } _ { \\Delta } ^ { \\prime } } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } } \\\\ { } & { = | \\mathcal { Z } _ { \\Delta } | \\cdot | \\mathcal { I } _ { \\Delta } ^ { \\prime } | \\cdot | \\mathcal { I } _ { \\Delta } ^ { \\prime } | \\cdot \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } } \\\\ { } & { = \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) ( 2 D - 1 ) ^ { 2 } \\| \\mathbf { d } \\| _ { 2 } ^ { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } , } \\end{array}", + "type": 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Inequality (52) is proved.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 473, + 479, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 385, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 503, + 387, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 387, + 517 + ], + "score": 1.0, + "content": "With these preparations, we can prove Theorem 3, Conclusion 2 now.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 503, + 387, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 292, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 293, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 293, + 546 + ], + "score": 1.0, + "content": "Proof of Theorem 3, Conclusion 2. Define set", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 532, + 293, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 551, + 428, + 573 + ], + "lines": [ + { + "bbox": [ + 182, + 551, + 428, + 573 + ], + "spans": [ + { + "bbox": [ + 182, + 551, + 428, + 573 + ], + "score": 0.91, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } } = \\Big \\{ \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } \\Big | \\mathbf { w } _ { m } \\cdot \\mathbf { d } _ { m } = 1 , \\forall m = 1 , \\cdots , M \\Big \\} .", + "type": "interline_equation", + "image_path": "1c7e9261265d7f20fbf76185a230fef6e2087f505c3a22a44a7e5df76270c7ea.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 182, + 551, + 428, + 573 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 272, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 273, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 131, + 594 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 579, + 186, + 590 + ], + "score": 0.93, + "content": "\\mathbf { d } \\in \\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 576, + 273, + 594 + ], + "score": 1.0, + "content": ", the set is nonempty:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 576, + 273, + 594 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 597, + 336, + 611 + ], + "lines": [ + { + "bbox": [ + 274, + 597, + 336, + 611 + ], + "spans": [ + { + "bbox": [ + 274, + 597, + 336, + 611 + ], + "score": 0.87, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } } \\neq \\emptyset .", + "type": "interline_equation", + "image_path": "b8bfd965b18f857304ea2f27899df09bb7aa1b7065c5f92938d88d1483aa80b4.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 274, + 597, + 336, + 611 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 338, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 336, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 175, + 634 + ], + "score": 1.0, + "content": "Define functions", + "type": "text" + }, + { + "bbox": [ + 175, + 617, + 336, + 632 + ], + "score": 0.91, + "content": "F _ { \\mathrm { c o n v } } ^ { N } : \\mathbb { R } ^ { D ^ { 2 } M } \\to \\mathbb { R } , F _ { \\mathrm { c i r } } ^ { N } : \\mathbb { R } ^ { D ^ { 2 } M } \\to \\mathbb { R } .", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 614, + 336, + 634 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 638, + 447, + 689 + ], + "lines": [ + { + "bbox": [ + 163, + 638, + 447, + 689 + ], + "spans": [ + { + "bbox": [ + 163, + 638, + 447, + 689 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) = \\displaystyle \\frac { 1 } { N + D - 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1 } \\Big | \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } \\Big \\| _ { F } - \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } \\big ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\Big \\| _ { F } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { 1 } { N + D - 1 } \\Big \\| ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\Big \\| _ { F } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { \\sqrt { \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) } ( 2 D - 1 ) } { N + D - 1 } \\| \\mathbf { d } \\| _ { 2 } \\| \\mathbf { w } \\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { ( 2 D - 1 ) \\sqrt { 2 ( D - 1 ) } } { \\sqrt { N + D - 1 } } \\| \\mathbf { d } \\| _ { 2 } \\| \\mathbf { w } \\| _ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "455c5dd273976382938b323bc7f54acb1ae96b25d32ffbc54054d87de9ab1e8a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 113, + 138, + 498, + 182.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 182.66666666666666, + 498, + 227.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 227.33333333333331, + 498, + 272.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 274, + 403, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 403, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 221, + 287 + ], + "score": 1.0, + "content": "Thus, there exists a constant", + "type": "text" + }, + { + "bbox": [ + 222, + 275, + 249, + 285 + ], + "score": 0.9, + "content": "B > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 273, + 351, + 287 + ], + "score": 1.0, + "content": ", which is independent of", + "type": "text" + }, + { + "bbox": [ + 351, + 275, + 361, + 284 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 273, + 403, + 287 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 289, + 464, + 317 + ], + "lines": [ + { + "bbox": [ + 146, + 289, + 464, + 317 + ], + "spans": [ + { + "bbox": [ + 146, + 289, + 464, + 317 + ], + "score": 0.92, + "content": "| F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) | \\leq \\frac { B } { \\sqrt { N } } \\operatorname* { s u p } _ { \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } } \\| \\mathbf { w } \\| , \\quad \\forall \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } .", + "type": "interline_equation", + "image_path": "4832443d38512316027642c072a501b7325bbfa969bc9aae1687ee7ea6b5dcd9.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 146, + 289, + 464, + 317 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 387, + 342 + ], + "lines": [ + { + "bbox": [ + 102, + 321, + 384, + 349 + ], + "spans": [ + { + "bbox": [ + 102, + 321, + 180, + 349 + ], + "score": 1.0, + "content": "Step 2: Proving", + "type": "text" + }, + { + "bbox": [ + 180, + 328, + 220, + 342 + ], + "score": 0.92, + "content": "F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 321, + 337, + 349 + ], + "score": 1.0, + "content": "epigraphically converges8 to", + "type": "text" + }, + { + "bbox": [ + 338, + 327, + 384, + 342 + ], + "score": 0.87, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } )", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 293, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 293, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 293, + 358 + ], + "score": 1.0, + "content": "We want to show, at each point w it holds that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 360, + 441, + 405 + ], + "lines": [ + { + "bbox": [ + 170, + 360, + 441, + 405 + ], + "spans": [ + { + "bbox": [ + 170, + 360, + 441, + 405 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { N \\infty } { \\operatorname* { l i m } \\operatorname* { i n f } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) \\geq F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\quad \\mathrm { f o r ~ e v e r y ~ s e q u e n c e ~ } \\mathbf { w } ^ { N } \\mathbf { w } } \\\\ & { \\underset { N \\infty } { \\operatorname* { l i m } \\operatorname* { s u p } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) \\leq F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\quad \\mathrm { f o r ~ s o m e ~ s e q u e n c e ~ } \\mathbf { w } ^ { N } \\mathbf { w } } \\end{array}", + "type": "interline_equation", + "image_path": "8f39f36af8809eab1def7e771bb9b2e78731f936aef06d8780023334d1939231.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 170, + 360, + 441, + 375.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 170, + 375.0, + 441, + 390.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 170, + 390.0, + 441, + 405.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 450, + 428 + ], + "lines": [ + { + "bbox": [ + 104, + 412, + 447, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 321, + 430 + ], + "score": 1.0, + "content": "Firstly, we prove (56). 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Since s, one h", + "type": "text" + }, + { + "bbox": [ + 291, + 434, + 327, + 445 + ], + "score": 0.9, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 425, + 407, + 460 + ], + "score": 1.0, + "content": "et, therfor all", + "type": "text" + }, + { + "bbox": [ + 446, + 425, + 447, + 460 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 448, + 434, + 464, + 444 + ], + "score": 0.89, + "content": "N ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 425, + 510, + 460 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 445, + 446, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 172, + 458 + ], + "score": 0.88, + "content": "\\mathbf { w } ^ { N } \\notin \\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 446, + 238, + 457 + ], + "score": 0.91, + "content": "N \\geq N ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 445, + 380, + 458 + ], + "score": 0.93, + "content": "F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) = + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 446, + 446, + 457 + ], + "score": 0.91, + "content": "N \\geq N ^ { + }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 461, + 390, + 482 + ], + "lines": [ + { + "bbox": [ + 219, + 461, + 390, + 482 + ], + "spans": [ + { + "bbox": [ + 219, + 461, + 390, + 482 + ], + "score": 0.91, + "content": "\\operatorname* { l i m } _ { N \\to \\infty } \\operatorname* { i n f } _ { C \\mathrm { c o n v } } ( \\mathbf { w } ^ { N } ) = F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) = + \\infty .", + "type": "interline_equation", + "image_path": "8d41c772478cb00f31875d5369a85cce1ab56b9e27d18f455389d595d9c7c4ee.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 219, + 461, + 390, + 482 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 506, + 524 + ], + "lines": [ + { + "bbox": [ + 104, + 490, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 122, + 505 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 122, + 492, + 180, + 502 + ], + "score": 0.87, + "content": "\\mathbf { v } \\in \\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 490, + 507, + 505 + ], + "score": 1.0, + "content": ", two cases should be considered. 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1 } \\Big | \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N + D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N + D - 1 } \\Big \\| _ { F } - \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c o n v } } ^ { N } \\big ) ^ { T } \\mathbf { W } _ { \\mathrm { c o n v } } ^ { N } \\Big \\| _ { F } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { 1 } { N + D - 1 } \\Big \\| ( \\Delta _ { \\mathbf { D } } ^ { N } ) ^ { T } \\Delta _ { \\mathbf { W } } ^ { N } \\Big \\| _ { F } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { \\sqrt { \\big ( 2 N ( D - 1 ) + ( D - 1 ) ^ { 2 } \\big ) } ( 2 D - 1 ) } { N + D - 1 } \\| \\mathbf { d } \\| _ { 2 } \\| \\mathbf { w } \\| _ { 2 } } \\\\ & { \\qquad \\leq \\displaystyle \\frac { ( 2 D - 1 ) \\sqrt { 2 ( D - 1 ) } } { \\sqrt { N + D - 1 } } \\| \\mathbf { d } \\| _ { 2 } \\| \\mathbf { w } \\| _ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "455c5dd273976382938b323bc7f54acb1ae96b25d32ffbc54054d87de9ab1e8a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 113, + 138, + 498, + 182.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 182.66666666666666, + 498, + 227.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 227.33333333333331, + 498, + 272.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 274, + 403, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 403, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 221, + 287 + ], + "score": 1.0, + "content": "Thus, there exists a constant", + "type": "text" + }, + { + "bbox": [ + 222, + 275, + 249, + 285 + ], + "score": 0.9, + "content": "B > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 273, + 351, + 287 + ], + "score": 1.0, + "content": ", which is independent of", + "type": "text" + }, + { + "bbox": [ + 351, + 275, + 361, + 284 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 273, + 403, + 287 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 273, + 403, + 287 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 289, + 464, + 317 + ], + "lines": [ + { + "bbox": [ + 146, + 289, + 464, + 317 + ], + "spans": [ + { + "bbox": [ + 146, + 289, + 464, + 317 + ], + "score": 0.92, + "content": "| F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ) - F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) | \\leq \\frac { B } { \\sqrt { N } } \\operatorname* { s u p } _ { \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } } \\| \\mathbf { w } \\| , \\quad \\forall \\mathbf { w } \\in X \\cap \\mathcal { W } _ { \\mathrm { n o r m a l } } .", + "type": "interline_equation", + "image_path": "4832443d38512316027642c072a501b7325bbfa969bc9aae1687ee7ea6b5dcd9.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 146, + 289, + 464, + 317 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 387, + 342 + ], + "lines": [ + { + "bbox": [ + 102, + 321, + 384, + 349 + ], + "spans": [ + { + "bbox": [ + 102, + 321, + 180, + 349 + ], + "score": 1.0, + "content": "Step 2: Proving", + "type": "text" + }, + { + "bbox": [ + 180, + 328, + 220, + 342 + ], + "score": 0.92, + "content": "F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 321, + 337, + 349 + ], + "score": 1.0, + "content": "epigraphically converges8 to", + "type": "text" + }, + { + "bbox": [ + 338, + 327, + 384, + 342 + ], + "score": 0.87, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } )", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 102, + 321, + 384, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 293, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 293, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 293, + 358 + ], + "score": 1.0, + "content": "We want to show, at each point w it holds that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 346, + 293, + 358 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 360, + 441, + 405 + ], + "lines": [ + { + "bbox": [ + 170, + 360, + 441, + 405 + ], + "spans": [ + { + "bbox": [ + 170, + 360, + 441, + 405 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { N \\infty } { \\operatorname* { l i m } \\operatorname* { i n f } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) \\geq F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\quad \\mathrm { f o r ~ e v e r y ~ s e q u e n c e ~ } \\mathbf { w } ^ { N } \\mathbf { w } } \\\\ & { \\underset { N \\infty } { \\operatorname* { l i m } \\operatorname* { s u p } } F _ { \\mathrm { c o n v } } ^ { N } ( \\mathbf { w } ^ { N } ) \\leq F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) \\quad \\mathrm { f o r ~ s o m e ~ s e q u e n c e ~ } \\mathbf { w } ^ { N } \\mathbf { w } } \\end{array}", + "type": "interline_equation", + "image_path": "8f39f36af8809eab1def7e771bb9b2e78731f936aef06d8780023334d1939231.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 170, + 360, + 441, + 375.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 170, + 375.0, + 441, + 390.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 170, + 390.0, + 441, + 405.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 450, + 428 + ], + "lines": [ + { + "bbox": [ + 104, + 412, + 447, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 321, + 430 + ], + "score": 1.0, + "content": "Firstly, we prove (56). 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All the other accumulation points of", + "type": "text" + }, + { + "bbox": [ + 303, + 167, + 330, + 197 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 308, + 150, + 404, + 164 + ], + "score": 0.81, + "content": "\\{ \\mathbf { w } ^ { N _ { k } } \\} _ { k = 0 } ^ { \\infty } \\subset \\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 176, + 387, + 188 + ], + "score": 0.89, + "content": "\\mathbf { w } \\notin \\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 167, + 419, + 197 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + }, + { + "bbox": [ + 404, + 142, + 410, + 171 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 407, + 162, + 482, + 176 + ], + "score": 0.91, + "content": "\\{ \\bar F _ { \\mathrm { c o n v } } ^ { \\bar { N } } ( \\mathbf { w } ^ { N } ) \\} _ { N = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 149, + 457, + 163 + ], + "score": 0.73, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 142, + 509, + 171 + ], + "score": 1.0, + "content": "is an accu-", + "type": "text" + }, + { + "bbox": [ + 482, + 155, + 510, + 182 + ], + "score": 1.0, + "content": "must", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 192, + 391, + 212 + ], + "lines": [ + { + "bbox": [ + 219, + 192, + 391, + 212 + ], + "spans": [ + { + "bbox": [ + 219, + 192, + 391, + 212 + ], + "score": 0.9, + "content": "\\operatorname* { l i m } _ { N \\to \\infty } \\operatorname* { i n f } _ { C \\mathrm { c o n v } } ( \\mathbf { w } ^ { N } ) = F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) < + \\infty .", + "type": "interline_equation", + "image_path": "874a1ff21ab2ba224ad5a9709a25eddfe9016eef6982e49611b41da05bcaae7c.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 219, + 192, + 391, + 212 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 504, + 247 + ], + "lines": [ + { + "bbox": [ + 104, + 221, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 241, + 237 + ], + "score": 1.0, + "content": "Secondly, we prove (57). We set", + "type": "text" + }, + { + "bbox": [ + 242, + 223, + 282, + 235 + ], + "score": 0.9, + "content": "\\mathbf { w } ^ { N } = \\mathbf { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 221, + 311, + 237 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 312, + 224, + 378, + 235 + ], + "score": 0.91, + "content": "N = 0 , 1 , 2 , \\cdots", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 221, + 506, + 237 + ], + "score": 1.0, + "content": ". 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( \\mathbf { D } _ { \\mathrm { c i r } } ^ { 2 D - 1 } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\right. _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "da043a519a3b7a06d3db139c755df78f74c01ac4f2a8530ee78d3374df231094.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 236, + 274, + 374, + 292 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 297, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 299, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 195, + 310 + ], + "score": 1.0, + "content": "We want to show that", + "type": "text" + }, + { + "bbox": [ + 195, + 297, + 220, + 308 + ], + "score": 0.93, + "content": "G ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 295, + 299, + 310 + ], + "score": 1.0, + "content": "is strongly convex.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 324, + 328 + ], + "lines": [ + { + "bbox": [ + 123, + 306, + 303, + 333 + ], + "spans": [ + { + "bbox": [ + 123, + 313, + 185, + 327 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { w } } _ { i } \\in \\mathbb { R } ^ { ( 2 D - 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Thus,", + "type": "text" + }, + { + "bbox": [ + 395, + 643, + 425, + 655 + ], + "score": 0.9, + "content": "T ^ { T } Q T", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 641, + 506, + 657 + ], + "score": 1.0, + "content": "is positive definite,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 654, + 124, + 666 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 124, + 666 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 102, + 624, + 506, + 666 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 663, + 355, + 678 + ], + "lines": [ + { + "bbox": [ + 255, + 663, + 355, + 678 + ], + "spans": [ + { + "bbox": [ + 255, + 663, + 355, + 678 + ], + "score": 0.89, + "content": "G ( \\mathbf { w } ) = \\mathbf { w } ^ { T } ( T ^ { T } Q T ) \\mathbf { w }", + "type": "interline_equation", + "image_path": "6076e8bf61d6618cb3caba4ff468ef4f50d7ee57138ad5dcffb1307b652389ca.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 255, + 663, + 355, + 678 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 102, + 676, + 504, + 697 + ], + "spans": [ + { + "bbox": [ + 102, + 676, + 205, + 697 + ], + "score": 1.0, + "content": "is strongly convex. Then", + "type": "text" + }, + { + "bbox": [ + 205, + 680, + 357, + 694 + ], + "score": 0.92, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( { \\bf w } ) = \\sqrt { G ( { \\bf w } ) } + \\iota _ { \\mathcal { W } _ { \\mathrm { n o r m a l } } } ( { \\bf w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 676, + 470, + 697 + ], + "score": 1.0, + "content": "has only one minimizer, i.e.,", + "type": "text" + }, + { + "bbox": [ + 470, + 680, + 504, + 694 + ], + "score": 0.91, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 693, + 234, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 234, + 704 + ], + "score": 1.0, + "content": "involves only a unique element.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 102, + 676, + 504, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Now we check the conditions of Propositions 7.32(c) and 7.33 in (Rockafellar & Wets, 2009) to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 720, + 157, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 157, + 734 + ], + "score": 1.0, + "content": "apply them.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 130, + 80, + 314, + 96 + ], + "lines": [ + { + "bbox": [ + 128, + 74, + 319, + 103 + ], + "spans": [ + { + "bbox": [ + 128, + 79, + 163, + 94 + ], + "score": 1.0, + "content": "1. F N", + "type": "text" + }, + { + "bbox": [ + 142, + 81, + 211, + 95 + ], + "score": 0.93, + "content": "F _ { \\mathrm { c o n v } } ^ { N } \\xrightarrow [ ] { \\mathrm { e } } F _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 74, + 319, + 103 + ], + "score": 1.0, + "content": ". This is proved in Step 2.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 124, + 101, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 123, + 93, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 123, + 93, + 142, + 128 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 101, + 173, + 115 + ], + "score": 0.89, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 93, + 281, + 128 + ], + "score": 1.0, + "content": "vel bounded. Since must be level bounde", + "type": "text" + }, + { + "bbox": [ + 281, + 103, + 306, + 115 + ], + "score": 0.91, + "content": "G ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 93, + 392, + 128 + ], + "score": 1.0, + "content": "is strongly convex,", + "type": "text" + }, + { + "bbox": [ + 393, + 101, + 505, + 116 + ], + "score": 0.92, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } ( \\mathbf { w } ) ~ = ~ \\sqrt { G ( \\mathbf { w } ) } ~ +", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 142, + 115, + 193, + 127 + ], + "spans": [ + { + "bbox": [ + 142, + 115, + 193, + 127 + ], + "score": 0.88, + "content": "\\iota _ { \\mathcal { W } _ { \\mathrm { n o r m a l } } } ( \\mathbf { w } )", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 128, + 132, + 502, + 157 + ], + "lines": [ + { + "bbox": [ + 125, + 125, + 504, + 157 + ], + "spans": [ + { + "bbox": [ + 125, + 125, + 142, + 157 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 142, + 133, + 205, + 146 + ], + "score": 0.92, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\not \\equiv + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 125, + 236, + 157 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 236, + 135, + 273, + 146 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 125, + 370, + 157 + ], + "score": 1.0, + "content": "is nonempty (54), dom", + "type": "text" + }, + { + "bbox": [ + 370, + 133, + 423, + 147 + ], + "score": 0.69, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 } \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 125, + 427, + 157 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 428, + 133, + 457, + 147 + ], + "score": 0.66, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 134, + 504, + 146 + ], + "score": 1.0, + "content": "s not con-", + "type": "text" + }, + { + "bbox": [ + 458, + 125, + 464, + 157 + ], + "score": 1.0, + "content": "i", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 172, + 147, + 190, + 156 + ], + "spans": [ + { + "bbox": [ + 172, + 147, + 190, + 156 + ], + "score": 0.89, + "content": "+ \\infty", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 126, + 163, + 500, + 177 + ], + "lines": [ + { + "bbox": [ + 218, + 163, + 497, + 177 + ], + "spans": [ + { + "bbox": [ + 218, + 163, + 242, + 176 + ], + "score": 0.92, + "content": "F _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 163, + 497, + 177 + ], + "score": 0.91, + "content": "F _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 129, + 183, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 129, + 180, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 129, + 184, + 142, + 196 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 137, + 180, + 176, + 195 + ], + "score": 1.0, + "content": ". F 2D−1", + "type": "text" + }, + { + "bbox": [ + 171, + 181, + 212, + 198 + ], + "score": 1.0, + "content": "and F Ncon", + "type": "text" + }, + { + "bbox": [ + 216, + 184, + 505, + 197 + ], + "score": 1.0, + "content": "are all lower semi-continuous and proper. This condition follows from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 194, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 140, + 194, + 287, + 210 + ], + "score": 1.0, + "content": "the fact that the functions F 2D−1", + "type": "text" + }, + { + "bbox": [ + 284, + 195, + 324, + 209 + ], + "score": 1.0, + "content": "and F N", + "type": "text" + }, + { + "bbox": [ + 331, + 198, + 506, + 209 + ], + "score": 1.0, + "content": "are all continuous functions defined on a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 207, + 317, + 222 + ], + "spans": [ + { + "bbox": [ + 141, + 207, + 275, + 222 + ], + "score": 1.0, + "content": "nonempty closed convex domain", + "type": "text" + }, + { + "bbox": [ + 276, + 209, + 312, + 219 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\mathrm { n o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 207, + 317, + 222 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 506, + 281 + ], + "lines": [ + { + "bbox": [ + 104, + 228, + 507, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 266, + 246 + ], + "score": 1.0, + "content": "Applying Proposition 7.32(c), we have", + "type": "text" + }, + { + "bbox": [ + 266, + 230, + 300, + 243 + ], + "score": 0.93, + "content": "\\{ F _ { \\mathrm { c o n v } } ^ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 228, + 507, + 246 + ], + "score": 1.0, + "content": "is eventually level bounded. If we arbitrarily pick", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 101, + 236, + 509, + 261 + ], + "spans": [ + { + "bbox": [ + 101, + 236, + 114, + 261 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 242, + 172, + 255 + ], + "score": 0.91, + "content": "\\mathbf { w } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 236, + 204, + 261 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 204, + 244, + 224, + 255 + ], + "score": 0.88, + "content": "\\mathbf { w } _ { \\mathrm { c i r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 236, + 317, + 261 + ], + "score": 1.0, + "content": "convbe the unique point in", + "type": "text" + }, + { + "bbox": [ + 317, + 242, + 350, + 255 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 236, + 509, + 261 + ], + "score": 1.0, + "content": ". Applying Proposition 7.33, we have", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 253, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 159, + 267 + ], + "score": 0.89, + "content": "\\mathbf { w } ^ { N } \\to \\mathbf { w } _ { \\mathrm { c i r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 253, + 174, + 288 + ], + "score": 1.0, + "content": ". Bsets", + "type": "text" + }, + { + "bbox": [ + 212, + 253, + 216, + 288 + ], + "score": 1.0, + "content": "o:", + "type": "text" + }, + { + "bbox": [ + 329, + 253, + 506, + 288 + ], + "score": 1.0, + "content": "ts, 2009), we obtain the convergence of the.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 174, + 266, + 329, + 281 + ], + "spans": [ + { + "bbox": [ + 174, + 266, + 212, + 280 + ], + "score": 0.88, + "content": "\\{ \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 266, + 329, + 281 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { N \\to \\infty } \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 106, + 294, + 309, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 309, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 309, + 310 + ], + "score": 1.0, + "content": "D DISCUSSION OF DEFINITION 2 (11)", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 318, + 506, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 365, + 332 + ], + "score": 1.0, + "content": "In this section, we want to numerically show that, given typical", + "type": "text" + }, + { + "bbox": [ + 366, + 320, + 376, + 330 + ], + "score": 0.77, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 318, + 394, + 332 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 322, + 401, + 329 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 318, + 444, + 332 + ], + "score": 1.0, + "content": ", there is a", + "type": "text" + }, + { + "bbox": [ + 445, + 320, + 483, + 331 + ], + "score": 0.91, + "content": "\\bar { \\sigma } _ { \\operatorname* { m i n } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 235, + 343 + ], + "score": 1.0, + "content": "that a random generated matrix", + "type": "text" + }, + { + "bbox": [ + 235, + 330, + 318, + 342 + ], + "score": 0.92, + "content": "\\mathbf { W } \\in \\mathbf { \\bar { \\mathcal { W } } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 330, + 389, + 343 + ], + "score": 1.0, + "content": ". However, given", + "type": "text" + }, + { + "bbox": [ + 389, + 331, + 399, + 341 + ], + "score": 0.68, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 330, + 418, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 331, + 431, + 341 + ], + "score": 0.61, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ", it’s intractable to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 342, + 201, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 201, + 354 + ], + "score": 1.0, + "content": "completely check (11):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 354, + 421, + 376 + ], + "lines": [ + { + "bbox": [ + 189, + 354, + 421, + 376 + ], + "spans": [ + { + "bbox": [ + 189, + 354, + 421, + 376 + ], + "score": 0.91, + "content": "\\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } , \\forall \\mathbb { S } \\mathrm { ~ w i t h ~ } 2 \\leq | \\mathbb { S } | \\leq s .", + "type": "interline_equation", + "image_path": "07141b11e27bd14a85e864e17e896eb6aeae5c72a3f3d77710ec420be89f17eb.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 189, + 354, + 421, + 376 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 378, + 503, + 400 + ], + "lines": [ + { + "bbox": [ + 107, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 107, + 376, + 367, + 390 + ], + "score": 1.0, + "content": "The reason is that there are extremely large amount of possible", + "type": "text" + }, + { + "bbox": [ + 367, + 378, + 375, + 388 + ], + "score": 0.5, + "content": "\\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 376, + 479, + 390 + ], + "score": 1.0, + "content": "s. For example, we take", + "type": "text" + }, + { + "bbox": [ + 480, + 378, + 505, + 389 + ], + "score": 0.85, + "content": "M =", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 388, + 269, + 401 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 197, + 401 + ], + "score": 0.9, + "content": "2 5 0 , N = 5 0 0 , s = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 388, + 269, + 401 + ], + "score": 1.0, + "content": ". There are totally", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 402, + 375, + 430 + ], + "lines": [ + { + "bbox": [ + 236, + 402, + 375, + 430 + ], + "spans": [ + { + "bbox": [ + 236, + 402, + 375, + 430 + ], + "score": 0.93, + "content": "\\binom { 5 0 0 } { 5 0 } + \\binom { 5 0 0 } { 4 9 } + \\cdot \\cdot \\cdot + \\binom { 5 0 0 } { 2 }", + "type": "interline_equation", + "image_path": "f6643be7dea0c01988edc5faba3ed58b1fd06155a47ecc0ffae96d841bf2d70d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 236, + 402, + 375, + 430 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 438, + 444 + ], + "lines": [ + { + "bbox": [ + 104, + 430, + 438, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 430, + 141, + 446 + ], + "score": 1.0, + "content": "possible", + "type": "text" + }, + { + "bbox": [ + 142, + 432, + 153, + 442 + ], + "score": 0.6, + "content": "\\mathbb { S } s", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 430, + 195, + 446 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 195, + 432, + 247, + 444 + ], + "score": 0.94, + "content": "2 \\leq | \\mathbb { S } | \\leq s", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 430, + 426, + 446 + ], + "score": 1.0, + "content": ". It’s impossible to check (11) on all possible", + "type": "text" + }, + { + "bbox": [ + 427, + 432, + 438, + 442 + ], + "score": 0.31, + "content": "\\mathbb { S } s", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 406, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 408, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 234, + 462 + ], + "score": 1.0, + "content": "Instead of checking all possible", + "type": "text" + }, + { + "bbox": [ + 234, + 449, + 245, + 459 + ], + "score": 0.72, + "content": "\\mathbb { S } s", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 448, + 294, + 462 + ], + "score": 1.0, + "content": ", we sample", + "type": "text" + }, + { + "bbox": [ + 294, + 449, + 327, + 460 + ], + "score": 0.57, + "content": "5 0 0 0 \\mathbb { S } \\mathrm { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 448, + 408, + 462 + ], + "score": 1.0, + "content": "from the whole set:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 462, + 407, + 477 + ], + "lines": [ + { + "bbox": [ + 202, + 462, + 407, + 477 + ], + "spans": [ + { + "bbox": [ + 202, + 462, + 407, + 477 + ], + "score": 0.87, + "content": "\\mathcal S ^ { \\prime } \\subset S = \\{ \\mathbb S : \\mathbb S \\subset \\{ 1 , 2 , \\cdots , 5 0 0 \\} | 2 \\leq | S | \\leq s \\} ,", + "type": "interline_equation", + "image_path": "860264e3b1bc307d4d39e54d56156661aa9da6effedd3a98b89505a6f08bb178.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 202, + 462, + 407, + 477 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 478, + 461, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 461, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 133, + 493 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 479, + 144, + 488 + ], + "score": 0.87, + "content": "S ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 476, + 331, + 493 + ], + "score": 1.0, + "content": "is the set of all the samples. Then we estimate", + "type": "text" + }, + { + "bbox": [ + 331, + 480, + 349, + 489 + ], + "score": 0.91, + "content": "\\bar { \\sigma } _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 476, + 461, + 493 + ], + "score": 1.0, + "content": "with the following quantity:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 492, + 401, + 514 + ], + "lines": [ + { + "bbox": [ + 210, + 492, + 401, + 514 + ], + "spans": [ + { + "bbox": [ + 210, + 492, + 401, + 514 + ], + "score": 0.94, + "content": "\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } ) = \\operatorname* { m i n } _ { \\mathbb { S } \\in \\mathcal { S } ^ { \\prime } } \\left\\{ \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\right\\}", + "type": "interline_equation", + "image_path": "93eb1491bde8aa9fe638becf5b448d70ec1795a5b8a231562e871594acb35687.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 210, + 492, + 401, + 514 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 231, + 528 + ], + "score": 1.0, + "content": "Furthermore, we use the same", + "type": "text" + }, + { + "bbox": [ + 232, + 517, + 242, + 527 + ], + "score": 0.64, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 516, + 379, + 528 + ], + "score": 1.0, + "content": "as that in Section 5 and generate", + "type": "text" + }, + { + "bbox": [ + 380, + 516, + 420, + 527 + ], + "score": 0.44, + "content": "1 0 0 0 ~ \\mathbf { W } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "with each entry i.i.d", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "sampled from the normal distribution. Then we normalize each column of the generated Ws. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 441, + 551 + ], + "score": 1.0, + "content": "technique is commonly used in sparse coding. Finally, we report the distribution of", + "type": "text" + }, + { + "bbox": [ + 441, + 538, + 483, + 550 + ], + "score": 0.92, + "content": "\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 550, + 312, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 143, + 561 + ], + "score": 1.0, + "content": "the fixed", + "type": "text" + }, + { + "bbox": [ + 144, + 550, + 154, + 559 + ], + "score": 0.55, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 550, + 312, + 561 + ], + "score": 1.0, + "content": "and the 1000 sampled Ws in Figure 5.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 276, + 578 + ], + "score": 1.0, + "content": "Figure 5 demonstrates that, with the fixed", + "type": "text" + }, + { + "bbox": [ + 276, + 567, + 286, + 577 + ], + "score": 0.57, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 566, + 461, + 578 + ], + "score": 1.0, + "content": ", most of the random generated Ws have a", + "type": "text" + }, + { + "bbox": [ + 461, + 566, + 504, + 578 + ], + "score": 0.92, + "content": "\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "within the interval [0.25, 0.35]. Thus, the numerical results support our claim: with high probability,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 236, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 236, + 600 + ], + "score": 1.0, + "content": "a random generated W satisfies", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 601, + 398, + 624 + ], + "lines": [ + { + "bbox": [ + 211, + 601, + 398, + 624 + ], + "spans": [ + { + "bbox": [ + 211, + 601, + 398, + 624 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbb { S } \\in \\mathcal { S } } \\left\\{ \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\right\\} \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } > 0 ,", + "type": "interline_equation", + "image_path": "5710d2058752b6b4c88c514c9dd9973f4677cbad7589f17042c8a0c2e8d5fd21.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 211, + 601, + 398, + 624 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 221, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 218, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 135, + 642 + ], + "score": 1.0, + "content": "that is,", + "type": "text" + }, + { + "bbox": [ + 136, + 626, + 218, + 640 + ], + "score": 0.91, + "content": "\\mathbf { W } \\in \\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } )", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 105, + 655, + 432, + 668 + ], + "lines": [ + { + "bbox": [ + 104, + 654, + 433, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 433, + 669 + ], + "score": 1.0, + "content": "E EFFICIENT ALGORITHM TO CALCULATE ANALYTIC WEIGHTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 106, + 678, + 313, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 313, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 313, + 692 + ], + "score": 1.0, + "content": "E.1 AN EFFICIENT ALGORITHM TO SOLVE (16)", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 104, + 699, + 496, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 498, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 498, + 713 + ], + "score": 1.0, + "content": "In this section, we introduce an algorithm to solve (16) (we copy (16) below to facilitate reading):", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 713, + 444, + 735 + ], + "lines": [ + { + "bbox": [ + 164, + 713, + 444, + 735 + ], + "spans": [ + { + "bbox": [ + 164, + 713, + 444, + 735 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\left\\| \\mathbf { W } ^ { T } \\mathbf { D } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( \\mathbf { W } _ { : , m } \\right) ^ { T } \\mathbf { D } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M ,", + "type": "interline_equation", + "image_path": "54e994429acc9adf0525b7d249640894e7f774f59fc7639483fecff56ad39cdf.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 164, + 713, + 444, + 735 + ], + "spans": [], + "index": 38 + } + ] + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "26", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 130, + 80, + 314, + 96 + ], + "lines": [ + { + "bbox": [ + 128, + 74, + 319, + 103 + ], + "spans": [ + { + "bbox": [ + 128, + 79, + 163, + 94 + ], + "score": 1.0, + "content": "1. F N", + "type": "text" + }, + { + "bbox": [ + 142, + 81, + 211, + 95 + ], + "score": 0.93, + "content": "F _ { \\mathrm { c o n v } } ^ { N } \\xrightarrow [ ] { \\mathrm { e } } F _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 74, + 319, + 103 + ], + "score": 1.0, + "content": ". This is proved in Step 2.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 128, + 74, + 319, + 103 + ] + }, + { + "type": "text", + "bbox": [ + 124, + 101, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 123, + 93, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 123, + 93, + 142, + 128 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 101, + 173, + 115 + ], + "score": 0.89, + "content": "F _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 93, + 281, + 128 + ], + "score": 1.0, + "content": "vel bounded. 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F 2D−1", + "type": "text" + }, + { + "bbox": [ + 171, + 181, + 212, + 198 + ], + "score": 1.0, + "content": "and F Ncon", + "type": "text" + }, + { + "bbox": [ + 216, + 184, + 505, + 197 + ], + "score": 1.0, + "content": "are all lower semi-continuous and proper. 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If we arbitrarily pick", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 101, + 236, + 509, + 261 + ], + "spans": [ + { + "bbox": [ + 101, + 236, + 114, + 261 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 242, + 172, + 255 + ], + "score": 0.91, + "content": "\\mathbf { w } ^ { N } \\in \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 236, + 204, + 261 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 204, + 244, + 224, + 255 + ], + "score": 0.88, + "content": "\\mathbf { w } _ { \\mathrm { c i r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 236, + 317, + 261 + ], + "score": 1.0, + "content": "convbe the unique point in", + "type": "text" + }, + { + "bbox": [ + 317, + 242, + 350, + 255 + ], + "score": 0.92, + "content": "\\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 236, + 509, + 261 + ], + "score": 1.0, + "content": ". Applying Proposition 7.33, we have", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 253, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 159, + 267 + ], + "score": 0.89, + "content": "\\mathbf { w } ^ { N } \\to \\mathbf { w } _ { \\mathrm { c i r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 253, + 174, + 288 + ], + "score": 1.0, + "content": ". Bsets", + "type": "text" + }, + { + "bbox": [ + 212, + 253, + 216, + 288 + ], + "score": 1.0, + "content": "o:", + "type": "text" + }, + { + "bbox": [ + 329, + 253, + 506, + 288 + ], + "score": 1.0, + "content": "ts, 2009), we obtain the convergence of the.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 174, + 266, + 329, + 281 + ], + "spans": [ + { + "bbox": [ + 174, + 266, + 212, + 280 + ], + "score": 0.88, + "content": "\\{ \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 266, + 329, + 281 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { N \\to \\infty } \\mathcal { W } _ { \\mathrm { c o n v } } ^ { N } = \\mathcal { W } _ { \\mathrm { c i r } } ^ { 2 D - 1 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 101, + 228, + 509, + 288 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 294, + 309, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 309, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 309, + 310 + ], + "score": 1.0, + "content": "D DISCUSSION OF DEFINITION 2 (11)", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 318, + 506, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 365, + 332 + ], + "score": 1.0, + "content": "In this section, we want to numerically show that, given typical", + "type": "text" + }, + { + "bbox": [ + 366, + 320, + 376, + 330 + ], + "score": 0.77, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 318, + 394, + 332 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 322, + 401, + 329 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 318, + 444, + 332 + ], + "score": 1.0, + "content": ", there is a", + "type": "text" + }, + { + "bbox": [ + 445, + 320, + 483, + 331 + ], + "score": 0.91, + "content": "\\bar { \\sigma } _ { \\operatorname* { m i n } } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 235, + 343 + ], + "score": 1.0, + "content": "that a random generated matrix", + "type": "text" + }, + { + "bbox": [ + 235, + 330, + 318, + 342 + ], + "score": 0.92, + "content": "\\mathbf { W } \\in \\mathbf { \\bar { \\mathcal { W } } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 330, + 389, + 343 + ], + "score": 1.0, + "content": ". However, given", + "type": "text" + }, + { + "bbox": [ + 389, + 331, + 399, + 341 + ], + "score": 0.68, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 330, + 418, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 418, + 331, + 431, + 341 + ], + "score": 0.61, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ", it’s intractable to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 342, + 201, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 201, + 354 + ], + "score": 1.0, + "content": "completely check (11):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 318, + 506, + 354 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 354, + 421, + 376 + ], + "lines": [ + { + "bbox": [ + 189, + 354, + 421, + 376 + ], + "spans": [ + { + "bbox": [ + 189, + 354, + 421, + 376 + ], + "score": 0.91, + "content": "\\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } , \\forall \\mathbb { S } \\mathrm { ~ w i t h ~ } 2 \\leq | \\mathbb { S } | \\leq s .", + "type": "interline_equation", + "image_path": "07141b11e27bd14a85e864e17e896eb6aeae5c72a3f3d77710ec420be89f17eb.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 189, + 354, + 421, + 376 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 378, + 503, + 400 + ], + "lines": [ + { + "bbox": [ + 107, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 107, + 376, + 367, + 390 + ], + "score": 1.0, + "content": "The reason is that there are extremely large amount of possible", + "type": "text" + }, + { + "bbox": [ + 367, + 378, + 375, + 388 + ], + "score": 0.5, + "content": "\\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 376, + 479, + 390 + ], + "score": 1.0, + "content": "s. For example, we take", + "type": "text" + }, + { + "bbox": [ + 480, + 378, + 505, + 389 + ], + "score": 0.85, + "content": "M =", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 388, + 269, + 401 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 197, + 401 + ], + "score": 0.9, + "content": "2 5 0 , N = 5 0 0 , s = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 388, + 269, + 401 + ], + "score": 1.0, + "content": ". There are totally", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 107, + 376, + 505, + 401 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 402, + 375, + 430 + ], + "lines": [ + { + "bbox": [ + 236, + 402, + 375, + 430 + ], + "spans": [ + { + "bbox": [ + 236, + 402, + 375, + 430 + ], + "score": 0.93, + "content": "\\binom { 5 0 0 } { 5 0 } + \\binom { 5 0 0 } { 4 9 } + \\cdot \\cdot \\cdot + \\binom { 5 0 0 } { 2 }", + "type": "interline_equation", + "image_path": "f6643be7dea0c01988edc5faba3ed58b1fd06155a47ecc0ffae96d841bf2d70d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 236, + 402, + 375, + 430 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 438, + 444 + ], + "lines": [ + { + "bbox": [ + 104, + 430, + 438, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 430, + 141, + 446 + ], + "score": 1.0, + "content": "possible", + "type": "text" + }, + { + "bbox": [ + 142, + 432, + 153, + 442 + ], + "score": 0.6, + "content": "\\mathbb { S } s", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 430, + 195, + 446 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 195, + 432, + 247, + 444 + ], + "score": 0.94, + "content": "2 \\leq | \\mathbb { S } | \\leq s", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 430, + 426, + 446 + ], + "score": 1.0, + "content": ". It’s impossible to check (11) on all possible", + "type": "text" + }, + { + "bbox": [ + 427, + 432, + 438, + 442 + ], + "score": 0.31, + "content": "\\mathbb { S } s", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 430, + 438, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 406, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 408, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 234, + 462 + ], + "score": 1.0, + "content": "Instead of checking all possible", + "type": "text" + }, + { + "bbox": [ + 234, + 449, + 245, + 459 + ], + "score": 0.72, + "content": "\\mathbb { S } s", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 448, + 294, + 462 + ], + "score": 1.0, + "content": ", we sample", + "type": "text" + }, + { + "bbox": [ + 294, + 449, + 327, + 460 + ], + "score": 0.57, + "content": "5 0 0 0 \\mathbb { S } \\mathrm { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 448, + 408, + 462 + ], + "score": 1.0, + "content": "from the whole set:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 448, + 408, + 462 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 462, + 407, + 477 + ], + "lines": [ + { + "bbox": [ + 202, + 462, + 407, + 477 + ], + "spans": [ + { + "bbox": [ + 202, + 462, + 407, + 477 + ], + "score": 0.87, + "content": "\\mathcal S ^ { \\prime } \\subset S = \\{ \\mathbb S : \\mathbb S \\subset \\{ 1 , 2 , \\cdots , 5 0 0 \\} | 2 \\leq | S | \\leq s \\} ,", + "type": "interline_equation", + "image_path": "860264e3b1bc307d4d39e54d56156661aa9da6effedd3a98b89505a6f08bb178.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 202, + 462, + 407, + 477 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 478, + 461, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 461, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 133, + 493 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 479, + 144, + 488 + ], + "score": 0.87, + "content": "S ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 476, + 331, + 493 + ], + "score": 1.0, + "content": "is the set of all the samples. Then we estimate", + "type": "text" + }, + { + "bbox": [ + 331, + 480, + 349, + 489 + ], + "score": 0.91, + "content": "\\bar { \\sigma } _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 476, + 461, + 493 + ], + "score": 1.0, + "content": "with the following quantity:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 476, + 461, + 493 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 492, + 401, + 514 + ], + "lines": [ + { + "bbox": [ + 210, + 492, + 401, + 514 + ], + "spans": [ + { + "bbox": [ + 210, + 492, + 401, + 514 + ], + "score": 0.94, + "content": "\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } ) = \\operatorname* { m i n } _ { \\mathbb { S } \\in \\mathcal { S } ^ { \\prime } } \\left\\{ \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\right\\}", + "type": "interline_equation", + "image_path": "93eb1491bde8aa9fe638becf5b448d70ec1795a5b8a231562e871594acb35687.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 210, + 492, + 401, + 514 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 231, + 528 + ], + "score": 1.0, + "content": "Furthermore, we use the same", + "type": "text" + }, + { + "bbox": [ + 232, + 517, + 242, + 527 + ], + "score": 0.64, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 516, + 379, + 528 + ], + "score": 1.0, + "content": "as that in Section 5 and generate", + "type": "text" + }, + { + "bbox": [ + 380, + 516, + 420, + 527 + ], + "score": 0.44, + "content": "1 0 0 0 ~ \\mathbf { W } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "with each entry i.i.d", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "sampled from the normal distribution. Then we normalize each column of the generated Ws. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 441, + 551 + ], + "score": 1.0, + "content": "technique is commonly used in sparse coding. Finally, we report the distribution of", + "type": "text" + }, + { + "bbox": [ + 441, + 538, + 483, + 550 + ], + "score": 0.92, + "content": "\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 550, + 312, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 143, + 561 + ], + "score": 1.0, + "content": "the fixed", + "type": "text" + }, + { + "bbox": [ + 144, + 550, + 154, + 559 + ], + "score": 0.55, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 550, + 312, + 561 + ], + "score": 1.0, + "content": "and the 1000 sampled Ws in Figure 5.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 516, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 276, + 578 + ], + "score": 1.0, + "content": "Figure 5 demonstrates that, with the fixed", + "type": "text" + }, + { + "bbox": [ + 276, + 567, + 286, + 577 + ], + "score": 0.57, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 566, + 461, + 578 + ], + "score": 1.0, + "content": ", most of the random generated Ws have a", + "type": "text" + }, + { + "bbox": [ + 461, + 566, + 504, + 578 + ], + "score": 0.92, + "content": "\\bar { \\sigma } ^ { \\prime } ( \\mathbf { D } , \\mathbf { W } )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "within the interval [0.25, 0.35]. Thus, the numerical results support our claim: with high probability,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 236, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 236, + 600 + ], + "score": 1.0, + "content": "a random generated W satisfies", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 566, + 505, + 600 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 601, + 398, + 624 + ], + "lines": [ + { + "bbox": [ + 211, + 601, + 398, + 624 + ], + "spans": [ + { + "bbox": [ + 211, + 601, + 398, + 624 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbb { S } \\in \\mathcal { S } } \\left\\{ \\sigma _ { \\operatorname* { m i n } } \\Big ( \\mathbf { I } - ( \\mathbf { W } _ { : , \\mathbb { S } } ) ^ { T } \\mathbf { D } _ { : , \\mathbb { S } } \\Big ) \\right\\} \\geq \\bar { \\sigma } _ { \\operatorname* { m i n } } > 0 ,", + "type": "interline_equation", + "image_path": "5710d2058752b6b4c88c514c9dd9973f4677cbad7589f17042c8a0c2e8d5fd21.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 211, + 601, + 398, + 624 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 221, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 218, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 135, + 642 + ], + "score": 1.0, + "content": "that is,", + "type": "text" + }, + { + "bbox": [ + 136, + 626, + 218, + 640 + ], + "score": 0.91, + "content": "\\mathbf { W } \\in \\bar { \\mathcal { W } } ( \\mathbf { D } , s , \\bar { \\sigma } _ { \\operatorname* { m i n } } )", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 624, + 218, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 655, + 432, + 668 + ], + "lines": [ + { + "bbox": [ + 104, + 654, + 433, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 433, + 669 + ], + "score": 1.0, + "content": "E EFFICIENT ALGORITHM TO CALCULATE ANALYTIC WEIGHTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 106, + 678, + 313, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 313, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 313, + 692 + ], + "score": 1.0, + "content": "E.1 AN EFFICIENT ALGORITHM TO SOLVE (16)", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 104, + 699, + 496, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 498, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 498, + 713 + ], + "score": 1.0, + "content": "In this section, we introduce an algorithm to solve (16) (we copy (16) below to facilitate reading):", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 699, + 498, + 713 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 713, + 444, + 735 + ], + "lines": [ + { + "bbox": [ + 164, + 713, + 444, + 735 + ], + "spans": [ + { + "bbox": [ + 164, + 713, + 444, + 735 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\left\\| \\mathbf { W } ^ { T } \\mathbf { D } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( \\mathbf { W } _ { : , m } \\right) ^ { T } \\mathbf { D } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M ,", + "type": "interline_equation", + "image_path": "54e994429acc9adf0525b7d249640894e7f774f59fc7639483fecff56ad39cdf.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 164, + 713, + 444, + 735 + ], + "spans": [], + "index": 38 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 153, + 83, + 459, + 198 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 153, + 83, + 459, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 153, + 83, + 459, + 198 + ], + "spans": [ 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\\mathbf { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 205, + 487, + 220 + ], + "score": 1.0, + "content": "on random generated Ws.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 320, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 320, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 320, + 251 + ], + "score": 1.0, + "content": "By the definition of the Frobenius norm, it holds that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 254, + 394, + 270 + ], + "lines": [ + { + "bbox": [ + 216, + 254, + 394, + 270 + ], + "spans": [ + { + "bbox": [ + 216, + 254, + 394, + 270 + ], + "score": 0.92, + "content": "\\| \\mathbf { W } ^ { T } \\mathbf { D } \\| _ { F } ^ { 2 } = \\| ( \\mathbf { W } ^ { T } \\mathbf { D } ) ^ { T } \\| _ { F } ^ { 2 } = \\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "3521703d63a855830bfac0099139782dc210146e4a0443fcf2b2a55ea229b331.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 216, + 254, + 394, + 270 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 274, + 281, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 280, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 280, + 286 + ], + "score": 1.0, + "content": "Thus, the above problem is equivalent with", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 289, + 445, + 312 + ], + "lines": [ + { + "bbox": [ + 165, + 289, + 445, + 312 + ], + "spans": [ + { + "bbox": [ + 165, + 289, + 445, + 312 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } \\left\\| \\mathbf { D } ^ { T } \\mathbf { W } \\right\\| _ { F } ^ { 2 } , \\quad \\mathrm { s . t . } \\left( \\mathbf { D } _ { : , m } \\right) ^ { T } \\mathbf { W } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M .", + "type": "interline_equation", + "image_path": "fd51096bf293bfd8357e5592ab63420954a8148a0a1dd7f7661ed556112c8b3e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 165, + 289, + 445, + 312 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 504, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 504, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 459, + 332 + ], + "score": 1.0, + "content": "We apply projected gradient descent (PGD) to solve the above problem. The gradient of", + "type": "text" + }, + { + "bbox": [ + 460, + 317, + 504, + 330 + ], + "score": 0.93, + "content": "\\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 326, + 295, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 116, + 344 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 329, + 218, + 342 + ], + "score": 0.92, + "content": "\\nabla \\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 } = \\mathbf { D } \\mathbf { D } ^ { T } \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 326, + 295, + 344 + ], + "score": 1.0, + "content": ". Denote the set by", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 346, + 434, + 362 + ], + "lines": [ + { + "bbox": [ + 177, + 346, + 434, + 362 + ], + "spans": [ + { + "bbox": [ + 177, + 346, + 434, + 362 + ], + "score": 0.88, + "content": "\\mathcal { W } = \\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } | ( \\mathbf { D } _ { : , m } ) ^ { T } \\mathbf { W } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M . \\}", + "type": "interline_equation", + "image_path": "b041b6efdf8d30dd617d402f804c675234db7280653b230fc9008f9cf73b613e.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 177, + 346, + 434, + 362 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 302, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 304, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 207, + 380 + ], + "score": 1.0, + "content": "Then the projection onto", + "type": "text" + }, + { + "bbox": [ + 207, + 366, + 219, + 376 + ], + "score": 0.83, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 363, + 304, + 380 + ], + "score": 1.0, + "content": "can be calculated by", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 504, + 412 + ], + "lines": [ + { + "bbox": [ + 111, + 380, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 111, + 380, + 504, + 396 + ], + "score": 0.85, + "content": "\\mathrm { \\mathrm { \\mathrm { ~ \\ p ~ } } } _ { \\mathrm { r o j } _ { \\mathcal { W } } } ( { \\bf W } ) = { \\bf W } + \\Delta { \\bf W } , \\Delta { \\bf W } = \\left[ ( 1 - ( { \\bf D } _ { : , 1 } ) ^ { T } { \\bf W } _ { : , 1 } ) { \\bf W } _ { : , 1 } , \\ \\cdots , \\ ( 1 - ( { \\bf D } _ { : , M } ) ^ { T } { \\bf W } _ { : , M } ) { \\bf W } _ { : , M } \\right]", + "type": "inline_equation", + "image_path": "460c23b47cdc3904e7aae93c0b6079da811e090bc15287fe58698ce4f4b6667e.jpg" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 398, + 461, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 461, + 413 + ], + "score": 1.0, + "content": "With these formulas, we are able to write down the PGD, which is listed in Algorithm 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 336, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 337, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 337, + 439 + ], + "score": 1.0, + "content": "Algorithm 1: Projected gradient descent for solving (16)", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 236, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 236, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 236, + 454 + ], + "score": 1.0, + "content": "Input: Dictionary D ∈ RN×M .", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 450, + 210, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 165, + 465 + ], + "score": 1.0, + "content": "Initialize: Let", + "type": "text" + }, + { + "bbox": [ + 166, + 452, + 205, + 463 + ], + "score": 0.9, + "content": "\\dot { \\mathbf { W } } ^ { 0 } = \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 450, + 210, + 465 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 100, + 464, + 268, + 475 + ], + "lines": [ + { + "bbox": [ + 97, + 462, + 267, + 476 + ], + "spans": [ + { + "bbox": [ + 97, + 462, + 121, + 476 + ], + "score": 1.0, + "content": "1 for", + "type": "text" + }, + { + "bbox": [ + 122, + 464, + 180, + 475 + ], + "score": 0.87, + "content": "j = 0 , 1 , 2 , \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 462, + 267, + 476 + ], + "score": 1.0, + "content": "until convergence do", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 98, + 494, + 124, + 504 + ], + "lines": [ + { + "bbox": [ + 95, + 492, + 126, + 506 + ], + "spans": [ + { + "bbox": [ + 95, + 492, + 126, + 506 + ], + "score": 1.0, + "content": "3 end", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 270, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 273, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 143, + 519 + ], + "score": 1.0, + "content": "Output:", + "type": "text" + }, + { + "bbox": [ + 144, + 505, + 163, + 516 + ], + "score": 0.84, + "content": "\\mathbf { W } ^ { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 504, + 193, + 519 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 194, + 506, + 201, + 516 + ], + "score": 0.8, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 504, + 273, + 519 + ], + "score": 1.0, + "content": "is the last iterate.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 351, + 553 + ], + "score": 1.0, + "content": "In each step, calculating the gradient has the complexity of", + "type": "text" + }, + { + "bbox": [ + 352, + 540, + 393, + 552 + ], + "score": 0.93, + "content": "O ( N ^ { 2 } M )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 538, + 429, + 553 + ], + "score": 1.0, + "content": "because", + "type": "text" + }, + { + "bbox": [ + 429, + 539, + 455, + 551 + ], + "score": 0.88, + "content": "\\mathbf { D } \\mathbf { D } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "can be pre-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 280, + 564 + ], + "score": 1.0, + "content": "computed. Calculating the projection takes", + "type": "text" + }, + { + "bbox": [ + 280, + 551, + 316, + 563 + ], + "score": 0.92, + "content": "O ( N M )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "time consumptions. Due to the objective func-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "tion to minimize in (16) is restricted strongly convex, Algorithm 1 is linear convergent (Zhang &", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 205, + 586 + ], + "score": 1.0, + "content": "Cheng, 2015). To get an", + "type": "text" + }, + { + "bbox": [ + 205, + 576, + 210, + 583 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 573, + 330, + 586 + ], + "score": 1.0, + "content": "-accurate solution, PGD takes", + "type": "text" + }, + { + "bbox": [ + 330, + 573, + 380, + 585 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\log ( 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "steps. Thus, the complexity of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 582, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 168, + 597 + ], + "score": 1.0, + "content": "Algorithm 1 is", + "type": "text" + }, + { + "bbox": [ + 168, + 584, + 243, + 596 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\log ( 1 / \\epsilon ) N ^ { 2 } M )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 582, + 506, + 597 + ], + "score": 1.0, + "content": ". We should note that the bounds given in Table 1 are the number", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "of parameters to train, not the training complexity. The training complexity can be estimated by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 605, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 196, + 619 + ], + "score": 1.0, + "content": "“Number of iterations", + "type": "text" + }, + { + "bbox": [ + 197, + 607, + 207, + 616 + ], + "score": 0.76, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 605, + 362, + 619 + ], + "score": 1.0, + "content": "complexity of back-propagation”, i.e.,", + "type": "text" + }, + { + "bbox": [ + 362, + 606, + 421, + 618 + ], + "score": 0.9, + "content": "\\bar { O } ( I B \\bar { K } N \\bar { M } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 605, + 449, + 619 + ], + "score": 1.0, + "content": ",where", + "type": "text" + }, + { + "bbox": [ + 450, + 606, + 456, + 616 + ], + "score": 0.73, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 605, + 505, + 619 + ], + "score": 1.0, + "content": "is the num-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 220, + 630 + ], + "score": 1.0, + "content": "ber of iterations for training,", + "type": "text" + }, + { + "bbox": [ + 221, + 617, + 230, + 627 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 616, + 316, + 630 + ], + "score": 1.0, + "content": "is the batch size , and", + "type": "text" + }, + { + "bbox": [ + 316, + 617, + 326, + 627 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "is the number of layers. Actually, Algorithm", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 331, + 641 + ], + "score": 1.0, + "content": "1 (Stage 1) only takes a few seconds on an example of", + "type": "text" + }, + { + "bbox": [ + 331, + 628, + 394, + 639 + ], + "score": 0.9, + "content": "\\mathbf { D } : 2 5 0 \\times 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 627, + 506, + 641 + ], + "score": 1.0, + "content": ", while the training process", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 639, + 347, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 347, + 652 + ], + "score": 1.0, + "content": "(Stage 2) of, for example, ALISTA, takes around 0.1 hours.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 106, + 663, + 313, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 313, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 313, + 676 + ], + "score": 1.0, + "content": "E.2 AN EFFICIENT ALGORITHM TO SOLVE (24)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 105, + 683, + 496, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 683, + 498, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 498, + 697 + ], + "score": 1.0, + "content": "In this section, we introduce an algorithm to solve (24) (we copy (24) below to facilitate reading):", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 700, + 397, + 735 + ], + "lines": [ + { + "bbox": [ + 212, + 700, + 397, + 735 + ], + "spans": [ + { + "bbox": [ + 212, + 700, + 397, + 735 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } } \\Big \\| \\big ( \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\big ) ^ { T } 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The gradient of", + "type": "text" + }, + { + "bbox": [ + 460, + 317, + 504, + 330 + ], + "score": 0.93, + "content": "\\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 326, + 295, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 116, + 344 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 329, + 218, + 342 + ], + "score": 0.92, + "content": "\\nabla \\| \\mathbf { D } ^ { T } \\mathbf { W } \\| _ { F } ^ { 2 } = \\mathbf { D } \\mathbf { D } ^ { T } \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 326, + 295, + 344 + ], + "score": 1.0, + "content": ". Denote the set by", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 315, + 504, + 344 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 346, + 434, + 362 + ], + "lines": [ + { + "bbox": [ + 177, + 346, + 434, + 362 + ], + "spans": [ + { + "bbox": [ + 177, + 346, + 434, + 362 + ], + "score": 0.88, + "content": "\\mathcal { W } = \\{ \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } | ( \\mathbf { D } _ { : , m } ) ^ { T } \\mathbf { W } _ { : , m } = 1 , \\forall m = 1 , 2 , \\cdots , M . \\}", + "type": "interline_equation", + "image_path": "b041b6efdf8d30dd617d402f804c675234db7280653b230fc9008f9cf73b613e.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 177, + 346, + 434, + 362 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 302, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 304, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 207, + 380 + ], + "score": 1.0, + "content": "Then the projection onto", + "type": "text" + }, + { + "bbox": [ + 207, + 366, + 219, + 376 + ], + "score": 0.83, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 363, + 304, + 380 + ], + "score": 1.0, + "content": "can be calculated by", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 363, + 304, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 504, + 412 + ], + "lines": [ + { + "bbox": [ + 111, + 380, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 111, + 380, + 504, + 396 + ], + "score": 0.85, + "content": "\\mathrm { \\mathrm { \\mathrm { ~ \\ p ~ } } } _ { \\mathrm { r o j } _ { \\mathcal { W } } } ( { \\bf W } ) = { \\bf W } + \\Delta { \\bf W } , \\Delta { \\bf W } = \\left[ ( 1 - ( { \\bf D } _ { : , 1 } ) ^ { T } { \\bf W } _ { : , 1 } ) { \\bf W } _ { : , 1 } , \\ \\cdots , \\ ( 1 - ( { \\bf D } _ { : , M } ) ^ { T } { \\bf W } _ { : , M } ) { \\bf W } _ { : , M } \\right]", + "type": "inline_equation", + "image_path": "460c23b47cdc3904e7aae93c0b6079da811e090bc15287fe58698ce4f4b6667e.jpg" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 398, + 461, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 461, + 413 + ], + "score": 1.0, + "content": "With these formulas, we are able to write down the PGD, which is listed in Algorithm 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 380, + 504, + 413 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 336, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 337, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 337, + 439 + ], + "score": 1.0, + "content": "Algorithm 1: Projected gradient descent for solving (16)", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "list", + "bbox": [ + 106, + 441, + 236, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 236, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 236, + 454 + ], + "score": 1.0, + "content": "Input: Dictionary D ∈ RN×M .", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 450, + 210, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 165, + 465 + ], + "score": 1.0, + "content": "Initialize: Let", + "type": "text" + }, + { + "bbox": [ + 166, + 452, + 205, + 463 + ], + "score": 0.9, + "content": "\\dot { \\mathbf { W } } ^ { 0 } = \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 450, + 210, + 465 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 439, + 236, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 464, + 268, + 475 + ], + "lines": [ + { + "bbox": [ + 97, + 462, + 267, + 476 + ], + "spans": [ + { + "bbox": [ + 97, + 462, + 121, + 476 + ], + "score": 1.0, + "content": "1 for", + "type": "text" + }, + { + "bbox": [ + 122, + 464, + 180, + 475 + ], + "score": 0.87, + "content": "j = 0 , 1 , 2 , \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 462, + 267, + 476 + ], + "score": 1.0, + "content": "until convergence do", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 97, + 462, + 267, + 476 + ] + }, + { + "type": "title", + "bbox": [ + 98, + 494, + 124, + 504 + ], + "lines": [ + { + "bbox": [ + 95, + 492, + 126, + 506 + ], + "spans": [ + { + "bbox": [ + 95, + 492, + 126, + 506 + ], + "score": 1.0, + "content": "3 end", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 270, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 273, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 143, + 519 + ], + "score": 1.0, + "content": "Output:", + "type": "text" + }, + { + "bbox": [ + 144, + 505, + 163, + 516 + ], + "score": 0.84, + "content": "\\mathbf { W } ^ { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 504, + 193, + 519 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 194, + 506, + 201, + 516 + ], + "score": 0.8, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 504, + 273, + 519 + ], + "score": 1.0, + "content": "is the last iterate.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 504, + 273, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 351, + 553 + ], + "score": 1.0, + "content": "In each step, calculating the gradient has the complexity of", + "type": "text" + }, + { + "bbox": [ + 352, + 540, + 393, + 552 + ], + "score": 0.93, + "content": "O ( N ^ { 2 } M )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 538, + 429, + 553 + ], + "score": 1.0, + "content": "because", + "type": "text" + }, + { + "bbox": [ + 429, + 539, + 455, + 551 + ], + "score": 0.88, + "content": "\\mathbf { D } \\mathbf { D } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "can be pre-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 280, + 564 + ], + "score": 1.0, + "content": "computed. Calculating the projection takes", + "type": "text" + }, + { + "bbox": [ + 280, + 551, + 316, + 563 + ], + "score": 0.92, + "content": "O ( N M )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "time consumptions. Due to the objective func-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "tion to minimize in (16) is restricted strongly convex, Algorithm 1 is linear convergent (Zhang &", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 205, + 586 + ], + "score": 1.0, + "content": "Cheng, 2015). To get an", + "type": "text" + }, + { + "bbox": [ + 205, + 576, + 210, + 583 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 573, + 330, + 586 + ], + "score": 1.0, + "content": "-accurate solution, PGD takes", + "type": "text" + }, + { + "bbox": [ + 330, + 573, + 380, + 585 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\log ( 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "steps. Thus, the complexity of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 582, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 168, + 597 + ], + "score": 1.0, + "content": "Algorithm 1 is", + "type": "text" + }, + { + "bbox": [ + 168, + 584, + 243, + 596 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\log ( 1 / \\epsilon ) N ^ { 2 } M )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 582, + 506, + 597 + ], + "score": 1.0, + "content": ". We should note that the bounds given in Table 1 are the number", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "of parameters to train, not the training complexity. The training complexity can be estimated by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 605, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 196, + 619 + ], + "score": 1.0, + "content": "“Number of iterations", + "type": "text" + }, + { + "bbox": [ + 197, + 607, + 207, + 616 + ], + "score": 0.76, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 605, + 362, + 619 + ], + "score": 1.0, + "content": "complexity of back-propagation”, i.e.,", + "type": "text" + }, + { + "bbox": [ + 362, + 606, + 421, + 618 + ], + "score": 0.9, + "content": "\\bar { O } ( I B \\bar { K } N \\bar { M } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 605, + 449, + 619 + ], + "score": 1.0, + "content": ",where", + "type": "text" + }, + { + "bbox": [ + 450, + 606, + 456, + 616 + ], + "score": 0.73, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 605, + 505, + 619 + ], + "score": 1.0, + "content": "is the num-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 220, + 630 + ], + "score": 1.0, + "content": "ber of iterations for training,", + "type": "text" + }, + { + "bbox": [ + 221, + 617, + 230, + 627 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 616, + 316, + 630 + ], + "score": 1.0, + "content": "is the batch size , and", + "type": "text" + }, + { + "bbox": [ + 316, + 617, + 326, + 627 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "is the number of layers. Actually, Algorithm", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 331, + 641 + ], + "score": 1.0, + "content": "1 (Stage 1) only takes a few seconds on an example of", + "type": "text" + }, + { + "bbox": [ + 331, + 628, + 394, + 639 + ], + "score": 0.9, + "content": "\\mathbf { D } : 2 5 0 \\times 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 627, + 506, + 641 + ], + "score": 1.0, + "content": ", while the training process", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 639, + 347, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 347, + 652 + ], + "score": 1.0, + "content": "(Stage 2) of, for example, ALISTA, takes around 0.1 hours.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 538, + 506, + 652 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 663, + 313, + 675 + ], + "lines": [ + { + "bbox": [ 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\\big ( \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\big ) ^ { T } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\Big \\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "9fe9c001e54b9c083124e44b29b653f316878f14140e9920e36660cd1f03207d.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 700, + 397, + 717.5 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 212, + 717.5, + 397, + 735.0 + ], + "spans": [], + "index": 33 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 331, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 331, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 331, + 97 + ], + "score": 1.0, + "content": "Similarly, by (58), the above problem is equivalent with", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 99, + 398, + 135 + ], + "lines": 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circular convolution is very efficient to calculate in the frequency domain, we consider", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 326, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 326, + 162 + ], + "score": 1.0, + "content": "solving (59) utilizing the fast Fourier transform (FFT).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 505, + 202 + ], + "lines": [ + { + "bbox": [ + 104, + 165, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 165, + 252, + 182 + ], + "score": 1.0, + "content": "Firstly, we introduce the operators", + "type": "text" + }, + { + "bbox": [ + 252, + 165, + 327, + 179 + ], + "score": 0.8, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) , \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 165, + 506, + 182 + ], + "score": 1.0, + "content": "in the frequency domain. To simplify the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 102, + 175, + 508, + 194 + ], + "spans": [ + { + "bbox": [ + 102, + 175, + 253, + 194 + ], + "score": 1.0, + "content": "notation, we denote the operators as", + "type": "text" + }, + { + "bbox": [ + 253, + 179, + 273, + 191 + ], + "score": 0.92, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 175, + 291, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 179, + 314, + 191 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 175, + 384, + 194 + ], + "score": 1.0, + "content": "respectively. Let", + "type": "text" + }, + { + "bbox": [ + 384, + 180, + 394, + 189 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 175, + 508, + 194 + ], + "score": 1.0, + "content": "be the FFT operator. Thus,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 189, + 227, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 152, + 203 + ], + "score": 0.92, + "content": "{ \\bf b } = { \\bf D } _ { \\mathrm { c i r } } ^ { N } { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 189, + 227, + 204 + ], + "score": 1.0, + "content": "is equivalent with", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 201, + 354, + 216 + ], + "lines": [ + { + "bbox": [ + 256, + 201, + 354, + 216 + ], + "spans": [ + { + "bbox": [ + 256, + 201, + 354, + 216 + ], + "score": 0.91, + "content": "\\mathcal { F } \\mathbf { b } = \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\mathcal { F } \\mathbf { x } .", + "type": "interline_equation", + "image_path": "039ad5f5277ea925816fb197cc37b4eca4e1c4dc9e0e4bae721f0c81c9e90458.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 256, + 201, + 354, + 216 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 219, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 103, + 216, + 507, + 237 + ], + "spans": [ + { + "bbox": [ + 103, + 216, + 122, + 237 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 218, + 204, + 232 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { b } } = \\mathcal { F } \\mathbf { b } , \\hat { \\mathbf { x } } = \\mathcal { F } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 216, + 353, + 237 + ], + "score": 1.0, + "content": "be the frequency domain signals, let", + "type": "text" + }, + { + "bbox": [ + 354, + 219, + 433, + 233 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 216, + 507, + 237 + ], + "score": 1.0, + "content": "be the frequency", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 270, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 270, + 244 + ], + "score": 1.0, + "content": "domain operator. The above equation is:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 280, + 248, + 330, + 264 + ], + "lines": [ + { + "bbox": [ + 280, + 248, + 330, + 264 + ], + "spans": [ + { + "bbox": [ + 280, + 248, + 330, + 264 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { b } } = \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } \\hat { \\mathbf { x } } .", + "type": "interline_equation", + "image_path": "9d261e7075b84fbff77469776a876f02d5da1d4123f2b75134300775b6f96731.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 280, + 248, + 330, + 264 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 269, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 508, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 239, + 288 + ], + "score": 1.0, + "content": "The frequency domain operator", + "type": "text" + }, + { + "bbox": [ + 240, + 269, + 259, + 283 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 267, + 456, + 288 + ], + "score": 1.0, + "content": "is much cheaper to calculate than the operator", + "type": "text" + }, + { + "bbox": [ + 457, + 270, + 476, + 282 + ], + "score": 0.9, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 267, + 508, + 288 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 474, + 294 + ], + "score": 1.0, + "content": "spacial domain because it is block diagonal (Wohlberg, 2016). Specifically, we zero pad d to", + "type": "text" + }, + { + "bbox": [ + 475, + 282, + 504, + 292 + ], + "score": 0.9, + "content": "N \\times N", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 292, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 292, + 158, + 307 + ], + "score": 1.0, + "content": "and do FFT:", + "type": "text" + }, + { + "bbox": [ + 158, + 293, + 305, + 307 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { d } } _ { m } = \\mathrm { F F T } \\left( \\mathrm { z e r o p a d } ( \\mathbf { d } _ { m } , N - D ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 292, + 506, + 307 + ], + "score": 1.0, + "content": ", then the above operator can be explicitly written", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 307, + 121, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 121, + 317 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 313, + 346, + 347 + ], + "lines": [ + { + "bbox": [ + 263, + 313, + 346, + 347 + ], + "spans": [ + { + "bbox": [ + 263, + 313, + 346, + 347 + ], + "score": 0.94, + "content": "\\hat { \\mathbf { b } } = \\sum _ { m = 1 } ^ { M } \\overline { { \\hat { \\mathbf { d } } _ { m } } } \\odot \\hat { \\mathbf { x } } _ { m } ,", + "type": "interline_equation", + "image_path": "12be63c474cd1c81de195338a6598529dc8968cca41de5de787ffc3f770437e2.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 313, + 346, + 330.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 263, + 330.0, + 346, + 347.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 304, + 367 + ], + "score": 1.0, + "content": "where ¯· means complex conjugate. This is due to", + "type": "text" + }, + { + "bbox": [ + 304, + 349, + 324, + 363 + ], + "score": 0.91, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 347, + 507, + 367 + ], + "score": 1.0, + "content": "is actually cross-correlation, not convolution", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "(see (18)). Cross-correlation is equal to the transpose of convolution. Thus, there should be complex", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 249, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 249, + 384 + ], + "score": 1.0, + "content": "conjugate in the frequency domain.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 163, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 164, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 164, + 401 + ], + "score": 1.0, + "content": "Further, since", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 404, + 475, + 455 + ], + "lines": [ + { + "bbox": [ + 136, + 404, + 475, + 455 + ], + "spans": [ + { + "bbox": [ + 136, + 404, + 475, + 455 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| ( \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } ) ^ { H } \\hat { \\mathbf { W } } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } = \\| ( \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } ) ^ { H } \\mathcal { F } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\| _ { F } ^ { 2 } = \\| \\mathcal { F } ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\| _ { F } ^ { 2 } } \\\\ & { \\qquad = \\| ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } \\| _ { F } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "e12680ad09d0d0f98d4aedbd125c2b7f2850f2946a995d425347c5adbe4b1daa.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 136, + 404, + 475, + 421.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 136, + 421.0, + 475, + 438.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 136, + 438.0, + 475, + 455.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 234, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 234, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 234, + 470 + ], + "score": 1.0, + "content": "problem (59) is equivalent with", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 474, + 382, + 510 + ], + "lines": [ + { + "bbox": [ + 228, + 474, + 382, + 510 + ], + "spans": [ + { + "bbox": [ + 228, + 474, + 382, + 510 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } } \\Big \\| \\big ( \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } \\big ) ^ { H } \\hat { \\mathbf { W } } _ { \\mathrm { c i r } } ^ { N } \\Big \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "b4cea4b4e9874943a856d3f0ab81f0a268ba37f82054674be17328ed0b36d38c.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 474, + 382, + 492.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 228, + 492.0, + 382, + 510.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "which can be efficiently solved by the frequency domain ISTA in (Liu et al., 2017). The details are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 206, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 206, + 537 + ], + "score": 1.0, + "content": "outlined in Algorithm 2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 106, + 553, + 447, + 566 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 448, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 448, + 569 + ], + "score": 1.0, + "content": "F VISUALIZATION OF THE ANALYTIC CONVOLUTIONAL WEIGHTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 104, + 576, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 101, + 570, + 509, + 601 + ], + "spans": [ + { + "bbox": [ + 101, + 570, + 241, + 601 + ], + "score": 1.0, + "content": "Fig. 6 visualizes the dictionary d A-LISTA simulation of Section 5", + "type": "text" + }, + { + "bbox": [ + 242, + 578, + 291, + 589 + ], + "score": 0.86, + "content": "( 7 \\times 7 \\times 6 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 570, + 357, + 601 + ], + "score": 1.0, + "content": "and the weights ned by Algorith", + "type": "text" + }, + { + "bbox": [ + 357, + 577, + 399, + 590 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { w } } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 570, + 509, + 601 + ], + "score": 1.0, + "content": ", used in the convolutionalendix E.2.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 615, + 406, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 407, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 407, + 630 + ], + "score": 1.0, + "content": "G ALGORITHM DETAILS OF TRAINING ROBUST ALISTA", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 238, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 238, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 238, + 653 + ], + "score": 1.0, + "content": "G.1 MODEL ARCHITECTURE", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "Inspired by the a similar unrolling and truncating fashion in LISTA, we can approximately solve the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "coherence minimization problem (16) using a similar finite-layer neural network that is unfolded", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "from iterative algorithms. Because the linear constraints in (16) are hard to enforce in deep neural", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 694, + 300, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 300, + 707 + ], + "score": 1.0, + "content": "networks, we first relax it to the following form:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 711, + 378, + 735 + ], + "lines": [ + { + "bbox": [ + 233, + 711, + 378, + 735 + ], + "spans": [ + { + "bbox": [ + 233, + 711, + 378, + 735 + ], + "score": 0.92, + "content": "\\underset { \\mathbf { W } \\in \\mathbb { R } ^ { N \\times M } } { \\arg \\operatorname* { m i n } } \\left\\| \\mathbf { Q } \\odot ( \\mathbf { D } ^ { T } \\mathbf { W } - \\pmb { I } _ { M } ) \\right\\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "caac860c850f61fc6948bd272dd22b53e02f403f87c096e91c207278bd3fb79c.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 233, + 711, + 378, + 735 + ], + "spans": [], + "index": 38 + } + ] + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "28", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 331, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 331, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 331, + 97 + ], + "score": 1.0, + "content": "Similarly, by (58), the above problem is equivalent with", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 331, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 99, + 398, + 135 + ], + "lines": [ + { + "bbox": [ + 212, + 99, + 398, + 135 + ], + "spans": [ + { + "bbox": [ + 212, + 99, + 398, + 135 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { D ^ { 2 } M } } \\Big \\| \\big ( \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) \\big ) ^ { T } \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } ) \\Big \\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "3fd8c908a02a39c82d31ea67e04659f56160e67538ffb9a7fbed32ea7475ca81.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 99, + 398, + 117.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 212, + 117.0, + 398, + 135.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "Since the circular convolution is very efficient to calculate in the frequency domain, we consider", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 326, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 326, + 162 + ], + "score": 1.0, + "content": "solving (59) utilizing the fast Fourier transform (FFT).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 106, + 139, + 505, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 505, + 202 + ], + "lines": [ + { + "bbox": [ + 104, + 165, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 165, + 252, + 182 + ], + "score": 1.0, + "content": "Firstly, we introduce the operators", + "type": "text" + }, + { + "bbox": [ + 252, + 165, + 327, + 179 + ], + "score": 0.8, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { d } ) , \\mathbf { W } _ { \\mathrm { c i r } } ^ { N } ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 165, + 506, + 182 + ], + "score": 1.0, + "content": "in the frequency domain. To simplify the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 102, + 175, + 508, + 194 + ], + "spans": [ + { + "bbox": [ + 102, + 175, + 253, + 194 + ], + "score": 1.0, + "content": "notation, we denote the operators as", + "type": "text" + }, + { + "bbox": [ + 253, + 179, + 273, + 191 + ], + "score": 0.92, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 175, + 291, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 179, + 314, + 191 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 175, + 384, + 194 + ], + "score": 1.0, + "content": "respectively. Let", + "type": "text" + }, + { + "bbox": [ + 384, + 180, + 394, + 189 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 175, + 508, + 194 + ], + "score": 1.0, + "content": "be the FFT operator. Thus,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 189, + 227, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 152, + 203 + ], + "score": 0.92, + "content": "{ \\bf b } = { \\bf D } _ { \\mathrm { c i r } } ^ { N } { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 189, + 227, + 204 + ], + "score": 1.0, + "content": "is equivalent with", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 102, + 165, + 508, + 204 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 201, + 354, + 216 + ], + "lines": [ + { + "bbox": [ + 256, + 201, + 354, + 216 + ], + "spans": [ + { + "bbox": [ + 256, + 201, + 354, + 216 + ], + "score": 0.91, + "content": "\\mathcal { F } \\mathbf { b } = \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H } \\mathcal { F } \\mathbf { x } .", + "type": "interline_equation", + "image_path": "039ad5f5277ea925816fb197cc37b4eca4e1c4dc9e0e4bae721f0c81c9e90458.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 256, + 201, + 354, + 216 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 219, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 103, + 216, + 507, + 237 + ], + "spans": [ + { + "bbox": [ + 103, + 216, + 122, + 237 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 218, + 204, + 232 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { b } } = \\mathcal { F } \\mathbf { b } , \\hat { \\mathbf { x } } = \\mathcal { F } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 216, + 353, + 237 + ], + "score": 1.0, + "content": "be the frequency domain signals, let", + "type": "text" + }, + { + "bbox": [ + 354, + 219, + 433, + 233 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } = \\mathcal { F } \\mathbf { D } _ { \\mathrm { c i r } } ^ { N } \\mathcal { F } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 216, + 507, + 237 + ], + "score": 1.0, + "content": "be the frequency", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 270, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 270, + 244 + ], + "score": 1.0, + "content": "domain operator. The above equation is:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 216, + 507, + 244 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 280, + 248, + 330, + 264 + ], + "lines": [ + { + "bbox": [ + 280, + 248, + 330, + 264 + ], + "spans": [ + { + "bbox": [ + 280, + 248, + 330, + 264 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { b } } = \\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N } \\hat { \\mathbf { x } } .", + "type": "interline_equation", + "image_path": "9d261e7075b84fbff77469776a876f02d5da1d4123f2b75134300775b6f96731.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 280, + 248, + 330, + 264 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 269, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 508, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 239, + 288 + ], + "score": 1.0, + "content": "The frequency domain operator", + "type": "text" + }, + { + "bbox": [ + 240, + 269, + 259, + 283 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { D } } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 267, + 456, + 288 + ], + "score": 1.0, + "content": "is much cheaper to calculate than the operator", + "type": "text" + }, + { + "bbox": [ + 457, + 270, + 476, + 282 + ], + "score": 0.9, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 267, + 508, + 288 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 474, + 294 + ], + "score": 1.0, + "content": "spacial domain because it is block diagonal (Wohlberg, 2016). Specifically, we zero pad d to", + "type": "text" + }, + { + "bbox": [ + 475, + 282, + 504, + 292 + ], + "score": 0.9, + "content": "N \\times N", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 292, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 292, + 158, + 307 + ], + "score": 1.0, + "content": "and do FFT:", + "type": "text" + }, + { + "bbox": [ + 158, + 293, + 305, + 307 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { d } } _ { m } = \\mathrm { F F T } \\left( \\mathrm { z e r o p a d } ( \\mathbf { d } _ { m } , N - D ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 292, + 506, + 307 + ], + "score": 1.0, + "content": ", then the above operator can be explicitly written", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 307, + 121, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 121, + 317 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 267, + 508, + 317 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 313, + 346, + 347 + ], + "lines": [ + { + "bbox": [ + 263, + 313, + 346, + 347 + ], + "spans": [ + { + "bbox": [ + 263, + 313, + 346, + 347 + ], + "score": 0.94, + "content": "\\hat { \\mathbf { b } } = \\sum _ { m = 1 } ^ { M } \\overline { { \\hat { \\mathbf { d } } _ { m } } } \\odot \\hat { \\mathbf { x } } _ { m } ,", + "type": "interline_equation", + "image_path": "12be63c474cd1c81de195338a6598529dc8968cca41de5de787ffc3f770437e2.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 313, + 346, + 330.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 263, + 330.0, + 346, + 347.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 304, + 367 + ], + "score": 1.0, + "content": "where ¯· means complex conjugate. This is due to", + "type": "text" + }, + { + "bbox": [ + 304, + 349, + 324, + 363 + ], + "score": 0.91, + "content": "\\mathbf { D } _ { \\mathrm { c i r } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 347, + 507, + 367 + ], + "score": 1.0, + "content": "is actually cross-correlation, not convolution", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "(see (18)). Cross-correlation is equal to the transpose of convolution. 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The details are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 206, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 206, + 537 + ], + "score": 1.0, + "content": "outlined in Algorithm 2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 514, + 505, + 537 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 553, + 447, + 566 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 448, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 448, + 569 + ], + "score": 1.0, + "content": "F VISUALIZATION OF THE ANALYTIC CONVOLUTIONAL WEIGHTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 104, + 576, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 101, + 570, + 509, + 601 + ], + "spans": [ + { + "bbox": [ + 101, + 570, + 241, + 601 + ], + "score": 1.0, + "content": "Fig. 6 visualizes the dictionary d A-LISTA simulation of Section 5", + "type": "text" + }, + { + "bbox": [ + 242, + 578, + 291, + 589 + ], + "score": 0.86, + "content": "( 7 \\times 7 \\times 6 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 570, + 357, + 601 + ], + "score": 1.0, + "content": "and the weights ned by Algorith", + "type": "text" + }, + { + "bbox": [ + 357, + 577, + 399, + 590 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { w } } \\in \\mathcal { W } _ { \\mathrm { c i r } } ^ { 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 570, + 509, + 601 + ], + "score": 1.0, + "content": ", used in the convolutionalendix E.2.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 101, + 570, + 509, + 601 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 615, + 406, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 407, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 407, + 630 + ], + "score": 1.0, + "content": "G ALGORITHM DETAILS OF TRAINING ROBUST ALISTA", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 238, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 238, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 238, + 653 + ], + "score": 1.0, + "content": "G.1 MODEL ARCHITECTURE", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 505, + 706 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "Inspired by the a similar unrolling and truncating fashion in LISTA, we can approximately solve the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "coherence minimization problem (16) using a similar finite-layer neural network that is unfolded", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "from iterative algorithms. 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One layer of this model is shown in Fig. 7(a).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 647, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "The illustration of the whole feed-forward robust model is shown in Fig. 7(b). The two parts, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "encoder and the decoder, can be jointly trained to gain the most from data-driven learning. We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 721, + 492, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 492, + 733 + ], + "score": 1.0, + "content": "further adopt pre-training and curriculum learning to stabilize training, as to be discussed below.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 699, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 83, + 500, + 204 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 83, + 500, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 83, + 500, + 204 + ], + "spans": [ + { + "bbox": [ + 117, + 83, + 500, + 204 + ], + "score": 0.959, + "type": "image", + "image_path": "6776e69f5cec6be82f55156bb9af67711806163f340570dab157b388fb069086.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 83, + 500, + 123.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 123.33333333333334, + 500, + 163.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 163.66666666666669, + 500, + 204.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 222, + 216, + 388, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 222, + 215, + 389, + 229 + ], + "spans": [ + { + "bbox": [ + 222, + 215, + 389, + 229 + ], + "score": 1.0, + "content": "Figure 7: Feed-Forward Analytic LISTA.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 107, + 251, + 213, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 214, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 214, + 264 + ], + "score": 1.0, + "content": "G.2 MODEL TRAINING", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 108, + 273, + 504, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 286 + ], + "score": 1.0, + "content": "To stabilize the training process, we train the model in two stages: the pre-training stage and cur-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 283, + 226, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 226, + 297 + ], + "score": 1.0, + "content": "riculum (joint) training stage.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 309, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 324 + ], + "score": 1.0, + "content": "Pre-Training Stage. We first pre-train the encoder and the decoder individually. The pre-training", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "of the decoder, e.g., ALISTA, follows the standard training procedure in Section 5.1, without both-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 210, + 344 + ], + "score": 1.0, + "content": "ering the perturbations of", + "type": "text" + }, + { + "bbox": [ + 211, + 332, + 221, + 342 + ], + "score": 0.78, + "content": "_ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 332, + 505, + 344 + ], + "score": 1.0, + "content": ". On the other hand, the encoder will always see perturbed dictionaries", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 342, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 107, + 343, + 167, + 355 + ], + "score": 0.91, + "content": "\\tilde { \\cal D } = { \\cal D } + \\varepsilon _ { \\cal D }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 342, + 197, + 358 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 198, + 345, + 212, + 355 + ], + "score": 0.85, + "content": "\\varepsilon _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 342, + 506, + 358 + ], + "score": 1.0, + "content": "’s entries are sampled from i.i.d. normal distribution with zero mean and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 354, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 107, + 355, + 126, + 368 + ], + "score": 0.91, + "content": "\\sigma _ { p r e } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 354, + 440, + 370 + ], + "score": 1.0, + "content": "variance, and update its weight to minimize loss function defined by (60). The", + "type": "text" + }, + { + "bbox": [ + 440, + 357, + 459, + 367 + ], + "score": 0.86, + "content": "\\sigma _ { p r e }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 354, + 506, + 370 + ], + "score": 1.0, + "content": "is a hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "parameter that we manually select for the pre-training stage, with a default value of 0.01. We use an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 376, + 492, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 433, + 391 + ], + "score": 1.0, + "content": "exponentially decaying learning rate for encoder pre-training with an initial value", + "type": "text" + }, + { + "bbox": [ + 434, + 376, + 487, + 389 + ], + "score": 0.92, + "content": "\\alpha _ { p r e } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 376, + 492, + 391 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "Curriculum (Joint) Training Stage. After the pre-training stage, we concatenate these two parts", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "score": 1.0, + "content": "and do joint training. However, a direct end-to-end tuning was observed to cause much instability,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "due to the randomness in weights. Inspired by the curriculum learning technique, we first perturb", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 433, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 450 + ], + "score": 1.0, + "content": "the dictionaries with smaller standard deviations and gradually increase the perturbation level during", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 445, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 353, + 461 + ], + "score": 1.0, + "content": "training. Specifically, starting from a small standard deviation", + "type": "text" + }, + { + "bbox": [ + 353, + 447, + 387, + 457 + ], + "score": 0.91, + "content": "\\sigma ^ { t } = \\sigma ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 445, + 506, + 461 + ], + "score": 1.0, + "content": ", the curriculum joint training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 458, + 255, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 255, + 470 + ], + "score": 1.0, + "content": "procedure repeats the routine below:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 131, + 481, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 129, + 479, + 508, + 500 + ], + "spans": [ + { + "bbox": [ + 129, + 479, + 362, + 500 + ], + "score": 1.0, + "content": "• First uniformly sample a batch of standard deviations", + "type": "text" + }, + { + "bbox": [ + 362, + 482, + 396, + 495 + ], + "score": 0.92, + "content": "\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { B _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 479, + 420, + 500 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 421, + 482, + 446, + 495 + ], + "score": 0.91, + "content": "[ 0 , \\sigma ^ { t } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 479, + 478, + 500 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 478, + 483, + 494, + 494 + ], + "score": 0.89, + "content": "B _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 479, + 508, + 500 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 142, + 494, + 379, + 506 + ], + "score": 1.0, + "content": "the batch size for perturbations of the original dictionary", + "type": "text" + }, + { + "bbox": [ + 380, + 495, + 389, + 504 + ], + "score": 0.41, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 494, + 505, + 506 + ], + "score": 1.0, + "content": ". 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Then", + "type": "text" + }, + { + "bbox": [ + 263, + 586, + 318, + 600 + ], + "score": 0.92, + "content": "( \\mathbf { y } _ { i , j } , \\mathbf { x } _ { j } , \\tilde { \\mathbf { D } } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 586, + 506, + 601 + ], + "score": 1.0, + "content": "forms a tripelet of training sample. 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Both networks have 5 layers, with the same dictionary D obtained from", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "the training set by solving (24). We reconstruct the denoised images using by convolving the learned", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 175 + ], + "score": 1.0, + "content": "feature maps with the original dictionary D. The mean-square-error (MSE) between denoised and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 461, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 461, + 185 + ], + "score": 1.0, + "content": "clean images are adopted as the network training loss, as inspired by (Zhou et al., 2018).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 189, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 269, + 203 + ], + "score": 1.0, + "content": "Six popular benchmark images (adding", + "type": "text" + }, + { + "bbox": [ + 270, + 190, + 299, + 200 + ], + "score": 0.88, + "content": "\\sigma = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 189, + 505, + 203 + ], + "score": 1.0, + "content": "noise) are tested and reported in Table 4. The A-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "PSNR denotes the average PSNR over all images and the A-Times represents the average inference", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "time (in seconds) for denoising one image. 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The results show that Conv", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "score": 1.0, + "content": "LISTA and Conv ALISTA (without heavy tuning done for their optimal performance) can perform", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "comparably with KSVD and outperforms CSC-GR, but with tremendously faster inference speeds", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "than KSVD/CSC-GR. More importantly, Conv LISTA and Conv ALISTA only have marginal per-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 437, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 437, + 290 + ], + "score": 1.0, + "content": "formance differences, validating again the analytic weights in convolutional cases.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "table", + "bbox": [ + 106, + 326, + 505, + 397 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 299, + 503, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 298, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 504, + 312 + ], + "score": 1.0, + "content": "Table 4: Peak Signal to Noise Ratio (PSNR) Comparision between Conv LISTA and Conv ALISTA.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 106, + 326, + 505, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 397 + ], + "score": 0.981, + "html": "
ModelImage PSNR (dB)A-PSNRA-Time
LennaHousePepperCoupleBoatsBarbara
KSVD31.0333.2430.9731.7131.0030.4731.4024.70
CSC-GR28.4129.1127.3929.3128.3527.1928.297.56
Conv LISTA31.2632.7731.0031.8930.7829.5331.210.012
Conv ALISTA31.0132.4630.8131.8530.5829.7231.070.014
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This comparison could consolidate our claim that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "the outstanding adaptiveness to dictionary perturbations of robust ALISTA is brought by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 142, + 720, + 415, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 415, + 733 + ], + "score": 1.0, + "content": "its encoder-decoder structure rather than its learning capacity alone.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + } + ], + "page_idx": 30, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 105, + 81, + 463, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 464, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 464, + 95 + ], + "score": 1.0, + "content": "H RESULTS OF NATURAL IMAGE DENOISING USING CONV ALISTA", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 184 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 120 + ], + "score": 1.0, + "content": "The natural image denoising experiment is conducted on the same BSD 500 dataset using the 400-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 504, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 504, + 130 + ], + "score": 1.0, + "content": "image training set, 50-image validation set and 50-image test set. We convert them all to grayscale,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 160, + 141 + ], + "score": 1.0, + "content": "and then add", + "type": "text" + }, + { + "bbox": [ + 160, + 129, + 192, + 139 + ], + "score": 0.88, + "content": "\\sigma = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "Gaussian i.i.d. noise. We train both Conv LISTA (i.e., model (20)) and Conv", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "ALISTA (i.e., model (26)). Both networks have 5 layers, with the same dictionary D obtained from", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "the training set by solving (24). We reconstruct the denoised images using by convolving the learned", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 175 + ], + "score": 1.0, + "content": "feature maps with the original dictionary D. The mean-square-error (MSE) between denoised and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 461, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 461, + 185 + ], + "score": 1.0, + "content": "clean images are adopted as the network training loss, as inspired by (Zhou et al., 2018).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 106, + 506, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 189, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 269, + 203 + ], + "score": 1.0, + "content": "Six popular benchmark images (adding", + "type": "text" + }, + { + "bbox": [ + 270, + 190, + 299, + 200 + ], + "score": 0.88, + "content": "\\sigma = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 189, + 505, + 203 + ], + "score": 1.0, + "content": "noise) are tested and reported in Table 4. The A-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "PSNR denotes the average PSNR over all images and the A-Times represents the average inference", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "time (in seconds) for denoising one image. We compare Conv LISTA and Conv ALISTA, as well", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 236 + ], + "score": 1.0, + "content": "as the classical KSVD denoising algorithm (Elad & Aharon, 2006) and the recent CSC denoising", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "algorithm with gradient regularization (CSC-GR) (Wohlberg, 2018). The results show that Conv", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "score": 1.0, + "content": "LISTA and Conv ALISTA (without heavy tuning done for their optimal performance) can perform", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "comparably with KSVD and outperforms CSC-GR, but with tremendously faster inference speeds", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "than KSVD/CSC-GR. More importantly, Conv LISTA and Conv ALISTA only have marginal per-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 437, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 437, + 290 + ], + "score": 1.0, + "content": "formance differences, validating again the analytic weights in convolutional cases.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 189, + 506, + 290 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 326, + 505, + 397 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 299, + 503, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 298, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 504, + 312 + ], + "score": 1.0, + "content": "Table 4: Peak Signal to Noise Ratio (PSNR) Comparision between Conv LISTA and Conv ALISTA.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 106, + 326, + 505, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 397 + ], + "score": 0.981, + "html": "
ModelImage PSNR (dB)A-PSNRA-Time
LennaHousePepperCoupleBoatsBarbara
KSVD31.0333.2430.9731.7131.0030.4731.4024.70
CSC-GR28.4129.1127.3929.3128.3527.1928.297.56
Conv LISTA31.2632.7731.0031.8930.7829.5331.210.012
Conv ALISTA31.0132.4630.8131.8530.5829.7231.070.014
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Therefore, they suggested that we increased the number of layers", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "in TiLISTA, ALISTA and the baseline model LISTA-CPSS in (Chen et al., 2018) to see if their", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "performance in this above evaluation setting can be improved in that way. Note that LISTA-CPSS", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "has tens of layers, hence actually containing more parameters than robust ALISTA. In addition, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "reviewers also suggested a set of ablation studies to investigate whether more layers in the encoder", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "can endorse the model better adaptivity to higher level perturbations. We conduct the suggested", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 309, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 309, + 539 + ], + "score": 1.0, + "content": "experiments and present the results in this section.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 449, + 506, + 539 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 552, + 398, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 399, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 399, + 566 + ], + "score": 1.0, + "content": "I.1 NUMBER OF LAYERS IN TILISTA, ALISTA AND LISTA-CPSS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "As the reviewers pointed out, the comparison we present in Section. 5.3 might be unfair because", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 583, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 505, + 599 + ], + "score": 1.0, + "content": "robust ALISTA contains much more parameters (because it contains a 4-layer encoder) comparing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "to ALISTA, which only learns two series of scalars, and TiLISTA which has just one more matrix", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "score": 1.0, + "content": "weight than ALISTA. Therefore, we add the following experiments to consolidate our claim on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 618, + 280, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 280, + 630 + ], + "score": 1.0, + "content": "effectiveness of the robust ALISTA model:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 574, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 639, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 134, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 134, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "• we increase the number of layers of TiLISTA and ALISTA which are then trained in the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 142, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "same data augmentation setting as we do in Section. 5.3, to see if they could yield compar-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 661, + 338, + 675 + ], + "spans": [ + { + "bbox": [ + 141, + 661, + 338, + 675 + ], + "score": 1.0, + "content": "itive robustness against dictionary perturbations;", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 140, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 140, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "we also compare the robust ALISTA with the baseline LISTA-CPSS model in Chen et al.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(2018), which contains tens of layers of independent weight matrices, thus having even", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 141, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "more parameters than robust ALISTA. This comparison could consolidate our claim that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "the outstanding adaptiveness to dictionary perturbations of robust ALISTA is brought by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 142, + 720, + 415, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 415, + 733 + ], + "score": 1.0, + "content": "its encoder-decoder structure rather than its learning capacity alone.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 134, + 639, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The results are shown in Table. 5, where the performances are measured with NMSE in dB, which", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "score": 1.0, + "content": "is defined in Section 5.1. The “Augmented” prefix means the models are trained in the data aug-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 183, + 117 + ], + "score": 1.0, + "content": "mentation setting.", + "type": "text" + }, + { + "bbox": [ + 183, + 106, + 191, + 114 + ], + "score": 0.72, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "is the standard deviation of the Gaussian distribution that is used to generate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 226, + 128 + ], + "score": 1.0, + "content": "the dictioanry perturbations.", + "type": "text" + }, + { + "bbox": [ + 226, + 116, + 235, + 125 + ], + "score": 0.68, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "stands for the number of layers (in the case of robust ALISTA it", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "score": 1.0, + "content": "means the nubmer of layers of the ALISTA decoder, with a 4-layer encoder). We follow the training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 435, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 316, + 150 + ], + "score": 1.0, + "content": "strategy and settings explained in Appendix G, with", + "type": "text" + }, + { + "bbox": [ + 316, + 138, + 370, + 149 + ], + "score": 0.91, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 137, + 435, + 150 + ], + "score": 1.0, + "content": "during training.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "On one hand, the comparison of performances of ALISTA, TiLISTA and LISTA-CPSS shows results", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "that are consistent to the intuition that larger parameter space yields larger learning capacity, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 287, + 189 + ], + "score": 1.0, + "content": "therefore, better adaptiveness (LISTA-CPSS", + "type": "text" + }, + { + "bbox": [ + 288, + 177, + 298, + 187 + ], + "score": 0.44, + "content": ">", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 175, + 338, + 189 + ], + "score": 1.0, + "content": "TiLISTA", + "type": "text" + }, + { + "bbox": [ + 338, + 177, + 349, + 187 + ], + "score": 0.76, + "content": ">", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "ALISTA). On the other hand, we can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "also find that ALISTA with more layers has worse performance. We think this observation is also", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "reasonable for two reasons: 1) adding more layers in ALISTA does not enlarge the parameter volume", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "significantly because it has only two scalar parameters in each layer; noting that ALISTA uses a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "fixed, analytically solved weight matrix, if this weight matrix is not compatible with the perturbed", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "dictionary, more layers can even hurt the performance instead of improving. Lastly, it’s clearly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "shown that robust ALISTA outperforms LISTA-CPSS, even if it contains less parameters. This", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "proves that the encoding process that adaptively transforms the perturbed dictiories is necessary to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 353, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 353, + 277 + ], + "score": 1.0, + "content": "achieve good robustness against perturbations in dictionaries.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "table", + "bbox": [ + 133, + 286, + 477, + 421 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 133, + 286, + 477, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 286, + 477, + 421 + ], + "spans": [ + { + "bbox": [ + 133, + 286, + 477, + 421 + ], + "score": 0.985, + "html": "
σ of perturbationsduring testing0.00010.0010.010.0150.020.025
AugmentedALISTAT=16-26.58-25.87-15.49-11.71-8.84-6.74
T=20-24.43-24.46-15.39-11.77-8.94-6.82
T=24-24.12-24.00-15.45-11.68-8.81-6.70
AugmentedTiLISTAT=16-27.76-27.18-16.83-12.95-9.81-7.55
T=20-28.13-28.54-17.15-12.98-9.83-7.58
T=24-26.08-27.27-17.34-13.14-9.91-7.61
AugmentedLISTA-CPSST=16-27.93-27.18-16.96-12.99-9.93-7.70
T=20-28.17-27.33-16.95-13.00-9.94-7.71
T=24-30.30-29.24-16.86-12.97-9.94-7.70
Robust ALISTAT=16-62.47-62.41-62.02-61.50-60.67-45.00
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A", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "natural intuition is that, adding more layers to the encoder can increase its ability to sustain larger", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 527, + 252, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 252, + 537 + ], + "score": 1.0, + "content": "perturbation levels. But is this true?", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 554 + ], + "score": 1.0, + "content": "Therefore, we train another two robust ALISTA models, with a 5-layer and a 6-layer encoders", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "respectively and 16-layer ALISTA decoders for both, and compare them with the originally reported", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 564, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 504, + 576 + ], + "score": 1.0, + "content": "robust ALISTA model with a 4-layer encoder and a 16-layer ALISTA decoders. All three models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "use one pretrained decoder, and pretrain their encoders using the same method (see Appendix G).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 585, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 158, + 601 + ], + "score": 1.0, + "content": "We only use", + "type": "text" + }, + { + "bbox": [ + 158, + 587, + 214, + 598 + ], + "score": 0.9, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 585, + 506, + 601 + ], + "score": 1.0, + "content": "during training. One thing to notice is that we observe unstable training", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 596, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 103, + 596, + 289, + 611 + ], + "score": 1.0, + "content": "process if we use default initial learning rates", + "type": "text" + }, + { + "bbox": [ + 289, + 597, + 343, + 610 + ], + "score": 0.92, + "content": "\\alpha _ { p r e } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 596, + 457, + 611 + ], + "score": 1.0, + "content": "in the pre-training stage and", + "type": "text" + }, + { + "bbox": [ + 458, + 597, + 504, + 609 + ], + "score": 0.91, + "content": "\\alpha _ { e } = 1 0 ^ { - } \\overline { { 6 } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 608, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 103, + 608, + 421, + 624 + ], + "score": 1.0, + "content": "in the joint training stage when encoders have 5 or 6 layers. Therefore, we use", + "type": "text" + }, + { + "bbox": [ + 421, + 609, + 475, + 623 + ], + "score": 0.95, + "content": "\\alpha _ { p r e } ^ { \\prime } = 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 608, + 507, + 624 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "6-layer encoder in the pre-training stage, and in the joint training stage use a decreased and uniform", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 185, + 645 + ], + "score": 1.0, + "content": "initial learning rate", + "type": "text" + }, + { + "bbox": [ + 185, + 631, + 231, + 644 + ], + "score": 0.93, + "content": "\\alpha _ { e } ^ { \\prime } = 1 0 ^ { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "for the three encoders while keeping the default initial learning rate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 641, + 354, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 152, + 654 + ], + "score": 0.92, + "content": "\\alpha _ { d } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 641, + 354, + 656 + ], + "score": 1.0, + "content": "for decoders. The other settings remain the same.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "Results are shown in Table. 6. The performances are measured with NMSE in dB, defined in Section", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "5.1. From the table we can see that encoders do show better robustness when they have more layers,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 220, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 220, + 694 + ], + "score": 1.0, + "content": "i.e. larger learning capacity.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "page_idx": 31, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The results are shown in Table. 5, where the performances are measured with NMSE in dB, which", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 505, + 108 + ], + "score": 1.0, + "content": "is defined in Section 5.1. The “Augmented” prefix means the models are trained in the data aug-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 183, + 117 + ], + "score": 1.0, + "content": "mentation setting.", + "type": "text" + }, + { + "bbox": [ + 183, + 106, + 191, + 114 + ], + "score": 0.72, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "is the standard deviation of the Gaussian distribution that is used to generate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 226, + 128 + ], + "score": 1.0, + "content": "the dictioanry perturbations.", + "type": "text" + }, + { + "bbox": [ + 226, + 116, + 235, + 125 + ], + "score": 0.68, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "stands for the number of layers (in the case of robust ALISTA it", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "score": 1.0, + "content": "means the nubmer of layers of the ALISTA decoder, with a 4-layer encoder). We follow the training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 435, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 316, + 150 + ], + "score": 1.0, + "content": "strategy and settings explained in Appendix G, with", + "type": "text" + }, + { + "bbox": [ + 316, + 138, + 370, + 149 + ], + "score": 0.91, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 137, + 435, + 150 + ], + "score": 1.0, + "content": "during training.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 82, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "On one hand, the comparison of performances of ALISTA, TiLISTA and LISTA-CPSS shows results", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "that are consistent to the intuition that larger parameter space yields larger learning capacity, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 287, + 189 + ], + "score": 1.0, + "content": "therefore, better adaptiveness (LISTA-CPSS", + "type": "text" + }, + { + "bbox": [ + 288, + 177, + 298, + 187 + ], + "score": 0.44, + "content": ">", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 175, + 338, + 189 + ], + "score": 1.0, + "content": "TiLISTA", + "type": "text" + }, + { + "bbox": [ + 338, + 177, + 349, + 187 + ], + "score": 0.76, + "content": ">", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "ALISTA). On the other hand, we can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "also find that ALISTA with more layers has worse performance. We think this observation is also", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "reasonable for two reasons: 1) adding more layers in ALISTA does not enlarge the parameter volume", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "significantly because it has only two scalar parameters in each layer; noting that ALISTA uses a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "fixed, analytically solved weight matrix, if this weight matrix is not compatible with the perturbed", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "dictionary, more layers can even hurt the performance instead of improving. Lastly, it’s clearly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "shown that robust ALISTA outperforms LISTA-CPSS, even if it contains less parameters. This", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "proves that the encoding process that adaptively transforms the perturbed dictiories is necessary to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 353, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 353, + 277 + ], + "score": 1.0, + "content": "achieve good robustness against perturbations in dictionaries.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 154, + 506, + 277 + ] + }, + { + "type": "table", + "bbox": [ + 133, + 286, + 477, + 421 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 133, + 286, + 477, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 286, + 477, + 421 + ], + "spans": [ + { + "bbox": [ + 133, + 286, + 477, + 421 + ], + "score": 0.985, + "html": "
σ of perturbationsduring testing0.00010.0010.010.0150.020.025
AugmentedALISTAT=16-26.58-25.87-15.49-11.71-8.84-6.74
T=20-24.43-24.46-15.39-11.77-8.94-6.82
T=24-24.12-24.00-15.45-11.68-8.81-6.70
AugmentedTiLISTAT=16-27.76-27.18-16.83-12.95-9.81-7.55
T=20-28.13-28.54-17.15-12.98-9.83-7.58
T=24-26.08-27.27-17.34-13.14-9.91-7.61
AugmentedLISTA-CPSST=16-27.93-27.18-16.96-12.99-9.93-7.70
T=20-28.17-27.33-16.95-13.00-9.94-7.71
T=24-30.30-29.24-16.86-12.97-9.94-7.70
Robust ALISTAT=16-62.47-62.41-62.02-61.50-60.67-45.00
", + "type": "table", + "image_path": "9e51f42941c85eec4bc501a0e6e57efe6443b7d15da3f37f93a74e0a94a2358f.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 133, + 286, + 477, + 331.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 133, + 331.0, + 477, + 376.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 133, + 376.0, + 477, + 421.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 106, + 429, + 504, + 452 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "score": 1.0, + "content": "Table 5: The results (recovery NMSE in dB) of ablation study on the influence of model capacity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 441, + 358, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 358, + 452 + ], + "score": 1.0, + "content": "towards the model robustness against dictionary perturbations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + } + ], + "index": 19.25 + }, + { + "type": "title", + "bbox": [ + 107, + 472, + 431, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 433, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 433, + 486 + ], + "score": 1.0, + "content": "I.2 ABLATION STUDY ON THE DEPTHS OF ENCODERS IN ROBUST ALISTA", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "Another constructive suggestion from the reviewers is to design an ablation study to investigate the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "influence of the depth of encoders in robust ALISTA on its adaptivity to dictionary perturbations. A", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "natural intuition is that, adding more layers to the encoder can increase its ability to sustain larger", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 527, + 252, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 252, + 537 + ], + "score": 1.0, + "content": "perturbation levels. But is this true?", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 493, + 505, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 554 + ], + "score": 1.0, + "content": "Therefore, we train another two robust ALISTA models, with a 5-layer and a 6-layer encoders", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "respectively and 16-layer ALISTA decoders for both, and compare them with the originally reported", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 564, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 504, + 576 + ], + "score": 1.0, + "content": "robust ALISTA model with a 4-layer encoder and a 16-layer ALISTA decoders. All three models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "use one pretrained decoder, and pretrain their encoders using the same method (see Appendix G).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 585, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 158, + 601 + ], + "score": 1.0, + "content": "We only use", + "type": "text" + }, + { + "bbox": [ + 158, + 587, + 214, + 598 + ], + "score": 0.9, + "content": "\\sigma _ { m a x } = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 585, + 506, + 601 + ], + "score": 1.0, + "content": "during training. One thing to notice is that we observe unstable training", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 596, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 103, + 596, + 289, + 611 + ], + "score": 1.0, + "content": "process if we use default initial learning rates", + "type": "text" + }, + { + "bbox": [ + 289, + 597, + 343, + 610 + ], + "score": 0.92, + "content": "\\alpha _ { p r e } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 596, + 457, + 611 + ], + "score": 1.0, + "content": "in the pre-training stage and", + "type": "text" + }, + { + "bbox": [ + 458, + 597, + 504, + 609 + ], + "score": 0.91, + "content": "\\alpha _ { e } = 1 0 ^ { - } \\overline { { 6 } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 608, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 103, + 608, + 421, + 624 + ], + "score": 1.0, + "content": "in the joint training stage when encoders have 5 or 6 layers. Therefore, we use", + "type": "text" + }, + { + "bbox": [ + 421, + 609, + 475, + 623 + ], + "score": 0.95, + "content": "\\alpha _ { p r e } ^ { \\prime } = 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 608, + 507, + 624 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "6-layer encoder in the pre-training stage, and in the joint training stage use a decreased and uniform", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 185, + 645 + ], + "score": 1.0, + "content": "initial learning rate", + "type": "text" + }, + { + "bbox": [ + 185, + 631, + 231, + 644 + ], + "score": 0.93, + "content": "\\alpha _ { e } ^ { \\prime } = 1 0 ^ { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "for the three encoders while keeping the default initial learning rate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 641, + 354, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 152, + 654 + ], + "score": 0.92, + "content": "\\alpha _ { d } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 641, + 354, + 656 + ], + "score": 1.0, + "content": "for decoders. The other settings remain the same.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 103, + 543, + 507, + 656 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "Results are shown in Table. 6. The performances are measured with NMSE in dB, defined in Section", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "5.1. From the table we can see that encoders do show better robustness when they have more layers,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 220, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 220, + 694 + ], + "score": 1.0, + "content": "i.e. larger learning capacity.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 659, + 506, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 148, + 360, + 464, + 419 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 148, + 360, + 464, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 148, + 360, + 464, + 419 + ], + "spans": [ + { + "bbox": [ + 148, + 360, + 464, + 419 + ], + "score": 0.972, + "html": "
# Encoder Layerso of perturbations during testing
0.00010.0010.010.0150.020.025
4-68.57-68.56-67.94-66.86-64.84-56.63
5-69.34-69.34-69.02-68.49-67.20-65.55
6-70.38-70.33-69.92-69.22-67.72-65.60
", + "type": "table", + "image_path": "a307eaddd20854b14eefd776a6c2f72b38e953b1403d43011fdfc01e5a888ad9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 148, + 360, + 464, + 379.6666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 148, + 379.6666666666667, + 464, + 399.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 148, + 399.33333333333337, + 464, + 419.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "Table 6: The results of ablation study on the influence of the depths of encoder towards the model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 437, + 282, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 282, + 450 + ], + "score": 1.0, + "content": "robustness against dictionary perturbations.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "page_idx": 32, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "33", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 148, + 360, + 464, + 419 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 148, + 360, + 464, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 148, + 360, + 464, + 419 + ], + "spans": [ + { + "bbox": [ + 148, + 360, + 464, + 419 + ], + "score": 0.972, + "html": "
# Encoder Layerso of perturbations during testing
0.00010.0010.010.0150.020.025
4-68.57-68.56-67.94-66.86-64.84-56.63
5-69.34-69.34-69.02-68.49-67.20-65.55
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N=3N=5N= 10N=15N = 20
0.18920.08500.02840.01610.0113
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lwir- w*||2/1lw*||2
N=10N= 11N = 12N=13N=15N = 20
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\\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | _ { 1 } } \\\\ & { = \\tilde { \\mu } \\gamma ^ { ( k ) } ( \\big | \\mathbb { S } \\big | - 1 ) \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | _ { 1 } + \\theta ^ { ( k ) } \\big | \\mathbb { S } \\big | + \\big | 1 - \\gamma ^ { ( k ) } \\big | \\big | \\big | { \\mathbf { x } } ^ { ( k ) } - { \\mathbf { x } } ^ { * } \\big | \\big | . } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 547, + 472, + 888, + 472, + 888, + 514, + 547, + 514 + ], + "score": 0.94, + "latex": "\\| \\mathbf { x } ^ { ( k ) } - \\mathbf { x } ^ { * } \\| _ { 1 } = \\| \\mathbf { x } _ { \\mathbb { S } } ^ { ( k ) } - \\mathbf { x } _ { \\mathbb { S } } ^ { * } \\| _ { 1 }" + }, + { + "category_id": 13, + "poly": [ + 341, + 1070, + 822, + 1070, + 822, + 1111, + 341, + 1111 + ], + "score": 0.94, + "latex": "c ^ { ( \\tau ) } = - \\log \\left( ( 2 \\tilde { \\mu } s - 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ModelImage PSNR (dB)A-PSNRA-Time
LennaHousePepperCoupleBoatsBarbara
KSVD31.0333.2430.9731.7131.0030.4731.4024.70
CSC-GR28.4129.1127.3929.3128.3527.1928.297.56
Conv LISTA31.2632.7731.0031.8930.7829.5331.210.012
Conv ALISTA31.0132.4630.8131.8530.5829.7231.070.014
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σ of perturbationsduring testing0.00010.0010.010.0150.020.025
AugmentedALISTAT=16-26.58-25.87-15.49-11.71-8.84-6.74
T=20-24.43-24.46-15.39-11.77-8.94-6.82
T=24-24.12-24.00-15.45-11.68-8.81-6.70
AugmentedTiLISTAT=16-27.76-27.18-16.83-12.95-9.81-7.55
T=20-28.13-28.54-17.15-12.98-9.83-7.58
T=24-26.08-27.27-17.34-13.14-9.91-7.61
AugmentedLISTA-CPSST=16-27.93-27.18-16.96-12.99-9.93-7.70
T=20-28.17-27.33-16.95-13.00-9.94-7.71
T=24-30.30-29.24-16.86-12.97-9.94-7.70
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We observe videos are naturally accompanied by abundant text information such as YouTube titles and Instagram captions. In this paper, we leverage this visual-textual connection to learn spatiotemporal features in an efficient weakly-supervised manner. We present a general cross-modal pair discrimination (CPD) framework to capture this correlation between a video and its associated text. We train our CPD models on both standard video dataset (Kinetics-210k) and uncurated web video dataset (Instagram-300k) to demonstrate its effectiveness. Without further fine-tuning, the learnt models obtain competitive results for action classification on Kinetics under the linear classification protocol. Moreover, our visual model provides an effective initialization to fine-tune on downstream tasks, which yields a remarkable performance gain for action recognition on UCF101 and HMDB51, compared with the existing state-of-the-art self-supervised training methods. In addition, our CPD demonstrates that pre-training a relatively small dataset is able to yield a comparable performance to those methods of using order magnitude more data, which is meaningful and practicable for the scenarios with limited computational facilities. + +# 1 INTRODUCTION + +Deep learning has made a remarkable progress for visual recognition in both image and video domain (Krizhevsky et al., 2012; He et al., 2016; Carreira & Zisserman, 2017; Feichtenhofer et al., 2018) by training powerful neural networks on large-scale manually annotated datasets (e.g., ImageNet (Deng et al., 2009) and Kinetics (Kay et al., 2017)). More importantly, it is well-established that this supervised pre-training on large-scale datasets would benefit the downstream tasks (e.g., object detection (Ren et al., 2015), pose estimation (He et al., 2017), and temporal action detection (Zhao et al., 2017)), in particular when the target datasets are relatively small. Yet, annotating a large-scale dataset for training such deep neural networks is costly and time-consuming, and even more challenging for video due to its various temporal structure and complex semantics. As a result, the existing video datasets size is still smaller than ImageNet in terms of training samples and classes. On the other hand, videos typically contain richer structure with abundant side information such as motion (Diba et al., 2019; $\mathrm { N g }$ et al., 2018), audio (Arandjelovic & Zisserman, 2017; Korbar et al., 2018), and text (Miech et al., 2019; Sun et al., 2019b). So these expected these associated modalities are expected to provide useful cues to learn video representations in a more efficient way. + +Language or text is probably the most natural and easy way to describe the semantic information of a video, and the associated textual information could be easily acquired when collecting video dataset (Rohrbach et al., 2017; Miech et al., 2019) from Internet or Movie. We argue that this correlation between a clip and its associated text could serve as an alternative supervision to learn video representation from scratch. This is different from some recent works (Sun et al., 2019b; Miech et al., 2019), in which these abundant textual information has been used to learn a high-level visual-text embedding applied to text-to-video retrieval or video captioning. Intuitively, it is more challenging to learn a general visual representation solely from text information without any human annotation, for reasons such as large numbers of noise in text, lacking careful initialization, and being hard to design an effective objective. + +In this paper, we aim to learn effective video representation from noisy and diverse textual information, which could serves as the basis for a variety of downstream tasks. Basically, we learn a mapping of text and video into a shared embedding space and leverage their correlation as supervision signal. The technical difficulty is how to design an effective objective function, that is capable of modeling this complex visual-textual correlation and as well easily optimized by training from scratch on noisy datasets. Inspired by unsupervised feature learning in images (Wu et al., 2018; Tian et al., 2019), we present a cross-modal pair discrimination (CPD) framework, which tries to recognize each video and text pair into a class via a non-parametric classifier. To solve the computational issues imposed by the huge numbers of pair classes, we adapt noise-contrastive estimation technique (Gutmann & Hyvarinen, 2010) to approximate the original loss function. ¨ + +Specifically, we learn the CPD framework from web videos with the associated title or caption that could be directly crawled from web platforms such as YouTube (Kay et al., 2017) and Instagram (Duan et al., 2020). We utilize the off-the-shelf language models such as BERT (Devlin et al., 2019) or Word2vec (Mikolov et al., 2013) and devise a curriculum learning strategy to progressively train the video models. We first test the generalization ability of learned video representation by CPD on the Kinetics dataset (Kay et al., 2017) by using shallow classifiers such k-NN and linear classifier. It shows that our learned spatiotemporal features obtain promising results which are comparable to some supervised learning methods on the Kinetics dataset (Kay et al., 2017). Then, we investigate the generalization power of learned spatiotemporal features of CPD by fine-tuning on the Kinetics (Kay et al., 2017), UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011) datasets, demonstrating that our method obtain superior performance to previous state-of-the-art self-supervised methods and comparable performance to the very recent methods of using orders of magnitude more videos (70M-100M vs. 0.3M). + +# 2 RELATED WORK + +Self/Weakly Supervised Representation Learning. Self supervised representation was popular in both image and video domains by designing various proxy tasks. In image domain, for instance, these tasks could be predicting the image context (Doersch et al., 2015), counting the objects (Noroozi et al., 2017), converting gray images to color one (Zhang et al., 2016), keeping global and local consistency (Hjelm et al., 2019). In video domain, typical examples include frame prediction (Diba et al., 2019; Vondrick et al., 2016), optical flow estimation $\mathrm { N g }$ et al., 2018; Zhou et al., 2017; Jayaraman & Grauman, 2017), instance tracking (Wang & Gupta, 2015; Wang et al., 2019b), temporal order or structure prediction (Misra et al., 2016; Fernando et al., 2017; Wei et al., 2018; Xu et al., 2019a). These learnt representations may capture some aspects of low-level image or video structures, but are generally outperformed by those using cross modal information. + +Several cross-modal self-supervised tasks was proposed to enhance single-modality representation power and typical example is audio-visual representation learning (Aytar et al., 2016; Arandjelovic & Zisserman, 2017; Korbar et al., 2018). Meanwhile, some weakly-supervised methods were developed by utilizing web supervision obtained in an automatic way, such as query ID (Chen & Gupta, 2015; Ghadiyaram et al., 2019), and hashtag (Mahajan et al., 2018). Concurrent work (Miech et al., 2020) tried to learn video representations by using narration as supervision with instructional videos (e.g., HowTo100M (Miech et al., 2019)). However, they are limited by the video type. Our CPD is applicable to more general video type and we experiment with a much smaller dataset (0.3M vs. 100M) of both PGC and UGC videos, but achieves a similar performance on UCF101 and HMDB51. Concurrent work (Stroud et al., 2020) proposed a similar framework but required more training videos (0.3M vs. 70M) and richer textual information to obtain similar performance to ours. + +Motion, Audio, and Text. Multi-modal information in videos provides natural cues for learning deep models. Motion or temporal information has been studied as to design proxy tasks to assist cross-modal learning, such as optical flow or tracking (Ng et al., 2018; Wang & Gupta, 2015), frame prediction (Diba et al., 2019; Vondrick et al., 2016), or high-level temporal structure (Wei et al., 2018; Xu et al., 2019a; Fernando et al., 2017). As most video contain synchronized audio and visual signals, audio information has served another common modality to supervised visual learning (Aytar et al., 2016; Arandjelovic & Zisserman, 2017; Korbar et al., 2018). However, both motion and audio information seem to be low-level signals and may lack high-level semantic for cross-modal learning. + +Speech or text has been widely studied as another cross-modal setting in video learning (Sun et al., 2019b; Miech et al., 2019; Dong et al., 2019; Miech et al., 2018; Pan et al., 2016; Plummer et al., 2017). These works mainly aimed to learn a joint video-text embedding where visual and textual cues are adjacent if they are semantically. However, these works focused on learn high-level visualtextual embedding by using the off-the-shelf models as feature extractors. Instead, our proposed CPD framework addresses a different issue of video representation learning from scratch. + +![](images/8a67af6f74daf14c18ca4442d46bf1823105ebabb57dde5b598d121cf0b3a1cf.jpg) +Figure 1: The pipeline of our cross-modal pair discrimination (CPD) framework. First, the visual and text are fed into modality-specific networks for feature extraction. Then, the visual and textual features are mapped into a common 256-dimensional space. The cross-modal framework is learned via video and text pair discrimination, which tries to make corresponding pairs closer than other inconsistent pairs using a softmax criteria. The learnt spatiotemporal features could be deployed directly or fine-tuned for downstream tasks. + +# 3 CROSS-MODAL PAIR DISCRIMINATION + +In this section we provide an detailed description on our proposed cross-modal pair discrimination (CPD) for weakly supervised spatiotemporal feature learning. First, we present the whole framework and analyze its important properties. Then, we describe the training strategy of CPD framework. Finally, we introduce text and video feature extraction networks. + +# 3.1 FRAMEWORK AND ANALYSIS + +Our goal is to propose a weakly supervised representation learning method by exploiting the correlation between each video clip and its associated text information, which could be easily obtained from a variety of sources such as YouTube titles, Instagram captions and automatic speech recognition (ASR). It is generally assumed that these text information contains semantic information, but also might be noisy and irrelevant. Therefore, from technical perspective, we need to design an effective objective function and training strategy to capture this semantic correlation and as well also suppress the effect of noisy and irrelevant information. To this end, we devise a video-text pair discrimination objective and a curriculum learning strategy as follows. + +More formally, as shown in Figure 1, we aim to learn a modality-specific embedding function $\mathcal { F } _ { v }$ and $\mathcal { F } _ { t }$ for the visual and textual information from a set of $N$ video clips and their associated textual information $\{ ( v _ { i } , t _ { i } ) _ { i = 1 } \} ^ { N }$ . Let $\mathbf { f } _ { i } ^ { v }$ and $\mathbf { f } _ { i } ^ { t }$ denote $\mathcal { F } _ { v } ( v _ { i } )$ and $\mathcal { F } _ { t } ( t _ { i } )$ , respectively. These embedding functions would map these two modality into a common space (i.e., $f _ { i } ^ { v } \in \mathbb { R } ^ { d }$ and $f _ { i } ^ { v } \in \mathbb { R } ^ { d } .$ ), and related visual and text information should be close to each other. The embedding functions could be implemented by neural networks which will be clarified in next section. We first focus on how to devise objective function to optimize these embedding functions. Inspired by the work of unsupervised learning in images (Wu et al., 2018), we design a cross-modal pair discrimination objective to learn these two embedding functions. + +Self-instance discrimination. In the original instance-level discrimination framework ( $\mathrm { W u }$ et al., 2018), each image is treated as a distinct class and it would learn a classifier to categorize each image into its own class. This framework could be naturally extended into the setting of video and text pair by directly using feature concatenation, and we call this extension as self-instance discrimination. Formally, this video-text level instance discrimination objective could be implemented with the following softmax criterion: + +$$ +p ( i | ( v , t ) ) = \frac { \exp ( \mathbf { w } _ { i } ^ { v T } \mathbf { f } ^ { v } + \mathbf { w } _ { i } ^ { t T } \mathbf { f } ^ { t } ) } { \sum _ { j = 1 } ^ { N } \exp ( \mathbf { w } _ { j } ^ { v T } \mathbf { f } ^ { v } + \mathbf { w } _ { j } ^ { t T } \mathbf { f } ^ { t } ) } , +$$ + +where the $i ^ { t h }$ video-text pair define a class $i$ , $( \mathbf { w } _ { i } ^ { v } , \mathbf { w } _ { i } ^ { t } )$ is a weight for class $i$ , and the class number is equal to training sample number $N$ . This class weight represent a class prototype for each video-text instance and is probably not easy to optimize as we only have a single sample for each class. Thus, the above parametric classifier could be refined with the following non-parametric variant: + +$$ +p ( i | ( v , t ) ) = \frac { \exp ( \mathbf { f } _ { i } ^ { v T } \mathbf { f } ^ { v } / \tau + \mathbf { f } _ { i } ^ { t T } \mathbf { f } ^ { t } / \tau ) } { \sum _ { j = 1 } ^ { N } \exp ( \mathbf { f } _ { j } ^ { v T } \mathbf { f } ^ { v } / \tau + \mathbf { f } _ { j } ^ { t T } \mathbf { f } ^ { t } / \tau ) } , +$$ + +where $\tau$ is a temperature parameter to control the class concentration level and our training objective is to optimize the likelihood $\begin{array} { r } { \prod _ { i = 1 } ^ { N } p ( i | ( v _ { i } , t _ { i } ) ) } \end{array}$ . This straight forward extension shares the advantage of instance-level discrimination by directly modeling in the joint video-text space. Yet, in fact, the semantic information of text modality is higher than video pixels and we aims at learning video features with the supervision of textual information. To meet this requirement, we propose a refined objective function from the perspective of conditional distribution. + +Cross-pair discrimination. According to the above analysis, we design the objective function by considering conditional distribution $p ( i _ { t } | v )$ and $p ( i _ { v } | t )$ rather than implicitly modeling distribution $p ( v , t )$ . Specifically, we design the following conditional distribution: + +$$ +p ( i _ { t } | v ) = \frac { \exp ( \mathbf { f } _ { i } ^ { t T } \mathbf { f } ^ { v } / \tau ) } { \sum _ { j = 1 } ^ { N } \exp ( \mathbf { f } _ { j } ^ { t T } \mathbf { f } ^ { v } / \tau ) } , +$$ + +where $i ^ { t h }$ text define a text class $i _ { t }$ , and both $\mathbf { f } ^ { t }$ and $\mathbf { f } ^ { v }$ with unit-norm constraint. The conditional distribution $p ( i _ { v } | t )$ could be defined at the same way. We call this framework as cross-pair discrimination, and during training phase, the objective is to maximize the likelihood $\begin{array} { r } { \prod _ { i = 1 } ^ { N } p ( i _ { t } | v _ { i } ) \prod _ { i = 1 } ^ { N } p ( i _ { v } | t _ { i } ) } \end{array}$ . The key difference between Equation (2) and (3) is that we propose to use cross-correlation term $\mathbf { f } ^ { t T } \mathbf { f } ^ { v }$ to replace the self-correlation term $( \mathbf { f } ^ { v T } \mathbf { f } ^ { v } + \mathbf { f } ^ { t T } \mathbf { f } ^ { t } )$ . This cross correlation is more effective to capture the mutual information between visual and textual information, and thereby better at guiding the spatiotemporal feature learning from video with text information as supervision. + +Ranking loss. There is some common ranking loss for cross-modal matching. To well study the effectiveness of proposed cross-modal pair discrimination objective, we also compare with a baseline of ranking loss, which is defined as follows: + +$$ +\mathcal { L } ( v _ { i } , t _ { i } ) = \frac { 1 } { n - 1 } \sum _ { j \neq i } \operatorname* { m a x } ( 0 , \delta + \mathcal { S } ( \mathbf { f } _ { j } ^ { t } , \mathbf { f } _ { i } ^ { v } ) - \mathcal { S } ( \mathbf { f } _ { i } ^ { t } , \mathbf { f } _ { i } ^ { v } ) ) , +$$ + +where each video $v _ { i }$ has a associated text $t _ { i }$ and unrelated text $t _ { j }$ from current batch. ${ \cal { S } } ( \mathbf { f } _ { j } ^ { t } , \mathbf { f } _ { i } ^ { v } )$ is the cosine similarity, $n$ is the batch size and $\delta$ is a margin. We apply Equation (4) in both ways of video with its associated text and text with its video. In experiment, we empirically compare this ranking loss with our designed cross-pair discrimination objective. + +# 3.2 TRAINING CPD + +The training of CPD framework needs to address two technical issues: (1) large number of video-text pair classes; (2) optimization difficulty on noisy video-text datasets by training from scratch. + +Noise-contrastive estimation. In training stage, we adopt noise-contrastive estimation technique (Gutmann & Hyvarinen, 2010) to approximate Equation (3) to solve the computational issues ¨ by the huge numbers of pairs. The basic idea is to transform the multi-class classification problem in Equation (3) into a set of binary classification problem. In the binary classification task, the task is to distinguish between data sample and noise sample. The approximate training objective is to minimize the following loss function: + +$$ +\mathcal { L } = - \mathbb { E } _ { P ( v ) } \left\{ \mathbb { E } _ { P _ { d } ( i _ { t } | v ) } [ \log h ( i _ { t } , v ) ] + m \mathbb { E } _ { P _ { n } ( i _ { t } ^ { \prime } | v ) } [ \log \left( 1 - h ( i _ { t } ^ { \prime } , v ) \right) ] \right\} , +$$ + +where $\begin{array} { r } { h ( i _ { t } , v ) = \frac { p ( i _ { t } | v ) } { p ( i _ { t } | v ) + m p _ { n } ( i _ { t } | v ) } } \end{array}$ , $P _ { d } ( i _ { t } | v )$ is the actual data distribution and $P _ { n } ( i _ { t } ^ { \prime } | v )$ is the uniform distribution for noise, and $m$ denotes the noise frequency. To compute $p ( i _ { t } | v )$ efficiently and avoid large memory consumption, following $\mathbf { W } \mathbf { u }$ et al., 2018), we maintain a memory bank to store the visual and textual features for each training pair. The memory bank is updated dynamically during the training procedure. + +Curriculum learning. To handle the optimization difficulty of directly training from scratch on noisy video-text dataset, we present a curriculum training strategy by resorting to the existing unsupervised pre-trained language models. To relieve the training difficulty, our curriculum learning strategy divides the training procedure into two stages. In the first stage, we fix the pre-trained language model and only update the parameters of visual model and embedding function. The motivation is that the language model is pre-trained well using corpus much larger than ours and the video model is totally trained from scratch. If we train both models simultaneously in the beginning, the random noise produced by video model will destroy the parameters of language model. In the second stage, after the good initialization of video model, we start to jointly train the visual-textual model with a smaller learning rate. + +# 3.3 ARCHITECTURE DESIGN + +Video architecture. For video representation, we use the 3D CNNs to extract spatiotemporal features from a video clip. Specifically, we randomly sample 8 frames from each video clip and sampling stride is 4. Following the implementation of slow stream in the recent SlowFast (Feichtenhofer et al., 2018), all filters from $c o n v _ { 1 }$ to $r e s _ { 3 }$ degenerate temporal convolutions into 2D convolution kernels and it only reserves 3D convolution kernels in $r e s _ { 4 }$ and $r e s _ { 5 }$ without temporal downsampling. We try two kinds of network architectures: (1) 3D ResNet34 trained on $1 1 2 \times 1 1 2 \times 8$ volumes and (2) 3D ResNet50 trained on $2 2 4 \times 2 2 4 \times 8$ volumes. The first tiny network is efficient for ablation study and then we transfer its optimal setting to the larger backbone and frame resolution. We also add a mapping layer to transform the visual features into 256-dimensional embedding space $\mathbf { f } ^ { v }$ and this 256-d vector is $\ell _ { 2 }$ -normalized. + +Text architecture. Our textual stream subnetwork is based on the off-the-shelf language models. We choose Word2vec (Mikolov et al., 2013) and DistilBERT (Devlin et al., 2019; Sanh et al., 2019) as our textual encoders. Word2vec is an unsupervised word encoder, pre-trained by reconstructing the surrounding words of the continue sentences. We average word vectors which are 300 dimensional as textual encoder. BERT (Devlin et al., 2019) encodes long sentences by predicting the missing words given their bidirectional context, and DistilBERT achieves comparable performance with a faster and lighter model via knowledge distillation (Hinton et al., 2015). We average word embeddings of title generated by DistilBERT and obtain 768 dimensional text feature. Finally, two fully connected layers with ReLU and Batch Normalization (Ioffe & Szegedy, 2015) are added to our textual encoder to obtain textual feature $\mathbf { f } ^ { t }$ in the common embedding space, which is also $\ell _ { 2 }$ -normalized. + +# 4 EXPERIMENTS + +In this section, we present the experimental results of our proposed CPD framework. First, we describe the training and evaluation datasets with implementation details. Then, we conduct ablation study on our proposed CPD framework. Finally, we verify the effectiveness of CPD from two aspects: weakly-supervised representation learning and representation transfer. + +# 4.1 DATASETS + +In our experiment, we pre-train our CPD framework on two video-text datasets: Kinetics-210k (Kay et al., 2017) and Instagram-300k (Duan et al., 2020). Then, we fine-tune the video model on three human action datasets: Kinetics400 (Kay et al., 2017), UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011). + +Kinetics-210k. Following the recent self-supervised methods (Wang et al., $2 0 1 9 \mathrm { a }$ ; Korbar et al., 2018; Han et al., 2019), we utilize Kinetics (Kay et al., 2017) dataset for weakly-supervised pretraining of CPD. It is often called Kinetics400 since it has 400 action classes, but we count training video number as we do not use any class information for weakly-supervised representation learning. Due to invalid urls and data cleaning, the collected dataset contains around 210k video-text pairs, and thus we call this dataset as Kinetics-210k. To construct video-text pairs, we equip each clip with the video title directly crawled from YouTube, termed as Kinetics-title. As the original title may be very noisy, we pre-process the text information in two ways. First, we delete special symbols and characters such as non-English words and emoji, termed as Kinetics-title-clean. Second, we use StanfordNLP (Qi et al., 2018) to obtain the dependency tree of sentences in titles and only reserve verbs and nouns, named Kinetics-title-tree. + +Instagram-300k. To avoid data bias in Kinetics caused by human annotation (i.e., trimmed videos with an action), we further verify the effectiveness our CPD model on an uncurated web video dataset (Duan et al., 2020). This new dataset is constructed from Instagram by searching action label of Kinetics-400 but without any manual filtering. Due to limited computation resource and also for fair comparison with pretraining on Kinetics-210k, we randomly sample 300k from the original web video dataset, termed as Instagram-300k. + +An important difference is that the these videos are with User Generated Content (UGC) and accompanied by captions uploaded by users. Therefore, its video content distribution is much different with those in Profession Generated Content (PGC) in UCF101 and HMDB51, and the text noise is also much higher. So, it is more challenging to train a pre-trained CPD model on Instagram-300k. + +UCF101 and HMDB51. We evaluate the generalization of our pre-trained models by fine-tuning on two small human action datasets: UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011), which contain $1 3 \mathrm { k }$ videos of 101 classes and 7k video of 51 classes respectively. We report ablation study on the first split and report average performance over three splits for fair comparison. + +# 4.2 IMPLEMENTATION DETAILS + +Weakly supervised learning of CPD. We train our CPD model on video-text datasets and use video-text retrieval on 1k unseen video-text pairs as validation set duration training. Specifically, 8 frames are sampled from each video clip and the sampling stride is 4. We use SGD to optimize our objective and the training parameters include a momentum of 0.9 and 1e-4 for weight decay. We set temperature parameter $\tau = 0 . 0 7$ and noise frequency $m$ to 4096. In the beginning, we fix the pre-trained language model and the learning rate is set as 0.2. When the retrieval performance on validation set saturates (170 epochs for 3D ResNet34 and 110 epochs for 3D ResNet50), we start to update the language model with learning rate of 3e-5 and decrease the rest learning rate to 0.02. The maximize training number is 250 epochs. For input size of $1 1 2 \times 1 1 2 \times 8$ , the mini-batch size is 64 clips per GPUs and 16 clips per GPUs for input size of $2 2 4 \times 2 2 4 \times 8$ . We use 8 GPUs for training. + +Evaluation on representation learning. We first verify our CPD learned representation by employing a shallow classifier on frozen features. Specifically, we utilize $\mathbf { k }$ -Nearest Neighbor (kNN) and linear classifier based on extracted features for classification. For video feature extraction, we sample 10 clips from each video and each clip contains 8 frames with 4 sampling stride. The 256- dimensional embedding feature and the output of global average pooling are extracted as features. The extracted features over 10 clips in a video are averaged as a video-level representation. We choose cosine distance as distance metric in kNN and set $k = 2 5$ . As for linear classifier, a fully connected layer after Batch Normalization is added with cross-entropy loss. We adopt Adam with learning rate of 1e-3 and reduce by a factor of 10 every 10 epochs, stopping at 30 epochs. + +Evaluation on representation transfer. A main goal of representation learning is to transfer them to downstream tasks. We fine-tune the learned spatiotemporal representation on the UCF101, HMDB51 and a small fraction of Kinetics400. During fine-tuning, 16 frames with stride 4 are sampled as input. We simply replace the embedding layer of video model with a new fully-connected layer and multi-way softmax for action recognition. The classifier is trained using the SGD optimizer with an initial learning rate 1e-2 and weight decay 5e-4. Learning rate is decreased twice by a factor of 10 when the validation loss saturates. During testing, for each video, we uniformly sample 10 clips and each clip contains 3 crops, following the common practice (Feichtenhofer et al., 2018). + +# 4.3 ABLATION STUDY + +In this study, we pre-train our CPD models on Kinetics-210k dataset and choose the task of representation transfer by fine tuning on UCF101 split 1 for evaluation. + +
TrainingstrategyAccuracy(%)
Random init.50.0
Direct fine-tuning81.3
Curr. learning182.2
Curr.learning284.2
+ +(b) Study on training strategies. + +
Textual encoderDataAccuracy(%)
Random Init.-50.0
Word2vecTree83.1
DistilBERTTree82.1
Word2vecClean82.5
DistilBERTClean84.2
+ +(c) Study on textual encoders. + +Table 1: Ablation study on UCF101 by fine tuning a pre-trained CPD model from Kinetics-210k. + +
Objective functionAccuracy(%)
Random init.50.0
Ranking loss79.9
Self-instance Dis.51.1
Cross-pair Dis.82.2
+ +(a) Study on loss functions. + +Table 2: Evaluation on weakly-supervised representation learning without fine-tuning. Top-1 classification accuracy is reported on Kinetics-400 validation set. + +
BackbonePre-trained Sup.Layer (Dim)KNNLC
3D-ConvNet (Kay et al.,2017)Kinetics-400Label--56.1
3D ResNet34 (Hara et al., 2018) 3D ResNet50 (ours)Kinetics-400 Label Kinetics-400 Label--60.1 73.2
ResNet50ImageNet Label-- 42.856.1
3DResNet34res5 (2048)
Instagram-300k Captionemb (256)34.537.3
3D ResNet34Instagram-300k Captionres5 (512)36.144.6
3D ResNet50Instagram-300k Captionemb (256)51.151.7
3DResNet50Instagram-300k Captionres5 (2048)51.155.4
3DResNet34Kinetics-210k Titleemb (256)49.950.8
3D ResNet34Kinetics-210k Titleres5 (512)50.153.3
3DResNet50Kinetics-210k Titleemb (256)58.059.6
3DResNet50Kinetics-210k Titleres5 (2048)58.263.8
+ +Objective function. We compare three objective functions for cross-modal pair discrimination described in Section 3.1. We pre-train models by utilizing DistilBERT as textual encoder without finetuning and the experimental results are reported in Table 1a. Self-instance discrimination almost has no contribution to learn effective representation as there is no cross-modal correlation modeling. Cross-pair discrimination gives a better performance than ranking loss as cross-pair discrimination can construct negative video-text pairs from entire dataset while ranking loss is only optimized by negative pairs from current batch. More theoretical analysis can be found in Section. A.1 of the Appendix. + +Curriculum learning. We design different training strategies from noisy video-text datasets. The first strategy is to fine-tune the pre-trained textual encoder directly at the beginning. Then we compare with stage I and stage II of curriculum learning proposed in Section 3.2. All these strategies are pre-trained on Kinetics-title-clean. The numerical results are summarized in Table 1b. Fixing the pre-trained language model gives better performance than direct fine-tuning at the beginning $( + 0 . 9 \% )$ . We ascribe this to the fact that the random noise produced by video model destroy the well pre-trained textual encoder at the beginning. Also, fine-tuning the language model after the video model is well initialized further boost the accuracy by $2 . 0 \%$ . + +Different textual information. In this experiment, we choose video-text pairs from Kinetics-titletree, Kinetics-title-clean datasets and utilize Word2vec and DistilBERT as a textual extractor. The experimental results are reported in Table 1c. For textual encoder, abundant and video-specific text information benefits to train our CPD model with stronger language model such as DistilBERT according to the performance difference between Kinetics-title-tree and Kinetics-title-clean $( 8 2 . 1 \%$ vs. $8 4 . 2 \%$ ). As for shallow textual encoder (e.g., Word2vec), simple text information from Kineticstitle-tree dataset gives better performance than abundant text information $( 8 3 . 1 \%$ vs. $8 2 . 5 \%$ ). From above observation, it can be concluded that Word2vec is more good at concise and accurate text while DistilBERT can handle more complex and noisy sentences which is close to realistic setting. Also, it is affordable to utilize strong language models due to our curriculum learning strategy and lightweight DistilBERT model. + +# 4.4 EVALUATION ON REPRESENTATION LEARNING + +To evaluate our learned representation, we report the classification performance on validation set of Kinetics via training shallow classifiers on frozen features as shown in Table 4.3. We perform kNN classifiers and linear classifiers (LC) on the embedding features or visual features from global average pooling after res5. In this shallow learning setting, we also compare with ImageNet pretraining representation (ResNet50) by using the same classifier. First, the representation learnt from + +Table 3: Evaluation on representation transfer by fine-tuning. We compare our CPD model with other methods trained on different type of supervision. + +
MethodSupervisionBackbonePre-trained DatasetUCF101HMDB51
RandomInit. (Hara etal.,2018)3DResNet1842.417.1
Kinetics Pre-trained (Hara etal., 2018)Action label3DResNet50Kinetics89.361.0
Supervised SOTA (Xie et al.,2018)Action labelS3DKinetics96.875.9
Shuffle& Learn (Misra et al.,2016)Order verificationCaffeNetUCF101/HMDB5150.218.1
OPN (Lee et al., 2017)Sequence orderVGGNetUCF101/HMDB5159.823.8
CMC (Tian et al., 2019)Optical flowCaffeNetUCF10155.3-
O3N (Fernando et al.,2017)Odd-one-outAlexNetUCF10160.332.5
MASN (Wang et al.,2019a)MotionC3DKinetics-40061.233.4
COP (Xu et al.,2019b)Clip order3D ResNet10UCF10164.929.5
DPC (Han et al.,2019)Prediction3DResNet34Kinetics-40075.735.7
CBT(Sun et al.,2019a)Audio(Text)/ContextS3DKinetics-60079.544.6
AVTS (Korbar et al.,2018)AudioI3DKinetics-60083.753.0
AVTS (Korbar et al., 2018)AudioMC3Audioset-1.8M89.061.6
XDC (Alwassel et al.,2019)AudioR(2+1)DKinetics-40084.247.1
XDC (Alwassel et al.,2019)AudioR(2+1)DIG-65M91.563.1
MIL-NCE (Miech et al., 2020)Audio(Text)S3DHT-100M91.361.0
TWS (Stroud et al.,2020)Text (Title,Des,Tag etc.)S3D-GWVT-70M90.365.3
CPD (Ours)Caption3DResNet50Instagram300k89.963.8
CPD (Ours)Title3DResNet50Kinetics210k90.563.6
+ +Kinetics-210k generally outperforms that of Instagram- $3 0 0 \mathrm { k }$ . The reason could be ascribed to the video distribution gap between UGC (Instagram) and PGC (Youtube), and also much noisier textual information in Instagram-300k. Second, we compare with ImageNet pretrained features, and our CPD representation is better under the same backbone. Finally, we compare with some end-toend trained representations with action labels, and there is still a performance gap between our representation and supervised end-to-end representation (e.g. $6 3 . 8 \%$ vs. $7 3 . 2 \%$ ). + +# 4.5 EVALUATION ON REPRESENTATION TRANSFER + +Transferring learned representation to downstream tasks is a main goal of representation learning. We transfer them to action recognition task on small datasets, namely UCF101 and HMDB51. We compare our CPD model pre-trained on Instagram-300k and Kinetics-210k with a randomly initialized network, self-supervised methods solely based on visual information, including Shuffle & Learn (Misra et al., 2016), CMC (Tian et al., 2019), MASN (Wang et al., 2019a), COP (Xu et al., 2019b), DPC (Han et al., 2019) and so on, and representation learning methods based on multimodal information (e.g., audio, text), including CBT (Sun et al., 2019a), AVTS (Korbar et al., 2018), XDC (Alwassel et al., 2019), MIL-NCE (Stroud et al., 2020), and TWS (Stroud et al., 2020). + +As shown in Table 3, our CPD models generally outperform those self-supervised learning approaches of only using visual information $( \geq 1 0 \%$ on UCF101 and $\geq 2 0 \%$ on HMDB51), which indicates that cross-modal information is useful cue for visual representation learning. Meanwhile, our CPD representations obtain comparable performance to the concurrent works (i.e., MIL-NCE and TWS) of using text as weak supervision. However, our CPD uses a much smaller pre-training dataset of around 0.3M videos, while the other methods uses 70M-100M videos. Training a CPD model on a such large-scale dataset is almost impossible for a university lab with limited computational facilities. Our work demonstrates that pre-training a relatively small video-text dataset is also possible to match the SOTA performance, and this is quite meaningful and practicable for university lab. Finally, we notice that the gap of CPD models learned from Instagram- ${ 3 0 0 } \mathrm { k }$ and Kinetics- ${ \it 2 0 0 k }$ is very small, indicating that our CPD model can effectively handle high noise in text. + +# 5 CONCLUSION + +In this paper, we have presented a general cross-modal pair discrimination (CPD) framework to capture the correlation between a video clip and its associated text from real word and adopt noisecontrastive estimation to approximate the objective. Without fine-tuning, the learned models obtain competitive results for action classification on Kinetics dataset with a shallow classifier. Also, our visual models provide an effective initialization to fine-tune on the datasets of downstream task, and matches the state-of-the-art performance with a much smaller pre-training dataset. + +# REFERENCES + +Humam Alwassel, Dhruv Mahajan, Lorenzo Torresani, Bernard Ghanem, and Du Tran. Selfsupervised learning by cross-modal audio-video clustering. 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Lowe. Unsupervised learning of depth and ego-motion from video. In 2017 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 6612–6619, 2017. + +# A TRAINING DETAILS OF CPD + +we adopt noise-contrastive estimation technique (NCE) to approximate objective function in Equation (5) in our main paper. The purpose of NCE is to transform the multi-class classification problem into a set of binary classification problems by comparing data distribution against noise distribution. So $p _ { n }$ is noise distribution and we formalize it as a uniform distribution: $\begin{array} { r } { { p } _ { n } = \frac { 1 } { N } } \end{array}$ , where $N$ is the number of video-text pairs. $h ( i _ { t } , v )$ is the posterior probability of feature from the data distribution which means video and text are matched. $m$ is the number of negative pairs and we set it as 4096. For each video feature $\mathbf { f } ^ { v }$ , we take its related text feature $\mathbf { f } _ { i } ^ { t }$ and sample 4096 unrelated text features $\mathbf { f } _ { j } ^ { t }$ which are all from memory bank. The FPS of training videos are 30. The code of CPD will be released. + +# A.1 ANALYSIS ON DIFFERENT LOSS FUNCTIONS + +More insight about why our loss is better than ranking loss could be found from gradient backpropagation. Let $\mathbf { f } ^ { t + }$ and $\mathbf { f } ^ { t - }$ represent the associated and unrelated text feature. For ranking loss, the negative gradient w.r.t $\mathbf { f } ^ { v }$ is $\bar { \mathbf { f } ^ { t + } } - \mathbf { f } ^ { t - }$ if $\mathcal { L } > 0$ else 0. For CPD loss, it is $[ 1 - h ( i _ { t } ^ { + } , v ) ] / \bar { \bf \Delta } \bar { \bf f } ^ { t + } -$ $\sum h ( i _ { t } ^ { - } , v ) / \tau \mathbf { f } ^ { t - }$ . We observe our loss assign different weights to different examples based on their posterior probability $h$ , which helps learn from hard examples while the ranking loss treats them equally. + +# B REPRESENTATION TRANSFER ON KINETICS + +
MethodThe Amount ofLabeled Data
1%10%20%
Fromscratch0.310.733.3
ImageNet Inflation12.836.845.7
Ours (Instagram-300k)18.741.347.4
Ours (Kinetics-210k)25.943.147.8
+ +Table 4: Results of classification with small amount of labeled data on Kinetics-400 validation set (showing top-1 accuracy). We utilize 3D ResNet34 as backbone and pre-train it on Kinetics-210k and Instagram-210k. + +Our weakly-supervised pre-trained representation can be an efficient initialization when training the model with only a small amount of labeled data. We randomly choose a small fraction of Kinetics400 training set as labeled data and fine-tune the pre-trained model on it. We report the performance of top-1 accuracy which is trained on labeled subset of $1 \%$ , $10 \%$ and $20 \%$ of the entire dataset in Table 4. We compare our method with training from scratch and ImageNet inflated model as baselines. Our method significantly surpasses the baselines on all present proportion of labeled subset especially when the amount of labeled data is extremely small. When only $1 \%$ of data is labeled, training from scratch can not learn anything yet our model achieves $1 8 . 7 \%$ and $2 5 . 9 \%$ top-1 accuracy. Both our CPD pre-trained models on Instagram and Kinetics outperform the ImageNet pre-trained models. + +# C EVALUATION ON ZERO-SHOT CLASSIFICATION + +We evaluate our visual-textual embedding of CPD model with zero-shot classification on UCF101 and Kinetics-400 without any fine-tuning in Table 5. We transform class labels and video clips into the same embedding space and recognize the video clip to its closest class with cosine distance. We compare our method with Mettes et al. (Mettes & Snoek, 2017) which realizes zero-shot localization and classification of human action in video via spatial-aware object embeddings on UCF101. Following (Mettes & Snoek, 2017), we select different classes for 10 times and average their accuracies for testing except the class number is 101. We outperform for every number of testing classes. For Kinetics-400, we achieve top-1 accuracy of $4 3 . 7 \%$ without fine-tuning and training label. In addition, top-1 accuracy of 20 random classes reaches to $7 4 . 4 \%$ , which shows the strong capability of our visual-textual embedding. + +Table 5: Top-1 accuracy of zero-shot classification on UCF-101 and Kinetics-400. We outperform other methods without any extra labeled data and training procedure after pre-training on Kinetics210k. + +
MethodsUCF-101Kinetics-400
TrainTestSplitAcc.TrainTestSplitAcc.
Mettes (Mettes& Snoek,2017)1101332.8--11
Ours(3D ResNet34)=101340.6=400138.2
Ours(3D ResNet50)-101339.9-400143.7
Mettes(Mettes& Snoek,2017)-501040.4-111
Ours(3D ResNet34)501047.2=1001055.3
Ours(3D ResNet50)501044.8=1001057.4
Mettes (Mettes& Snoek,2017)201051.21--=
Ours(3D ResNet34)201054.4=201073.1
Ours(3D ResNet50)201058.1=201074.4
+ +# D ANALYZE TEXT INFORMATION + +# D.1 ANALYSIS ON KINETICS TITLE + +Table 6: Analyze text information of Kinetics-210k datasets. At Least One: The proportion of text information that contains at least one word in action classes of Kinetics-400. All: The proportion of text information that contains the entire action class. $R e l$ : The proportion of word in text information that is relevant to action classes. + +
DatasetsAt Least One(%)A11(%)Rel(%)
Kinetics-title-tree90.544.346.3
Kinetics-title-clean91.638.426.0
+ +We provide an analysis of text information we used and the result in Table 6. First, there exists a large overlap between action class and text information (more than $90 \%$ for at least one word and more than $38 \%$ for complete action class). However, the titles also contain many other words and noisier information than action classes. Only $26 \%$ of words in Kinetics-title-clean are relevant to action classes. + +![](images/ad7bcc7280b2e78acea6087fc4a4e1cc11c2bec2263d73b9806c870843806321.jpg) +Figure 2: List of top 10 and bottom 10 kinetics classes sorted by the frequency of at least one word in label occurring in according title of Kinetics-title-clean dataset. Zoom in for more details. We also report the per-class accuracy of top 10 and bottom 10 classes sorted by word overlapping in Figure 2 and see that this accuracy is not positively correlated with word overlapping percentage. Finally, we provide some examples of videos and their titles from Kinetics-210k in Figure 3. + +# D.2 VISUALIZATION OF INSTAGRAM CAPTION + +Since videos from Instagram-300k are not annotated or filtered by human, both of their visual and textual information are very noisy. Figure 4 demonstrates some examples of videos and their associated captions. Figure 4a presents an example of high-quality video and relative accurate caption that both are about folding napkin. Many captions describe some useful information but also contain noisy text that is not related to video content (e.g., summerdays and gettingtattooed in Figure 4b and very long sentences in Figure 4d). In addition, there are some correct but not totally accurate descriptions. Figure 4c shows that the action in video is shot putting rather than spinning (appears in associated caption). Figure 4f illustrates that a person is climbing but its caption is mainly about high jumpping. Figure 4e shows that video content can also be noisy due to low video quality and shot transformation. + +![](images/18a06d7e24dcf2ca11f47c9494c912565a377463e4f0eff00aa54fe17b8bc0da.jpg) +Figure 3: Examples of video and title pairs from Kinetics-210k. + +Bird of Paradise Napkin Fold #napkinfolding #napkinart #napkin #yellow #napkinfold #birdofparadise #kidscrafts #diy #video #tutorial #decoration #tablesetting + +![](images/2bc24d3718235e6d923f7c302986676576fe887ce7fc1a26163532e11a745b6f.jpg) +Figure 4: Examples of videos and their associated captions from Instagram-300k. + +#roadtrip from #britishcolumbia to #alberta. drivingallnight #mountains #trees #summer #summerdays #summerlovin #highway #music #playlist #carpoolkaraoke #notreallyasinger #butitry #gettingtattooed #buyingatruck #duoting + +Day 1 of learning to spin $\circledast$ . A work in progress but having fun with it! + +Good morning beautiful people of #god #grateful for another day he has given me #prayingfor #guidance and #protection in #Jesusname I #pray #amen Early #dinnersettings today #potroastbeef #bakechicken #vegetables WHAT #RICEANDPEAS WOULD YOU EAT WITH THIS? #pyjamachef #foodbloger #pyjamachef #healthyeating #alltypeoffood #lovecooking + +My pets are unruly... Who wants them...??? #computerwork #selfishpets + +High Jump Challenge Very funny $\textcircled { \dag }$ ?? #boulder #highjump #fitness #climbing #ninjawarrior #ninjawarriorswitzerland #challenge #funny \ No newline at end of file diff --git a/parse/train/Bw7VC-DJUM/Bw7VC-DJUM_content_list.json b/parse/train/Bw7VC-DJUM/Bw7VC-DJUM_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..543460927aa054225d8dc32ad2f29e0ba38417e6 --- /dev/null +++ b/parse/train/Bw7VC-DJUM/Bw7VC-DJUM_content_list.json @@ -0,0 +1,1873 @@ +[ + { + "type": "text", + "text": "LEARNING SPATIOTEMPORAL FEATURES VIA VIDEO AND TEXT PAIR DISCRIMINATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Current video representations heavily rely on learning from manually annotated video datasets which are time-consuming and expensive to acquire. We observe videos are naturally accompanied by abundant text information such as YouTube titles and Instagram captions. In this paper, we leverage this visual-textual connection to learn spatiotemporal features in an efficient weakly-supervised manner. We present a general cross-modal pair discrimination (CPD) framework to capture this correlation between a video and its associated text. We train our CPD models on both standard video dataset (Kinetics-210k) and uncurated web video dataset (Instagram-300k) to demonstrate its effectiveness. Without further fine-tuning, the learnt models obtain competitive results for action classification on Kinetics under the linear classification protocol. Moreover, our visual model provides an effective initialization to fine-tune on downstream tasks, which yields a remarkable performance gain for action recognition on UCF101 and HMDB51, compared with the existing state-of-the-art self-supervised training methods. In addition, our CPD demonstrates that pre-training a relatively small dataset is able to yield a comparable performance to those methods of using order magnitude more data, which is meaningful and practicable for the scenarios with limited computational facilities. ", + "bbox": [ + 233, + 265, + 764, + 501 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 520, + 336, + 536 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep learning has made a remarkable progress for visual recognition in both image and video domain (Krizhevsky et al., 2012; He et al., 2016; Carreira & Zisserman, 2017; Feichtenhofer et al., 2018) by training powerful neural networks on large-scale manually annotated datasets (e.g., ImageNet (Deng et al., 2009) and Kinetics (Kay et al., 2017)). More importantly, it is well-established that this supervised pre-training on large-scale datasets would benefit the downstream tasks (e.g., object detection (Ren et al., 2015), pose estimation (He et al., 2017), and temporal action detection (Zhao et al., 2017)), in particular when the target datasets are relatively small. Yet, annotating a large-scale dataset for training such deep neural networks is costly and time-consuming, and even more challenging for video due to its various temporal structure and complex semantics. As a result, the existing video datasets size is still smaller than ImageNet in terms of training samples and classes. On the other hand, videos typically contain richer structure with abundant side information such as motion (Diba et al., 2019; $\\mathrm { N g }$ et al., 2018), audio (Arandjelovic & Zisserman, 2017; Korbar et al., 2018), and text (Miech et al., 2019; Sun et al., 2019b). So these expected these associated modalities are expected to provide useful cues to learn video representations in a more efficient way. ", + "bbox": [ + 174, + 547, + 825, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Language or text is probably the most natural and easy way to describe the semantic information of a video, and the associated textual information could be easily acquired when collecting video dataset (Rohrbach et al., 2017; Miech et al., 2019) from Internet or Movie. We argue that this correlation between a clip and its associated text could serve as an alternative supervision to learn video representation from scratch. This is different from some recent works (Sun et al., 2019b; Miech et al., 2019), in which these abundant textual information has been used to learn a high-level visual-text embedding applied to text-to-video retrieval or video captioning. Intuitively, it is more challenging to learn a general visual representation solely from text information without any human annotation, for reasons such as large numbers of noise in text, lacking careful initialization, and being hard to design an effective objective. ", + "bbox": [ + 174, + 750, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we aim to learn effective video representation from noisy and diverse textual information, which could serves as the basis for a variety of downstream tasks. Basically, we learn a mapping of text and video into a shared embedding space and leverage their correlation as supervision signal. The technical difficulty is how to design an effective objective function, that is capable of modeling this complex visual-textual correlation and as well easily optimized by training from scratch on noisy datasets. Inspired by unsupervised feature learning in images (Wu et al., 2018; Tian et al., 2019), we present a cross-modal pair discrimination (CPD) framework, which tries to recognize each video and text pair into a class via a non-parametric classifier. To solve the computational issues imposed by the huge numbers of pair classes, we adapt noise-contrastive estimation technique (Gutmann & Hyvarinen, 2010) to approximate the original loss function. ¨ ", + "bbox": [ + 173, + 895, + 821, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Specifically, we learn the CPD framework from web videos with the associated title or caption that could be directly crawled from web platforms such as YouTube (Kay et al., 2017) and Instagram (Duan et al., 2020). We utilize the off-the-shelf language models such as BERT (Devlin et al., 2019) or Word2vec (Mikolov et al., 2013) and devise a curriculum learning strategy to progressively train the video models. We first test the generalization ability of learned video representation by CPD on the Kinetics dataset (Kay et al., 2017) by using shallow classifiers such k-NN and linear classifier. It shows that our learned spatiotemporal features obtain promising results which are comparable to some supervised learning methods on the Kinetics dataset (Kay et al., 2017). Then, we investigate the generalization power of learned spatiotemporal features of CPD by fine-tuning on the Kinetics (Kay et al., 2017), UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011) datasets, demonstrating that our method obtain superior performance to previous state-of-the-art self-supervised methods and comparable performance to the very recent methods of using orders of magnitude more videos (70M-100M vs. 0.3M). ", + "bbox": [ + 174, + 222, + 825, + 401 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 412, + 344, + 428 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Self/Weakly Supervised Representation Learning. Self supervised representation was popular in both image and video domains by designing various proxy tasks. In image domain, for instance, these tasks could be predicting the image context (Doersch et al., 2015), counting the objects (Noroozi et al., 2017), converting gray images to color one (Zhang et al., 2016), keeping global and local consistency (Hjelm et al., 2019). In video domain, typical examples include frame prediction (Diba et al., 2019; Vondrick et al., 2016), optical flow estimation $\\mathrm { N g }$ et al., 2018; Zhou et al., 2017; Jayaraman & Grauman, 2017), instance tracking (Wang & Gupta, 2015; Wang et al., 2019b), temporal order or structure prediction (Misra et al., 2016; Fernando et al., 2017; Wei et al., 2018; Xu et al., 2019a). These learnt representations may capture some aspects of low-level image or video structures, but are generally outperformed by those using cross modal information. ", + "bbox": [ + 174, + 443, + 825, + 583 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Several cross-modal self-supervised tasks was proposed to enhance single-modality representation power and typical example is audio-visual representation learning (Aytar et al., 2016; Arandjelovic & Zisserman, 2017; Korbar et al., 2018). Meanwhile, some weakly-supervised methods were developed by utilizing web supervision obtained in an automatic way, such as query ID (Chen & Gupta, 2015; Ghadiyaram et al., 2019), and hashtag (Mahajan et al., 2018). Concurrent work (Miech et al., 2020) tried to learn video representations by using narration as supervision with instructional videos (e.g., HowTo100M (Miech et al., 2019)). However, they are limited by the video type. Our CPD is applicable to more general video type and we experiment with a much smaller dataset (0.3M vs. 100M) of both PGC and UGC videos, but achieves a similar performance on UCF101 and HMDB51. Concurrent work (Stroud et al., 2020) proposed a similar framework but required more training videos (0.3M vs. 70M) and richer textual information to obtain similar performance to ours. ", + "bbox": [ + 174, + 589, + 825, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Motion, Audio, and Text. Multi-modal information in videos provides natural cues for learning deep models. Motion or temporal information has been studied as to design proxy tasks to assist cross-modal learning, such as optical flow or tracking (Ng et al., 2018; Wang & Gupta, 2015), frame prediction (Diba et al., 2019; Vondrick et al., 2016), or high-level temporal structure (Wei et al., 2018; Xu et al., 2019a; Fernando et al., 2017). As most video contain synchronized audio and visual signals, audio information has served another common modality to supervised visual learning (Aytar et al., 2016; Arandjelovic & Zisserman, 2017; Korbar et al., 2018). However, both motion and audio information seem to be low-level signals and may lack high-level semantic for cross-modal learning. ", + "bbox": [ + 174, + 750, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Speech or text has been widely studied as another cross-modal setting in video learning (Sun et al., 2019b; Miech et al., 2019; Dong et al., 2019; Miech et al., 2018; Pan et al., 2016; Plummer et al., 2017). These works mainly aimed to learn a joint video-text embedding where visual and textual cues are adjacent if they are semantically. However, these works focused on learn high-level visualtextual embedding by using the off-the-shelf models as feature extractors. Instead, our proposed CPD framework addresses a different issue of video representation learning from scratch. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/8a67af6f74daf14c18ca4442d46bf1823105ebabb57dde5b598d121cf0b3a1cf.jpg", + "image_caption": [ + "Figure 1: The pipeline of our cross-modal pair discrimination (CPD) framework. First, the visual and text are fed into modality-specific networks for feature extraction. Then, the visual and textual features are mapped into a common 256-dimensional space. The cross-modal framework is learned via video and text pair discrimination, which tries to make corresponding pairs closer than other inconsistent pairs using a softmax criteria. The learnt spatiotemporal features could be deployed directly or fine-tuned for downstream tasks. " + ], + "image_footnote": [], + "bbox": [ + 243, + 104, + 753, + 261 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 367, + 823, + 396 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 CROSS-MODAL PAIR DISCRIMINATION ", + "text_level": 1, + "bbox": [ + 176, + 416, + 531, + 433 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section we provide an detailed description on our proposed cross-modal pair discrimination (CPD) for weakly supervised spatiotemporal feature learning. First, we present the whole framework and analyze its important properties. Then, we describe the training strategy of CPD framework. Finally, we introduce text and video feature extraction networks. ", + "bbox": [ + 174, + 446, + 825, + 503 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 FRAMEWORK AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 521, + 415, + 535 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our goal is to propose a weakly supervised representation learning method by exploiting the correlation between each video clip and its associated text information, which could be easily obtained from a variety of sources such as YouTube titles, Instagram captions and automatic speech recognition (ASR). It is generally assumed that these text information contains semantic information, but also might be noisy and irrelevant. Therefore, from technical perspective, we need to design an effective objective function and training strategy to capture this semantic correlation and as well also suppress the effect of noisy and irrelevant information. To this end, we devise a video-text pair discrimination objective and a curriculum learning strategy as follows. ", + "bbox": [ + 174, + 546, + 825, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "More formally, as shown in Figure 1, we aim to learn a modality-specific embedding function $\\mathcal { F } _ { v }$ and $\\mathcal { F } _ { t }$ for the visual and textual information from a set of $N$ video clips and their associated textual information $\\{ ( v _ { i } , t _ { i } ) _ { i = 1 } \\} ^ { N }$ . Let $\\mathbf { f } _ { i } ^ { v }$ and $\\mathbf { f } _ { i } ^ { t }$ denote $\\mathcal { F } _ { v } ( v _ { i } )$ and $\\mathcal { F } _ { t } ( t _ { i } )$ , respectively. These embedding functions would map these two modality into a common space (i.e., $f _ { i } ^ { v } \\in \\mathbb { R } ^ { d }$ and $f _ { i } ^ { v } \\in \\mathbb { R } ^ { d } .$ ), and related visual and text information should be close to each other. The embedding functions could be implemented by neural networks which will be clarified in next section. We first focus on how to devise objective function to optimize these embedding functions. Inspired by the work of unsupervised learning in images (Wu et al., 2018), we design a cross-modal pair discrimination objective to learn these two embedding functions. ", + "bbox": [ + 173, + 665, + 825, + 791 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Self-instance discrimination. In the original instance-level discrimination framework ( $\\mathrm { W u }$ et al., 2018), each image is treated as a distinct class and it would learn a classifier to categorize each image into its own class. This framework could be naturally extended into the setting of video and text pair by directly using feature concatenation, and we call this extension as self-instance discrimination. Formally, this video-text level instance discrimination objective could be implemented with the following softmax criterion: ", + "bbox": [ + 173, + 797, + 823, + 882 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a66c0eb0cdfc89197b3faed94529309a8a4b782ba3992eee2fd5c941ae99591d.jpg", + "text": "$$\np ( i | ( v , t ) ) = \\frac { \\exp ( \\mathbf { w } _ { i } ^ { v T } \\mathbf { f } ^ { v } + \\mathbf { w } _ { i } ^ { t T } \\mathbf { f } ^ { t } ) } { \\sum _ { j = 1 } ^ { N } \\exp ( \\mathbf { w } _ { j } ^ { v T } \\mathbf { f } ^ { v } + \\mathbf { w } _ { j } ^ { t T } \\mathbf { f } ^ { t } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 888, + 642, + 929 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the $i ^ { t h }$ video-text pair define a class $i$ , $( \\mathbf { w } _ { i } ^ { v } , \\mathbf { w } _ { i } ^ { t } )$ is a weight for class $i$ , and the class number is equal to training sample number $N$ . This class weight represent a class prototype for each video-text instance and is probably not easy to optimize as we only have a single sample for each class. Thus, the above parametric classifier could be refined with the following non-parametric variant: ", + "bbox": [ + 174, + 102, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/613e69cbdf5bd68b5717f7cbba772f967ea4a61996bfc845acc48ab252233f13.jpg", + "text": "$$\np ( i | ( v , t ) ) = \\frac { \\exp ( \\mathbf { f } _ { i } ^ { v T } \\mathbf { f } ^ { v } / \\tau + \\mathbf { f } _ { i } ^ { t T } \\mathbf { f } ^ { t } / \\tau ) } { \\sum _ { j = 1 } ^ { N } \\exp ( \\mathbf { f } _ { j } ^ { v T } \\mathbf { f } ^ { v } / \\tau + \\mathbf { f } _ { j } ^ { t T } \\mathbf { f } ^ { t } / \\tau ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 166, + 653, + 207 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\tau$ is a temperature parameter to control the class concentration level and our training objective is to optimize the likelihood $\\begin{array} { r } { \\prod _ { i = 1 } ^ { N } p ( i | ( v _ { i } , t _ { i } ) ) } \\end{array}$ . This straight forward extension shares the advantage of instance-level discrimination by directly modeling in the joint video-text space. Yet, in fact, the semantic information of text modality is higher than video pixels and we aims at learning video features with the supervision of textual information. To meet this requirement, we propose a refined objective function from the perspective of conditional distribution. ", + "bbox": [ + 173, + 212, + 825, + 299 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Cross-pair discrimination. According to the above analysis, we design the objective function by considering conditional distribution $p ( i _ { t } | v )$ and $p ( i _ { v } | t )$ rather than implicitly modeling distribution $p ( v , t )$ . Specifically, we design the following conditional distribution: ", + "bbox": [ + 174, + 305, + 825, + 348 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7c9244f9443c466379fbfbae05d4cae660e1931a95bf677199d255b05c4cbea0.jpg", + "text": "$$\np ( i _ { t } | v ) = \\frac { \\exp ( \\mathbf { f } _ { i } ^ { t T } \\mathbf { f } ^ { v } / \\tau ) } { \\sum _ { j = 1 } ^ { N } \\exp ( \\mathbf { f } _ { j } ^ { t T } \\mathbf { f } ^ { v } / \\tau ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 390, + 356, + 606, + 396 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $i ^ { t h }$ text define a text class $i _ { t }$ , and both $\\mathbf { f } ^ { t }$ and $\\mathbf { f } ^ { v }$ with unit-norm constraint. The conditional distribution $p ( i _ { v } | t )$ could be defined at the same way. We call this framework as cross-pair discrimination, and during training phase, the objective is to maximize the likelihood $\\begin{array} { r } { \\prod _ { i = 1 } ^ { N } p ( i _ { t } | v _ { i } ) \\prod _ { i = 1 } ^ { N } p ( i _ { v } | t _ { i } ) } \\end{array}$ . The key difference between Equation (2) and (3) is that we propose to use cross-correlation term $\\mathbf { f } ^ { t T } \\mathbf { f } ^ { v }$ to replace the self-correlation term $( \\mathbf { f } ^ { v T } \\mathbf { f } ^ { v } + \\mathbf { f } ^ { t T } \\mathbf { f } ^ { t } )$ . This cross correlation is more effective to capture the mutual information between visual and textual information, and thereby better at guiding the spatiotemporal feature learning from video with text information as supervision. ", + "bbox": [ + 173, + 402, + 825, + 520 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Ranking loss. There is some common ranking loss for cross-modal matching. To well study the effectiveness of proposed cross-modal pair discrimination objective, we also compare with a baseline of ranking loss, which is defined as follows: ", + "bbox": [ + 174, + 526, + 825, + 568 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1b5f1696c2abce4b0207d9385a7d6f9021c9a4050f4e71bd2774b9b967754b26.jpg", + "text": "$$\n\\mathcal { L } ( v _ { i } , t _ { i } ) = \\frac { 1 } { n - 1 } \\sum _ { j \\neq i } \\operatorname* { m a x } ( 0 , \\delta + \\mathcal { S } ( \\mathbf { f } _ { j } ^ { t } , \\mathbf { f } _ { i } ^ { v } ) - \\mathcal { S } ( \\mathbf { f } _ { i } ^ { t } , \\mathbf { f } _ { i } ^ { v } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 308, + 574, + 689, + 614 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where each video $v _ { i }$ has a associated text $t _ { i }$ and unrelated text $t _ { j }$ from current batch. ${ \\cal { S } } ( \\mathbf { f } _ { j } ^ { t } , \\mathbf { f } _ { i } ^ { v } )$ is the cosine similarity, $n$ is the batch size and $\\delta$ is a margin. We apply Equation (4) in both ways of video with its associated text and text with its video. In experiment, we empirically compare this ranking loss with our designed cross-pair discrimination objective. ", + "bbox": [ + 174, + 621, + 825, + 678 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 TRAINING CPD ", + "text_level": 1, + "bbox": [ + 174, + 695, + 325, + 709 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The training of CPD framework needs to address two technical issues: (1) large number of video-text pair classes; (2) optimization difficulty on noisy video-text datasets by training from scratch. ", + "bbox": [ + 174, + 720, + 825, + 750 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Noise-contrastive estimation. In training stage, we adopt noise-contrastive estimation technique (Gutmann & Hyvarinen, 2010) to approximate Equation (3) to solve the computational issues ¨ by the huge numbers of pairs. The basic idea is to transform the multi-class classification problem in Equation (3) into a set of binary classification problem. In the binary classification task, the task is to distinguish between data sample and noise sample. The approximate training objective is to minimize the following loss function: ", + "bbox": [ + 173, + 756, + 825, + 840 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5f210f5a681699b153b4a0a137df16a6d4be30a73e16e5cdb340d909e5632f75.jpg", + "text": "$$\n\\mathcal { L } = - \\mathbb { E } _ { P ( v ) } \\left\\{ \\mathbb { E } _ { P _ { d } ( i _ { t } | v ) } [ \\log h ( i _ { t } , v ) ] + m \\mathbb { E } _ { P _ { n } ( i _ { t } ^ { \\prime } | v ) } [ \\log \\left( 1 - h ( i _ { t } ^ { \\prime } , v ) \\right) ] \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 261, + 848, + 735, + 867 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { h ( i _ { t } , v ) = \\frac { p ( i _ { t } | v ) } { p ( i _ { t } | v ) + m p _ { n } ( i _ { t } | v ) } } \\end{array}$ , $P _ { d } ( i _ { t } | v )$ is the actual data distribution and $P _ { n } ( i _ { t } ^ { \\prime } | v )$ is the uniform distribution for noise, and $m$ denotes the noise frequency. To compute $p ( i _ { t } | v )$ efficiently and avoid large memory consumption, following $\\mathbf { W } \\mathbf { u }$ et al., 2018), we maintain a memory bank to store the visual and textual features for each training pair. The memory bank is updated dynamically during the training procedure. ", + "bbox": [ + 174, + 875, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Curriculum learning. To handle the optimization difficulty of directly training from scratch on noisy video-text dataset, we present a curriculum training strategy by resorting to the existing unsupervised pre-trained language models. To relieve the training difficulty, our curriculum learning strategy divides the training procedure into two stages. In the first stage, we fix the pre-trained language model and only update the parameters of visual model and embedding function. The motivation is that the language model is pre-trained well using corpus much larger than ours and the video model is totally trained from scratch. If we train both models simultaneously in the beginning, the random noise produced by video model will destroy the parameters of language model. In the second stage, after the good initialization of video model, we start to jointly train the visual-textual model with a smaller learning rate. ", + "bbox": [ + 174, + 138, + 825, + 277 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 ARCHITECTURE DESIGN ", + "text_level": 1, + "bbox": [ + 176, + 296, + 382, + 310 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Video architecture. For video representation, we use the 3D CNNs to extract spatiotemporal features from a video clip. Specifically, we randomly sample 8 frames from each video clip and sampling stride is 4. Following the implementation of slow stream in the recent SlowFast (Feichtenhofer et al., 2018), all filters from $c o n v _ { 1 }$ to $r e s _ { 3 }$ degenerate temporal convolutions into 2D convolution kernels and it only reserves 3D convolution kernels in $r e s _ { 4 }$ and $r e s _ { 5 }$ without temporal downsampling. We try two kinds of network architectures: (1) 3D ResNet34 trained on $1 1 2 \\times 1 1 2 \\times 8$ volumes and (2) 3D ResNet50 trained on $2 2 4 \\times 2 2 4 \\times 8$ volumes. The first tiny network is efficient for ablation study and then we transfer its optimal setting to the larger backbone and frame resolution. We also add a mapping layer to transform the visual features into 256-dimensional embedding space $\\mathbf { f } ^ { v }$ and this 256-d vector is $\\ell _ { 2 }$ -normalized. ", + "bbox": [ + 174, + 323, + 825, + 462 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Text architecture. Our textual stream subnetwork is based on the off-the-shelf language models. We choose Word2vec (Mikolov et al., 2013) and DistilBERT (Devlin et al., 2019; Sanh et al., 2019) as our textual encoders. Word2vec is an unsupervised word encoder, pre-trained by reconstructing the surrounding words of the continue sentences. We average word vectors which are 300 dimensional as textual encoder. BERT (Devlin et al., 2019) encodes long sentences by predicting the missing words given their bidirectional context, and DistilBERT achieves comparable performance with a faster and lighter model via knowledge distillation (Hinton et al., 2015). We average word embeddings of title generated by DistilBERT and obtain 768 dimensional text feature. Finally, two fully connected layers with ReLU and Batch Normalization (Ioffe & Szegedy, 2015) are added to our textual encoder to obtain textual feature $\\mathbf { f } ^ { t }$ in the common embedding space, which is also $\\ell _ { 2 }$ -normalized. ", + "bbox": [ + 174, + 469, + 825, + 622 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 643, + 326, + 660 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we present the experimental results of our proposed CPD framework. First, we describe the training and evaluation datasets with implementation details. Then, we conduct ablation study on our proposed CPD framework. Finally, we verify the effectiveness of CPD from two aspects: weakly-supervised representation learning and representation transfer. ", + "bbox": [ + 174, + 676, + 825, + 732 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 751, + 287, + 763 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In our experiment, we pre-train our CPD framework on two video-text datasets: Kinetics-210k (Kay et al., 2017) and Instagram-300k (Duan et al., 2020). Then, we fine-tune the video model on three human action datasets: Kinetics400 (Kay et al., 2017), UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011). ", + "bbox": [ + 174, + 776, + 823, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Kinetics-210k. Following the recent self-supervised methods (Wang et al., $2 0 1 9 \\mathrm { a }$ ; Korbar et al., 2018; Han et al., 2019), we utilize Kinetics (Kay et al., 2017) dataset for weakly-supervised pretraining of CPD. It is often called Kinetics400 since it has 400 action classes, but we count training video number as we do not use any class information for weakly-supervised representation learning. Due to invalid urls and data cleaning, the collected dataset contains around 210k video-text pairs, and thus we call this dataset as Kinetics-210k. To construct video-text pairs, we equip each clip with the video title directly crawled from YouTube, termed as Kinetics-title. As the original title may be very noisy, we pre-process the text information in two ways. First, we delete special symbols and characters such as non-English words and emoji, termed as Kinetics-title-clean. Second, we use StanfordNLP (Qi et al., 2018) to obtain the dependency tree of sentences in titles and only reserve verbs and nouns, named Kinetics-title-tree. ", + "bbox": [ + 174, + 840, + 823, + 922 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Instagram-300k. To avoid data bias in Kinetics caused by human annotation (i.e., trimmed videos with an action), we further verify the effectiveness our CPD model on an uncurated web video dataset (Duan et al., 2020). This new dataset is constructed from Instagram by searching action label of Kinetics-400 but without any manual filtering. Due to limited computation resource and also for fair comparison with pretraining on Kinetics-210k, we randomly sample 300k from the original web video dataset, termed as Instagram-300k. ", + "bbox": [ + 174, + 180, + 825, + 263 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "An important difference is that the these videos are with User Generated Content (UGC) and accompanied by captions uploaded by users. Therefore, its video content distribution is much different with those in Profession Generated Content (PGC) in UCF101 and HMDB51, and the text noise is also much higher. So, it is more challenging to train a pre-trained CPD model on Instagram-300k. ", + "bbox": [ + 174, + 271, + 823, + 327 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "UCF101 and HMDB51. We evaluate the generalization of our pre-trained models by fine-tuning on two small human action datasets: UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011), which contain $1 3 \\mathrm { k }$ videos of 101 classes and 7k video of 51 classes respectively. We report ablation study on the first split and report average performance over three splits for fair comparison. ", + "bbox": [ + 174, + 334, + 825, + 390 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 417, + 403, + 431 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Weakly supervised learning of CPD. We train our CPD model on video-text datasets and use video-text retrieval on 1k unseen video-text pairs as validation set duration training. Specifically, 8 frames are sampled from each video clip and the sampling stride is 4. We use SGD to optimize our objective and the training parameters include a momentum of 0.9 and 1e-4 for weight decay. We set temperature parameter $\\tau = 0 . 0 7$ and noise frequency $m$ to 4096. In the beginning, we fix the pre-trained language model and the learning rate is set as 0.2. When the retrieval performance on validation set saturates (170 epochs for 3D ResNet34 and 110 epochs for 3D ResNet50), we start to update the language model with learning rate of 3e-5 and decrease the rest learning rate to 0.02. The maximize training number is 250 epochs. For input size of $1 1 2 \\times 1 1 2 \\times 8$ , the mini-batch size is 64 clips per GPUs and 16 clips per GPUs for input size of $2 2 4 \\times 2 2 4 \\times 8$ . We use 8 GPUs for training. ", + "bbox": [ + 174, + 446, + 825, + 587 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Evaluation on representation learning. We first verify our CPD learned representation by employing a shallow classifier on frozen features. Specifically, we utilize $\\mathbf { k }$ -Nearest Neighbor (kNN) and linear classifier based on extracted features for classification. For video feature extraction, we sample 10 clips from each video and each clip contains 8 frames with 4 sampling stride. The 256- dimensional embedding feature and the output of global average pooling are extracted as features. The extracted features over 10 clips in a video are averaged as a video-level representation. We choose cosine distance as distance metric in kNN and set $k = 2 5$ . As for linear classifier, a fully connected layer after Batch Normalization is added with cross-entropy loss. We adopt Adam with learning rate of 1e-3 and reduce by a factor of 10 every 10 epochs, stopping at 30 epochs. ", + "bbox": [ + 173, + 593, + 825, + 719 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Evaluation on representation transfer. A main goal of representation learning is to transfer them to downstream tasks. We fine-tune the learned spatiotemporal representation on the UCF101, HMDB51 and a small fraction of Kinetics400. During fine-tuning, 16 frames with stride 4 are sampled as input. We simply replace the embedding layer of video model with a new fully-connected layer and multi-way softmax for action recognition. The classifier is trained using the SGD optimizer with an initial learning rate 1e-2 and weight decay 5e-4. Learning rate is decreased twice by a factor of 10 when the validation loss saturates. During testing, for each video, we uniformly sample 10 clips and each clip contains 3 crops, following the common practice (Feichtenhofer et al., 2018). ", + "bbox": [ + 174, + 726, + 825, + 837 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 176, + 866, + 338, + 878 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this study, we pre-train our CPD models on Kinetics-210k dataset and choose the task of representation transfer by fine tuning on UCF101 split 1 for evaluation. ", + "bbox": [ + 174, + 895, + 820, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/4bea579d226dfc281b0662c4193b0318c63c6991076162861141fc59bfad245f.jpg", + "table_caption": [], + "table_footnote": [ + "(b) Study on training strategies. " + ], + "table_body": "
TrainingstrategyAccuracy(%)
Random init.50.0
Direct fine-tuning81.3
Curr. learning182.2
Curr.learning284.2
", + "bbox": [ + 393, + 117, + 578, + 172 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/17e985e72931506a34d5805a3ea0f1ba91fbfcc978561deda2179967b7feff41.jpg", + "table_caption": [], + "table_footnote": [ + "(c) Study on textual encoders. " + ], + "table_body": "
Textual encoderDataAccuracy(%)
Random Init.-50.0
Word2vecTree83.1
DistilBERTTree82.1
Word2vecClean82.5
DistilBERTClean84.2
", + "bbox": [ + 584, + 113, + 805, + 178 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/6001b620a5e5ce95e095f6ab6221a1eece9cfae316af350e06e7aae9a5bd7efe.jpg", + "table_caption": [ + "Table 1: Ablation study on UCF101 by fine tuning a pre-trained CPD model from Kinetics-210k. " + ], + "table_footnote": [ + "(a) Study on loss functions. " + ], + "table_body": "
Objective functionAccuracy(%)
Random init.50.0
Ranking loss79.9
Self-instance Dis.51.1
Cross-pair Dis.82.2
", + "bbox": [ + 191, + 118, + 377, + 172 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/fc74469a73e784b9d6fce1297326013f235e9d67632cf22516a3d97ce9f7795b.jpg", + "table_caption": [ + "Table 2: Evaluation on weakly-supervised representation learning without fine-tuning. Top-1 classification accuracy is reported on Kinetics-400 validation set. " + ], + "table_footnote": [], + "table_body": "
BackbonePre-trained Sup.Layer (Dim)KNNLC
3D-ConvNet (Kay et al.,2017)Kinetics-400Label--56.1
3D ResNet34 (Hara et al., 2018) 3D ResNet50 (ours)Kinetics-400 Label Kinetics-400 Label--60.1 73.2
ResNet50ImageNet Label-- 42.856.1
3DResNet34res5 (2048)
Instagram-300k Captionemb (256)34.537.3
3D ResNet34Instagram-300k Captionres5 (512)36.144.6
3D ResNet50Instagram-300k Captionemb (256)51.151.7
3DResNet50Instagram-300k Captionres5 (2048)51.155.4
3DResNet34Kinetics-210k Titleemb (256)49.950.8
3D ResNet34Kinetics-210k Titleres5 (512)50.153.3
3DResNet50Kinetics-210k Titleemb (256)58.059.6
3DResNet50Kinetics-210k Titleres5 (2048)58.263.8
", + "bbox": [ + 266, + 234, + 728, + 371 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Objective function. We compare three objective functions for cross-modal pair discrimination described in Section 3.1. We pre-train models by utilizing DistilBERT as textual encoder without finetuning and the experimental results are reported in Table 1a. Self-instance discrimination almost has no contribution to learn effective representation as there is no cross-modal correlation modeling. Cross-pair discrimination gives a better performance than ranking loss as cross-pair discrimination can construct negative video-text pairs from entire dataset while ranking loss is only optimized by negative pairs from current batch. More theoretical analysis can be found in Section. A.1 of the Appendix. ", + "bbox": [ + 173, + 420, + 825, + 530 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Curriculum learning. We design different training strategies from noisy video-text datasets. The first strategy is to fine-tune the pre-trained textual encoder directly at the beginning. Then we compare with stage I and stage II of curriculum learning proposed in Section 3.2. All these strategies are pre-trained on Kinetics-title-clean. The numerical results are summarized in Table 1b. Fixing the pre-trained language model gives better performance than direct fine-tuning at the beginning $( + 0 . 9 \\% )$ . We ascribe this to the fact that the random noise produced by video model destroy the well pre-trained textual encoder at the beginning. Also, fine-tuning the language model after the video model is well initialized further boost the accuracy by $2 . 0 \\%$ . ", + "bbox": [ + 173, + 537, + 825, + 648 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Different textual information. In this experiment, we choose video-text pairs from Kinetics-titletree, Kinetics-title-clean datasets and utilize Word2vec and DistilBERT as a textual extractor. The experimental results are reported in Table 1c. For textual encoder, abundant and video-specific text information benefits to train our CPD model with stronger language model such as DistilBERT according to the performance difference between Kinetics-title-tree and Kinetics-title-clean $( 8 2 . 1 \\%$ vs. $8 4 . 2 \\%$ ). As for shallow textual encoder (e.g., Word2vec), simple text information from Kineticstitle-tree dataset gives better performance than abundant text information $( 8 3 . 1 \\%$ vs. $8 2 . 5 \\%$ ). From above observation, it can be concluded that Word2vec is more good at concise and accurate text while DistilBERT can handle more complex and noisy sentences which is close to realistic setting. Also, it is affordable to utilize strong language models due to our curriculum learning strategy and lightweight DistilBERT model. ", + "bbox": [ + 173, + 656, + 825, + 808 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4 EVALUATION ON REPRESENTATION LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 828, + 534, + 842 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To evaluate our learned representation, we report the classification performance on validation set of Kinetics via training shallow classifiers on frozen features as shown in Table 4.3. We perform kNN classifiers and linear classifiers (LC) on the embedding features or visual features from global average pooling after res5. In this shallow learning setting, we also compare with ImageNet pretraining representation (ResNet50) by using the same classifier. First, the representation learnt from ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/618e1387d8bc6b4c36e8552eb710ad50d51da89cea9dbb6050d87839f7d20c06.jpg", + "table_caption": [ + "Table 3: Evaluation on representation transfer by fine-tuning. We compare our CPD model with other methods trained on different type of supervision. " + ], + "table_footnote": [], + "table_body": "
MethodSupervisionBackbonePre-trained DatasetUCF101HMDB51
RandomInit. (Hara etal.,2018)3DResNet1842.417.1
Kinetics Pre-trained (Hara etal., 2018)Action label3DResNet50Kinetics89.361.0
Supervised SOTA (Xie et al.,2018)Action labelS3DKinetics96.875.9
Shuffle& Learn (Misra et al.,2016)Order verificationCaffeNetUCF101/HMDB5150.218.1
OPN (Lee et al., 2017)Sequence orderVGGNetUCF101/HMDB5159.823.8
CMC (Tian et al., 2019)Optical flowCaffeNetUCF10155.3-
O3N (Fernando et al.,2017)Odd-one-outAlexNetUCF10160.332.5
MASN (Wang et al.,2019a)MotionC3DKinetics-40061.233.4
COP (Xu et al.,2019b)Clip order3D ResNet10UCF10164.929.5
DPC (Han et al.,2019)Prediction3DResNet34Kinetics-40075.735.7
CBT(Sun et al.,2019a)Audio(Text)/ContextS3DKinetics-60079.544.6
AVTS (Korbar et al.,2018)AudioI3DKinetics-60083.753.0
AVTS (Korbar et al., 2018)AudioMC3Audioset-1.8M89.061.6
XDC (Alwassel et al.,2019)AudioR(2+1)DKinetics-40084.247.1
XDC (Alwassel et al.,2019)AudioR(2+1)DIG-65M91.563.1
MIL-NCE (Miech et al., 2020)Audio(Text)S3DHT-100M91.361.0
TWS (Stroud et al.,2020)Text (Title,Des,Tag etc.)S3D-GWVT-70M90.365.3
CPD (Ours)Caption3DResNet50Instagram300k89.963.8
CPD (Ours)Title3DResNet50Kinetics210k90.563.6
", + "bbox": [ + 178, + 99, + 813, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Kinetics-210k generally outperforms that of Instagram- $3 0 0 \\mathrm { k }$ . The reason could be ascribed to the video distribution gap between UGC (Instagram) and PGC (Youtube), and also much noisier textual information in Instagram-300k. Second, we compare with ImageNet pretrained features, and our CPD representation is better under the same backbone. Finally, we compare with some end-toend trained representations with action labels, and there is still a performance gap between our representation and supervised end-to-end representation (e.g. $6 3 . 8 \\%$ vs. $7 3 . 2 \\%$ ). ", + "bbox": [ + 174, + 364, + 825, + 448 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.5 EVALUATION ON REPRESENTATION TRANSFER ", + "text_level": 1, + "bbox": [ + 174, + 474, + 532, + 488 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Transferring learned representation to downstream tasks is a main goal of representation learning. We transfer them to action recognition task on small datasets, namely UCF101 and HMDB51. We compare our CPD model pre-trained on Instagram-300k and Kinetics-210k with a randomly initialized network, self-supervised methods solely based on visual information, including Shuffle & Learn (Misra et al., 2016), CMC (Tian et al., 2019), MASN (Wang et al., 2019a), COP (Xu et al., 2019b), DPC (Han et al., 2019) and so on, and representation learning methods based on multimodal information (e.g., audio, text), including CBT (Sun et al., 2019a), AVTS (Korbar et al., 2018), XDC (Alwassel et al., 2019), MIL-NCE (Stroud et al., 2020), and TWS (Stroud et al., 2020). ", + "bbox": [ + 174, + 503, + 825, + 614 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As shown in Table 3, our CPD models generally outperform those self-supervised learning approaches of only using visual information $( \\geq 1 0 \\%$ on UCF101 and $\\geq 2 0 \\%$ on HMDB51), which indicates that cross-modal information is useful cue for visual representation learning. Meanwhile, our CPD representations obtain comparable performance to the concurrent works (i.e., MIL-NCE and TWS) of using text as weak supervision. However, our CPD uses a much smaller pre-training dataset of around 0.3M videos, while the other methods uses 70M-100M videos. Training a CPD model on a such large-scale dataset is almost impossible for a university lab with limited computational facilities. Our work demonstrates that pre-training a relatively small video-text dataset is also possible to match the SOTA performance, and this is quite meaningful and practicable for university lab. Finally, we notice that the gap of CPD models learned from Instagram- ${ 3 0 0 } \\mathrm { k }$ and Kinetics- ${ \\it 2 0 0 k }$ is very small, indicating that our CPD model can effectively handle high noise in text. ", + "bbox": [ + 174, + 621, + 825, + 773 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 803, + 318, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we have presented a general cross-modal pair discrimination (CPD) framework to capture the correlation between a video clip and its associated text from real word and adopt noisecontrastive estimation to approximate the objective. Without fine-tuning, the learned models obtain competitive results for action classification on Kinetics dataset with a shallow classifier. Also, our visual models provide an effective initialization to fine-tune on the datasets of downstream task, and matches the state-of-the-art performance with a much smaller pre-training dataset. 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In 2017 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 6612–6619, 2017. ", + "bbox": [ + 174, + 722, + 826, + 766 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A TRAINING DETAILS OF CPD ", + "text_level": 1, + "bbox": [ + 178, + 102, + 444, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "we adopt noise-contrastive estimation technique (NCE) to approximate objective function in Equation (5) in our main paper. The purpose of NCE is to transform the multi-class classification problem into a set of binary classification problems by comparing data distribution against noise distribution. So $p _ { n }$ is noise distribution and we formalize it as a uniform distribution: $\\begin{array} { r } { { p } _ { n } = \\frac { 1 } { N } } \\end{array}$ , where $N$ is the number of video-text pairs. $h ( i _ { t } , v )$ is the posterior probability of feature from the data distribution which means video and text are matched. $m$ is the number of negative pairs and we set it as 4096. For each video feature $\\mathbf { f } ^ { v }$ , we take its related text feature $\\mathbf { f } _ { i } ^ { t }$ and sample 4096 unrelated text features $\\mathbf { f } _ { j } ^ { t }$ which are all from memory bank. The FPS of training videos are 30. The code of CPD will be released. ", + "bbox": [ + 174, + 133, + 825, + 258 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 ANALYSIS ON DIFFERENT LOSS FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 276, + 519, + 290 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "More insight about why our loss is better than ranking loss could be found from gradient backpropagation. Let $\\mathbf { f } ^ { t + }$ and $\\mathbf { f } ^ { t - }$ represent the associated and unrelated text feature. For ranking loss, the negative gradient w.r.t $\\mathbf { f } ^ { v }$ is $\\bar { \\mathbf { f } ^ { t + } } - \\mathbf { f } ^ { t - }$ if $\\mathcal { L } > 0$ else 0. For CPD loss, it is $[ 1 - h ( i _ { t } ^ { + } , v ) ] / \\bar { \\bf \\Delta } \\bar { \\bf f } ^ { t + } -$ $\\sum h ( i _ { t } ^ { - } , v ) / \\tau \\mathbf { f } ^ { t - }$ . We observe our loss assign different weights to different examples based on their posterior probability $h$ , which helps learn from hard examples while the ranking loss treats them equally. ", + "bbox": [ + 174, + 301, + 825, + 387 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B REPRESENTATION TRANSFER ON KINETICS ", + "text_level": 1, + "bbox": [ + 174, + 407, + 566, + 424 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/4e0c09c5ff7f8c972fb77fd12b966bb75037fc4a7dbb4278d67444654c5b60a0.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodThe Amount ofLabeled Data
1%10%20%
Fromscratch0.310.733.3
ImageNet Inflation12.836.845.7
Ours (Instagram-300k)18.741.347.4
Ours (Kinetics-210k)25.943.147.8
", + "bbox": [ + 320, + 441, + 671, + 525 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Table 4: Results of classification with small amount of labeled data on Kinetics-400 validation set (showing top-1 accuracy). We utilize 3D ResNet34 as backbone and pre-train it on Kinetics-210k and Instagram-210k. ", + "bbox": [ + 176, + 535, + 821, + 577 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Our weakly-supervised pre-trained representation can be an efficient initialization when training the model with only a small amount of labeled data. We randomly choose a small fraction of Kinetics400 training set as labeled data and fine-tune the pre-trained model on it. We report the performance of top-1 accuracy which is trained on labeled subset of $1 \\%$ , $10 \\%$ and $20 \\%$ of the entire dataset in Table 4. We compare our method with training from scratch and ImageNet inflated model as baselines. Our method significantly surpasses the baselines on all present proportion of labeled subset especially when the amount of labeled data is extremely small. When only $1 \\%$ of data is labeled, training from scratch can not learn anything yet our model achieves $1 8 . 7 \\%$ and $2 5 . 9 \\%$ top-1 accuracy. Both our CPD pre-trained models on Instagram and Kinetics outperform the ImageNet pre-trained models. ", + "bbox": [ + 174, + 593, + 825, + 733 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C EVALUATION ON ZERO-SHOT CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 753, + 591, + 768 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We evaluate our visual-textual embedding of CPD model with zero-shot classification on UCF101 and Kinetics-400 without any fine-tuning in Table 5. We transform class labels and video clips into the same embedding space and recognize the video clip to its closest class with cosine distance. We compare our method with Mettes et al. (Mettes & Snoek, 2017) which realizes zero-shot localization and classification of human action in video via spatial-aware object embeddings on UCF101. Following (Mettes & Snoek, 2017), we select different classes for 10 times and average their accuracies for testing except the class number is 101. We outperform for every number of testing classes. For Kinetics-400, we achieve top-1 accuracy of $4 3 . 7 \\%$ without fine-tuning and training label. In addition, top-1 accuracy of 20 random classes reaches to $7 4 . 4 \\%$ , which shows the strong capability of our visual-textual embedding. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/7cfe47029e02f00395c0f5758d19df92b231e41dd16bbfaad339dffa5baef2bb.jpg", + "table_caption": [ + "Table 5: Top-1 accuracy of zero-shot classification on UCF-101 and Kinetics-400. We outperform other methods without any extra labeled data and training procedure after pre-training on Kinetics210k. " + ], + "table_footnote": [], + "table_body": "
MethodsUCF-101Kinetics-400
TrainTestSplitAcc.TrainTestSplitAcc.
Mettes (Mettes& Snoek,2017)1101332.8--11
Ours(3D ResNet34)=101340.6=400138.2
Ours(3D ResNet50)-101339.9-400143.7
Mettes(Mettes& Snoek,2017)-501040.4-111
Ours(3D ResNet34)501047.2=1001055.3
Ours(3D ResNet50)501044.8=1001057.4
Mettes (Mettes& Snoek,2017)201051.21--=
Ours(3D ResNet34)201054.4=201073.1
Ours(3D ResNet50)201058.1=201074.4
", + "bbox": [ + 199, + 101, + 794, + 244 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D ANALYZE TEXT INFORMATION ", + "text_level": 1, + "bbox": [ + 176, + 319, + 462, + 335 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D.1 ANALYSIS ON KINETICS TITLE ", + "text_level": 1, + "bbox": [ + 176, + 352, + 431, + 366 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/8bf601fa3280166a0ad5676b0cfbe092f29ddbf72694f327e5dfe610f4db6a15.jpg", + "table_caption": [ + "Table 6: Analyze text information of Kinetics-210k datasets. At Least One: The proportion of text information that contains at least one word in action classes of Kinetics-400. All: The proportion of text information that contains the entire action class. $R e l$ : The proportion of word in text information that is relevant to action classes. " + ], + "table_footnote": [], + "table_body": "
DatasetsAt Least One(%)A11(%)Rel(%)
Kinetics-title-tree90.544.346.3
Kinetics-title-clean91.638.426.0
", + "bbox": [ + 307, + 382, + 686, + 424 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We provide an analysis of text information we used and the result in Table 6. First, there exists a large overlap between action class and text information (more than $90 \\%$ for at least one word and more than $38 \\%$ for complete action class). However, the titles also contain many other words and noisier information than action classes. Only $26 \\%$ of words in Kinetics-title-clean are relevant to action classes. ", + "bbox": [ + 173, + 505, + 825, + 575 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/ad7bcc7280b2e78acea6087fc4a4e1cc11c2bec2263d73b9806c870843806321.jpg", + "image_caption": [ + "Figure 2: List of top 10 and bottom 10 kinetics classes sorted by the frequency of at least one word in label occurring in according title of Kinetics-title-clean dataset. Zoom in for more details. We also report the per-class accuracy of top 10 and bottom 10 classes sorted by word overlapping in Figure 2 and see that this accuracy is not positively correlated with word overlapping percentage. Finally, we provide some examples of videos and their titles from Kinetics-210k in Figure 3. " + ], + "image_footnote": [], + "bbox": [ + 173, + 587, + 823, + 718 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D.2 VISUALIZATION OF INSTAGRAM CAPTION ", + "text_level": 1, + "bbox": [ + 173, + 814, + 508, + 828 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since videos from Instagram-300k are not annotated or filtered by human, both of their visual and textual information are very noisy. Figure 4 demonstrates some examples of videos and their associated captions. Figure 4a presents an example of high-quality video and relative accurate caption that both are about folding napkin. Many captions describe some useful information but also contain noisy text that is not related to video content (e.g., summerdays and gettingtattooed in Figure 4b and very long sentences in Figure 4d). In addition, there are some correct but not totally accurate descriptions. Figure 4c shows that the action in video is shot putting rather than spinning (appears in associated caption). Figure 4f illustrates that a person is climbing but its caption is mainly about high jumpping. Figure 4e shows that video content can also be noisy due to low video quality and shot transformation. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/18a06d7e24dcf2ca11f47c9494c912565a377463e4f0eff00aa54fe17b8bc0da.jpg", + "image_caption": [ + "Figure 3: Examples of video and title pairs from Kinetics-210k. " + ], + "image_footnote": [], + "bbox": [ + 205, + 125, + 787, + 679 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 746, + 825, + 801 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Bird of Paradise Napkin Fold #napkinfolding #napkinart #napkin #yellow #napkinfold #birdofparadise #kidscrafts #diy #video #tutorial #decoration #tablesetting ", + "bbox": [ + 602, + 171, + 763, + 239 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/2bc24d3718235e6d923f7c302986676576fe887ce7fc1a26163532e11a745b6f.jpg", + "image_caption": [ + "Figure 4: Examples of videos and their associated captions from Instagram-300k. " + ], + "image_footnote": [], + "bbox": [ + 209, + 167, + 575, + 842 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "#roadtrip from #britishcolumbia to #alberta. drivingallnight #mountains #trees #summer #summerdays #summerlovin #highway #music #playlist #carpoolkaraoke #notreallyasinger #butitry #gettingtattooed #buyingatruck #duoting ", + "bbox": [ + 602, + 260, + 776, + 349 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Day 1 of learning to spin $\\circledast$ . A work in progress but having fun with it! ", + "bbox": [ + 602, + 368, + 751, + 402 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Good morning beautiful people of #god #grateful for another day he has given me #prayingfor #guidance and #protection in #Jesusname I #pray #amen Early #dinnersettings today #potroastbeef #bakechicken #vegetables WHAT #RICEANDPEAS WOULD YOU EAT WITH THIS? #pyjamachef #foodbloger #pyjamachef #healthyeating #alltypeoffood #lovecooking ", + "bbox": [ + 602, + 477, + 777, + 570 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "My pets are unruly... 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Moreover, our visual model provides an effective", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 344 + ], + "score": 1.0, + "content": "initialization to fine-tune on downstream tasks, which yields a remarkable perfor-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 354 + ], + "score": 1.0, + "content": "mance gain for action recognition on UCF101 and HMDB51, compared with the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "existing state-of-the-art self-supervised training methods. In addition, our CPD", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 364, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 469, + 376 + ], + "score": 1.0, + "content": "demonstrates that pre-training a relatively small dataset is able to yield a compa-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 375, + 469, + 388 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 469, + 388 + ], + "score": 1.0, + "content": "rable performance to those methods of using order magnitude more data, which is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 386, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 470, + 399 + ], + "score": 1.0, + "content": "meaningful and practicable for the scenarios with limited computational facilities.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 212, + 470, + 399 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 412, + 206, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 208, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 208, + 428 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Deep learning has made a remarkable progress for visual recognition in both image and video do-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "main (Krizhevsky et al., 2012; He et al., 2016; Carreira & Zisserman, 2017; Feichtenhofer et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "2018) by training powerful neural networks on large-scale manually annotated datasets (e.g., Ima-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "geNet (Deng et al., 2009) and Kinetics (Kay et al., 2017)). More importantly, it is well-established", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 476, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 506, + 493 + ], + "score": 1.0, + "content": "that this supervised pre-training on large-scale datasets would benefit the downstream tasks (e.g.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "object detection (Ren et al., 2015), pose estimation (He et al., 2017), and temporal action detec-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "score": 1.0, + "content": "tion (Zhao et al., 2017)), in particular when the target datasets are relatively small. Yet, annotating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "a large-scale dataset for training such deep neural networks is costly and time-consuming, and even", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "more challenging for video due to its various temporal structure and complex semantics. As a re-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "sult, the existing video datasets size is still smaller than ImageNet in terms of training samples and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "classes. On the other hand, videos typically contain richer structure with abundant side information", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 243, + 569 + ], + "score": 1.0, + "content": "such as motion (Diba et al., 2019;", + "type": "text" + }, + { + "bbox": [ + 243, + 555, + 257, + 567 + ], + "score": 0.45, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "et al., 2018), audio (Arandjelovic & Zisserman, 2017; Korbar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "et al., 2018), and text (Miech et al., 2019; Sun et al., 2019b). So these expected these associated", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "modalities are expected to provide useful cues to learn video representations in a more efficient way.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 434, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "Language or text is probably the most natural and easy way to describe the semantic information", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "of a video, and the associated textual information could be easily acquired when collecting video", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "dataset (Rohrbach et al., 2017; Miech et al., 2019) from Internet or Movie. We argue that this", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "correlation between a clip and its associated text could serve as an alternative supervision to learn", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "video representation from scratch. This is different from some recent works (Sun et al., 2019b;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 650, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 660 + ], + "score": 1.0, + "content": "Miech et al., 2019), in which these abundant textual information has been used to learn a high-level", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "visual-text embedding applied to text-to-video retrieval or video captioning. Intuitively, it is more", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "challenging to learn a general visual representation solely from text information without any human", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "annotation, for reasons such as large numbers of noise in text, lacking careful initialization, and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 694, + 280, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 280, + 705 + ], + "score": 1.0, + "content": "being hard to design an effective objective.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 595, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "score": 1.0, + "content": "In this paper, we aim to learn effective video representation from noisy and diverse textual infor-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "mation, which could serves as the basis for a variety of downstream tasks. Basically, we learn a", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "mapping of text and video into a shared embedding space and leverage their correlation as supervi-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "sion signal. The technical difficulty is how to design an effective objective function, that is capable", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "of modeling this complex visual-textual correlation and as well easily optimized by training from", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "scratch on noisy datasets. Inspired by unsupervised feature learning in images (Wu et al., 2018;", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "Tian et al., 2019), we present a cross-modal pair discrimination (CPD) framework, which tries to", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "recognize each video and text pair into a class via a non-parametric classifier. To solve the compu-", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "tational issues imposed by the huge numbers of pair classes, we adapt noise-contrastive estimation", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 440, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 440, + 172 + ], + "score": 1.0, + "content": "technique (Gutmann & Hyvarinen, 2010) to approximate the original loss function. ¨", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 710, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "mapping of text and video into a shared embedding space and leverage their correlation as supervi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "sion signal. The technical difficulty is how to design an effective objective function, that is capable", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "of modeling this complex visual-textual correlation and as well easily optimized by training from", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "scratch on noisy datasets. Inspired by unsupervised feature learning in images (Wu et al., 2018;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "Tian et al., 2019), we present a cross-modal pair discrimination (CPD) framework, which tries to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "recognize each video and text pair into a class via a non-parametric classifier. To solve the compu-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "tational issues imposed by the huge numbers of pair classes, we adapt noise-contrastive estimation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 440, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 440, + 172 + ], + "score": 1.0, + "content": "technique (Gutmann & Hyvarinen, 2010) to approximate the original loss function. ¨", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "Specifically, we learn the CPD framework from web videos with the associated title or caption", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "that could be directly crawled from web platforms such as YouTube (Kay et al., 2017) and Insta-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "gram (Duan et al., 2020). We utilize the off-the-shelf language models such as BERT (Devlin et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "2019) or Word2vec (Mikolov et al., 2013) and devise a curriculum learning strategy to progressively", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "train the video models. We first test the generalization ability of learned video representation by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "CPD on the Kinetics dataset (Kay et al., 2017) by using shallow classifiers such k-NN and linear", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "classifier. It shows that our learned spatiotemporal features obtain promising results which are com-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "parable to some supervised learning methods on the Kinetics dataset (Kay et al., 2017). Then, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "investigate the generalization power of learned spatiotemporal features of CPD by fine-tuning on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "the Kinetics (Kay et al., 2017), UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "datasets, demonstrating that our method obtain superior performance to previous state-of-the-art", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "self-supervised methods and comparable performance to the very recent methods of using orders of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 298, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 298, + 320 + ], + "score": 1.0, + "content": "magnitude more videos (70M-100M vs. 0.3M).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 327, + 211, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 213, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 213, + 342 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "Self/Weakly Supervised Representation Learning. Self supervised representation was popular", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "in both image and video domains by designing various proxy tasks. In image domain, for in-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 374, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 504, + 387 + ], + "score": 1.0, + "content": "stance, these tasks could be predicting the image context (Doersch et al., 2015), counting the ob-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "jects (Noroozi et al., 2017), converting gray images to color one (Zhang et al., 2016), keeping global", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "and local consistency (Hjelm et al., 2019). 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These learnt representations may capture some aspects of low-level image or video", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 439, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 439, + 463 + ], + "score": 1.0, + "content": "structures, but are generally outperformed by those using cross modal information.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Several cross-modal self-supervised tasks was proposed to enhance single-modality representation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "power and typical example is audio-visual representation learning (Aytar et al., 2016; Arandjelovic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "& Zisserman, 2017; Korbar et al., 2018). 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Our CPD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "is applicable to more general video type and we experiment with a much smaller dataset (0.3M", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "vs. 100M) of both PGC and UGC videos, but achieves a similar performance on UCF101 and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "HMDB51. 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Motion or temporal information has been studied as to design proxy tasks to assist", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "cross-modal learning, such as optical flow or tracking (Ng et al., 2018; Wang & Gupta, 2015), frame", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "prediction (Diba et al., 2019; Vondrick et al., 2016), or high-level temporal structure (Wei et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "2018; Xu et al., 2019a; Fernando et al., 2017). As most video contain synchronized audio and visual", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "signals, audio information has served another common modality to supervised visual learning (Aytar", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "et al., 2016; Arandjelovic & Zisserman, 2017; Korbar et al., 2018). However, both motion and audio", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "information seem to be low-level signals and may lack high-level semantic for cross-modal learning.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Speech or text has been widely studied as another cross-modal setting in video learning (Sun et al.,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "2019b; Miech et al., 2019; Dong et al., 2019; Miech et al., 2018; Pan et al., 2016; Plummer et al.,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "2017). These works mainly aimed to learn a joint video-text embedding where visual and textual", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "cues are adjacent if they are semantically. 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We utilize the off-the-shelf language models such as BERT (Devlin et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "2019) or Word2vec (Mikolov et al., 2013) and devise a curriculum learning strategy to progressively", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "train the video models. We first test the generalization ability of learned video representation by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "CPD on the Kinetics dataset (Kay et al., 2017) by using shallow classifiers such k-NN and linear", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "classifier. It shows that our learned spatiotemporal features obtain promising results which are com-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "parable to some supervised learning methods on the Kinetics dataset (Kay et al., 2017). Then, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "investigate the generalization power of learned spatiotemporal features of CPD by fine-tuning on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "the Kinetics (Kay et al., 2017), UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al., 2011)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "datasets, demonstrating that our method obtain superior performance to previous state-of-the-art", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "self-supervised methods and comparable performance to the very recent methods of using orders of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 298, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 298, + 320 + ], + "score": 1.0, + "content": "magnitude more videos (70M-100M vs. 0.3M).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 175, + 506, + 320 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 327, + 211, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 213, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 213, + 342 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "Self/Weakly Supervised Representation Learning. Self supervised representation was popular", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "in both image and video domains by designing various proxy tasks. In image domain, for in-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 374, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 504, + 387 + ], + "score": 1.0, + "content": "stance, these tasks could be predicting the image context (Doersch et al., 2015), counting the ob-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "jects (Noroozi et al., 2017), converting gray images to color one (Zhang et al., 2016), keeping global", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "and local consistency (Hjelm et al., 2019). In video domain, typical examples include frame predic-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 390, + 419 + ], + "score": 1.0, + "content": "tion (Diba et al., 2019; Vondrick et al., 2016), optical flow estimation", + "type": "text" + }, + { + "bbox": [ + 391, + 407, + 405, + 418 + ], + "score": 0.63, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "et al., 2018; Zhou et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "2017; Jayaraman & Grauman, 2017), instance tracking (Wang & Gupta, 2015; Wang et al., 2019b),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "temporal order or structure prediction (Misra et al., 2016; Fernando et al., 2017; Wei et al., 2018; Xu", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "et al., 2019a). These learnt representations may capture some aspects of low-level image or video", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 439, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 439, + 463 + ], + "score": 1.0, + "content": "structures, but are generally outperformed by those using cross modal information.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 352, + 506, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Several cross-modal self-supervised tasks was proposed to enhance single-modality representation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "power and typical example is audio-visual representation learning (Aytar et al., 2016; Arandjelovic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "& Zisserman, 2017; Korbar et al., 2018). 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Our CPD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "is applicable to more general video type and we experiment with a much smaller dataset (0.3M", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "vs. 100M) of both PGC and UGC videos, but achieves a similar performance on UCF101 and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "HMDB51. Concurrent work (Stroud et al., 2020) proposed a similar framework but required more", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "training videos (0.3M vs. 70M) and richer textual information to obtain similar performance to ours.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 468, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "score": 1.0, + "content": "Motion, Audio, and Text. Multi-modal information in videos provides natural cues for learning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "deep models. Motion or temporal information has been studied as to design proxy tasks to assist", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "cross-modal learning, such as optical flow or tracking (Ng et al., 2018; Wang & Gupta, 2015), frame", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "prediction (Diba et al., 2019; Vondrick et al., 2016), or high-level temporal structure (Wei et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "2018; Xu et al., 2019a; Fernando et al., 2017). As most video contain synchronized audio and visual", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "signals, audio information has served another common modality to supervised visual learning (Aytar", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "et al., 2016; Arandjelovic & Zisserman, 2017; Korbar et al., 2018). However, both motion and audio", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "information seem to be low-level signals and may lack high-level semantic for cross-modal learning.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 592, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Speech or text has been widely studied as another cross-modal setting in video learning (Sun et al.,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "2019b; Miech et al., 2019; Dong et al., 2019; Miech et al., 2018; Pan et al., 2016; Plummer et al.,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "2017). These works mainly aimed to learn a joint video-text embedding where visual and textual", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "cues are adjacent if they are semantically. However, these works focused on learn high-level visual-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "textual embedding by using the off-the-shelf models as feature extractors. Instead, our proposed", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 302, + 465, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 465, + 315 + ], + "score": 1.0, + "content": "CPD framework addresses a different issue of video representation learning from scratch.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 688, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 149, + 83, + 461, + 207 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 149, + 83, + 461, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 83, + 461, + 207 + ], + "spans": [ + { + "bbox": [ + 149, + 83, + 461, + 207 + ], + "score": 0.971, + "type": "image", + "image_path": "8a67af6f74daf14c18ca4442d46bf1823105ebabb57dde5b598d121cf0b3a1cf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 149, + 83, + 461, + 124.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 149, + 124.33333333333334, + 461, + 165.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 149, + 165.66666666666669, + 461, + 207.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 215, + 506, + 281 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 215, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 227 + ], + "score": 1.0, + "content": "Figure 1: The pipeline of our cross-modal pair discrimination (CPD) framework. First, the visual", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "and text are fed into modality-specific networks for feature extraction. Then, the visual and textual", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "features are mapped into a common 256-dimensional space. The cross-modal framework is learned", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "via video and text pair discrimination, which tries to make corresponding pairs closer than other", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "inconsistent pairs using a softmax criteria. The learnt spatiotemporal features could be deployed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 270, + 283, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 283, + 282 + ], + "score": 1.0, + "content": "directly or fine-tuned for downstream tasks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "textual embedding by using the off-the-shelf models as feature extractors. Instead, our proposed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 302, + 465, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 465, + 315 + ], + "score": 1.0, + "content": "CPD framework addresses a different issue of video representation learning from scratch.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 325, + 343 + ], + "lines": [ + { + "bbox": [ + 104, + 328, + 326, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 326, + 345 + ], + "score": 1.0, + "content": "3 CROSS-MODAL PAIR DISCRIMINATION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "In this section we provide an detailed description on our proposed cross-modal pair discrimination", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "(CPD) for weakly supervised spatiotemporal feature learning. First, we present the whole framework", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "and analyze its important properties. Then, we describe the training strategy of CPD framework.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 388, + 366, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 366, + 400 + ], + "score": 1.0, + "content": "Finally, we introduce text and video feature extraction networks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 254, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 256, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 256, + 425 + ], + "score": 1.0, + "content": "3.1 FRAMEWORK AND ANALYSIS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "Our goal is to propose a weakly supervised representation learning method by exploiting the corre-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "lation between each video clip and its associated text information, which could be easily obtained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "score": 1.0, + "content": "from a variety of sources such as YouTube titles, Instagram captions and automatic speech recog-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "nition (ASR). It is generally assumed that these text information contains semantic information, but", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "also might be noisy and irrelevant. Therefore, from technical perspective, we need to design an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "effective objective function and training strategy to capture this semantic correlation and as well", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "also suppress the effect of noisy and irrelevant information. 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These embedding", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 394, + 574 + ], + "score": 1.0, + "content": "functions would map these two modality into a common space (i.e.,", + "type": "text" + }, + { + "bbox": [ + 394, + 560, + 436, + 573 + ], + "score": 0.92, + "content": "f _ { i } ^ { v } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 559, + 456, + 574 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 457, + 560, + 498, + 573 + ], + "score": 0.91, + "content": "f _ { i } ^ { v } \\in \\mathbb { R } ^ { d } .", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "and related visual and text information should be close to each other. The embedding functions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "could be implemented by neural networks which will be clarified in next section. We first focus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "on how to devise objective function to optimize these embedding functions. Inspired by the work", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "of unsupervised learning in images (Wu et al., 2018), we design a cross-modal pair discrimination", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 307, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 307, + 629 + ], + "score": 1.0, + "content": "objective to learn these two embedding functions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 463, + 645 + ], + "score": 1.0, + "content": "Self-instance discrimination. In the original instance-level discrimination framework (", + "type": "text" + }, + { + "bbox": [ + 463, + 633, + 479, + 644 + ], + "score": 0.32, + "content": "\\mathrm { W u }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "2018), each image is treated as a distinct class and it would learn a classifier to categorize each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "image into its own class. This framework could be naturally extended into the setting of video and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "text pair by directly using feature concatenation, and we call this extension as self-instance discrim-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ination. 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First, the visual", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "and text are fed into modality-specific networks for feature extraction. Then, the visual and textual", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "features are mapped into a common 256-dimensional space. The cross-modal framework is learned", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "via video and text pair discrimination, which tries to make corresponding pairs closer than other", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "inconsistent pairs using a softmax criteria. The learnt spatiotemporal features could be deployed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 270, + 283, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 283, + 282 + ], + "score": 1.0, + "content": "directly or fine-tuned for downstream tasks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 504, + 314 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 106, + 291, + 505, + 315 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 325, + 343 + ], + "lines": [ + { + "bbox": [ + 104, + 328, + 326, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 326, + 345 + ], + "score": 1.0, + "content": "3 CROSS-MODAL PAIR DISCRIMINATION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "In this section we provide an detailed description on our proposed cross-modal pair discrimination", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "(CPD) for weakly supervised spatiotemporal feature learning. First, we present the whole framework", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "and analyze its important properties. Then, we describe the training strategy of CPD framework.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 388, + 366, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 366, + 400 + ], + "score": 1.0, + "content": "Finally, we introduce text and video feature extraction networks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 354, + 505, + 400 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 254, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 256, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 256, + 425 + ], + "score": 1.0, + "content": "3.1 FRAMEWORK AND ANALYSIS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "Our goal is to propose a weakly supervised representation learning method by exploiting the corre-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "lation between each video clip and its associated text information, which could be easily obtained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 469 + ], + "score": 1.0, + "content": "from a variety of sources such as YouTube titles, Instagram captions and automatic speech recog-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "nition (ASR). It is generally assumed that these text information contains semantic information, but", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "also might be noisy and irrelevant. Therefore, from technical perspective, we need to design an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "effective objective function and training strategy to capture this semantic correlation and as well", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "also suppress the effect of noisy and irrelevant information. 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These embedding", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 394, + 574 + ], + "score": 1.0, + "content": "functions would map these two modality into a common space (i.e.,", + "type": "text" + }, + { + "bbox": [ + 394, + 560, + 436, + 573 + ], + "score": 0.92, + "content": "f _ { i } ^ { v } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 559, + 456, + 574 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 457, + 560, + 498, + 573 + ], + "score": 0.91, + "content": "f _ { i } ^ { v } \\in \\mathbb { R } ^ { d } .", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "and related visual and text information should be close to each other. The embedding functions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "could be implemented by neural networks which will be clarified in next section. We first focus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "on how to devise objective function to optimize these embedding functions. Inspired by the work", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "of unsupervised learning in images (Wu et al., 2018), we design a cross-modal pair discrimination", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 307, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 307, + 629 + ], + "score": 1.0, + "content": "objective to learn these two embedding functions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 527, + 506, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 463, + 645 + ], + "score": 1.0, + "content": "Self-instance discrimination. In the original instance-level discrimination framework (", + "type": "text" + }, + { + "bbox": [ + 463, + 633, + 479, + 644 + ], + "score": 0.32, + "content": "\\mathrm { W u }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "2018), each image is treated as a distinct class and it would learn a classifier to categorize each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "image into its own class. This framework could be naturally extended into the setting of video and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "text pair by directly using feature concatenation, and we call this extension as self-instance discrim-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ination. 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This class weight represent a class prototype for each video-text", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "instance and is probably not easy to optimize as we only have a single sample for each class. 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This straight forward extension shares the advantage", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "of instance-level discrimination by directly modeling in the joint video-text space. Yet, in fact, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "semantic information of text modality is higher than video pixels and we aims at learning video", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "features with the supervision of textual information. To meet this requirement, we propose a refined", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 373, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 373, + 238 + ], + "score": 1.0, + "content": "objective function from the perspective of conditional distribution.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "score": 1.0, + "content": "Cross-pair discrimination. According to the above analysis, we design the objective function by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 252, + 266 + ], + "score": 1.0, + "content": "considering conditional distribution", + "type": "text" + }, + { + "bbox": [ + 252, + 254, + 280, + 266 + ], + "score": 0.92, + "content": "p ( i _ { t } | v )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 254, + 299, + 266 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 254, + 327, + 266 + ], + "score": 0.92, + "content": "p ( i _ { v } | t )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "rather than implicitly modeling distribution", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 387, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 133, + 277 + ], + "score": 0.91, + "content": "p ( v , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 264, + 387, + 277 + ], + "score": 1.0, + "content": ". 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The con-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 193, + 343 + ], + "score": 1.0, + "content": "ditional distribution", + "type": "text" + }, + { + "bbox": [ + 193, + 331, + 221, + 343 + ], + "score": 0.92, + "content": "p ( i _ { v } | t )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "could be defined at the same way. We call this framework as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "cross-pair discrimination, and during training phase, the objective is to maximize the likelihood", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 353, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 107, + 353, + 219, + 368 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\prod _ { i = 1 } ^ { N } p ( i _ { t } | v _ { i } ) \\prod _ { i = 1 } ^ { N } p ( i _ { v } | t _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 353, + 507, + 370 + ], + "score": 1.0, + "content": ". The key difference between Equation (2) and (3) is that we propose to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 366, + 211, + 380 + ], + "score": 1.0, + "content": "use cross-correlation term", + "type": "text" + }, + { + "bbox": [ + 212, + 367, + 235, + 378 + ], + "score": 0.87, + "content": "\\mathbf { f } ^ { t T } \\mathbf { f } ^ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 366, + 376, + 380 + ], + "score": 1.0, + "content": "to replace the self-correlation term", + "type": "text" + }, + { + "bbox": [ + 376, + 366, + 439, + 379 + ], + "score": 0.88, + "content": "( \\mathbf { f } ^ { v T } \\mathbf { f } ^ { v } + \\mathbf { f } ^ { t T } \\mathbf { f } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 366, + 505, + 380 + ], + "score": 1.0, + "content": ". This cross cor-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "relation is more effective to capture the mutual information between visual and textual information,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "and thereby better at guiding the spatiotemporal feature learning from video with text information", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 168, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 168, + 412 + ], + "score": 1.0, + "content": "as supervision.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 504, + 429 + ], + "score": 1.0, + "content": "Ranking loss. There is some common ranking loss for cross-modal matching. 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We apply Equation (4) in both ways of video", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "with its associated text and text with its video. In experiment, we empirically compare this ranking", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 526, + 341, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 341, + 538 + ], + "score": 1.0, + "content": "loss with our designed cross-pair discrimination objective.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 551, + 199, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 200, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 200, + 564 + ], + "score": 1.0, + "content": "3.2 TRAINING CPD", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "The training of CPD framework needs to address two technical issues: (1) large number of video-text", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 477, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 477, + 595 + ], + "score": 1.0, + "content": "pair classes; (2) optimization difficulty on noisy video-text datasets by training from scratch.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "Noise-contrastive estimation. In training stage, we adopt noise-contrastive estimation tech-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "nique (Gutmann & Hyvarinen, 2010) to approximate Equation (3) to solve the computational issues ¨", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "by the huge numbers of pairs. The basic idea is to transform the multi-class classification problem", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 644 + ], + "score": 1.0, + "content": "in Equation (3) into a set of binary classification problem. In the binary classification task, the task", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "is to distinguish between data sample and noise sample. 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This straight forward extension shares the advantage", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "of instance-level discrimination by directly modeling in the joint video-text space. Yet, in fact, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "semantic information of text modality is higher than video pixels and we aims at learning video", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "features with the supervision of textual information. To meet this requirement, we propose a refined", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 373, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 373, + 238 + ], + "score": 1.0, + "content": "objective function from the perspective of conditional distribution.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 103, + 169, + 507, + 238 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 257 + ], + "score": 1.0, + "content": "Cross-pair discrimination. 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The con-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 193, + 343 + ], + "score": 1.0, + "content": "ditional distribution", + "type": "text" + }, + { + "bbox": [ + 193, + 331, + 221, + 343 + ], + "score": 0.92, + "content": "p ( i _ { v } | t )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "could be defined at the same way. We call this framework as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "cross-pair discrimination, and during training phase, the objective is to maximize the likelihood", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 353, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 107, + 353, + 219, + 368 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\prod _ { i = 1 } ^ { N } p ( i _ { t } | v _ { i } ) \\prod _ { i = 1 } ^ { N } p ( i _ { v } | t _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 353, + 507, + 370 + ], + "score": 1.0, + "content": ". The key difference between Equation (2) and (3) is that we propose to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 366, + 211, + 380 + ], + "score": 1.0, + "content": "use cross-correlation term", + "type": "text" + }, + { + "bbox": [ + 212, + 367, + 235, + 378 + ], + "score": 0.87, + "content": "\\mathbf { f } ^ { t T } \\mathbf { f } ^ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 366, + 376, + 380 + ], + "score": 1.0, + "content": "to replace the self-correlation term", + "type": "text" + }, + { + "bbox": [ + 376, + 366, + 439, + 379 + ], + "score": 0.88, + "content": "( \\mathbf { f } ^ { v T } \\mathbf { f } ^ { v } + \\mathbf { f } ^ { t T } \\mathbf { f } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 366, + 505, + 380 + ], + "score": 1.0, + "content": ". This cross cor-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "relation is more effective to capture the mutual information between visual and textual information,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "and thereby better at guiding the spatiotemporal feature learning from video with text information", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 168, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 168, + 412 + ], + "score": 1.0, + "content": "as supervision.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 318, + 507, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 504, + 429 + ], + "score": 1.0, + "content": "Ranking loss. There is some common ranking loss for cross-modal matching. To well study the ef-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 504, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 504, + 441 + ], + "score": 1.0, + "content": "fectiveness of proposed cross-modal pair discrimination objective, we also compare with a baseline", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 284, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 284, + 451 + ], + "score": 1.0, + "content": "of ranking loss, which is defined as follows:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 416, + 504, + 451 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 455, + 422, + 487 + ], + "lines": [ + { + "bbox": [ + 189, + 455, + 422, + 487 + ], + "spans": [ + { + "bbox": [ + 189, + 455, + 422, + 487 + ], + "score": 0.93, + "content": "\\mathcal { L } ( v _ { i } , t _ { i } ) = \\frac { 1 } { n - 1 } \\sum _ { j \\neq i } \\operatorname* { m a x } ( 0 , \\delta + \\mathcal { S } ( \\mathbf { f } _ { j } ^ { t } , \\mathbf { f } _ { i } ^ { v } ) - \\mathcal { S } ( \\mathbf { f } _ { i } ^ { t } , \\mathbf { f } _ { i } ^ { v } ) ) ,", + "type": "interline_equation", + "image_path": "1b5f1696c2abce4b0207d9385a7d6f9021c9a4050f4e71bd2774b9b967754b26.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 455, + 422, + 471.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 189, + 471.0, + 422, + 487.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 178, + 506 + ], + "score": 1.0, + "content": "where each video", + "type": "text" + }, + { + "bbox": [ + 178, + 495, + 187, + 504 + ], + "score": 0.85, + "content": "v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 492, + 271, + 506 + ], + "score": 1.0, + "content": "has a associated text", + "type": "text" + }, + { + "bbox": [ + 271, + 494, + 279, + 504 + ], + "score": 0.87, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 492, + 353, + 506 + ], + "score": 1.0, + "content": "and unrelated text", + "type": "text" + }, + { + "bbox": [ + 353, + 494, + 362, + 505 + ], + "score": 0.86, + "content": "t _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 492, + 442, + 506 + ], + "score": 1.0, + "content": "from current batch.", + "type": "text" + }, + { + "bbox": [ + 443, + 492, + 480, + 506 + ], + "score": 0.93, + "content": "{ \\cal { S } } ( \\mathbf { f } _ { j } ^ { t } , \\mathbf { f } _ { i } ^ { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 177, + 516 + ], + "score": 1.0, + "content": "cosine similarity,", + "type": "text" + }, + { + "bbox": [ + 177, + 506, + 184, + 514 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 504, + 267, + 516 + ], + "score": 1.0, + "content": "is the batch size and", + "type": "text" + }, + { + "bbox": [ + 268, + 505, + 273, + 514 + ], + "score": 0.84, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "is a margin. We apply Equation (4) in both ways of video", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "with its associated text and text with its video. In experiment, we empirically compare this ranking", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 526, + 341, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 341, + 538 + ], + "score": 1.0, + "content": "loss with our designed cross-pair discrimination objective.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 492, + 506, + 538 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 551, + 199, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 200, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 200, + 564 + ], + "score": 1.0, + "content": "3.2 TRAINING CPD", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "The training of CPD framework needs to address two technical issues: (1) large number of video-text", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 477, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 477, + 595 + ], + "score": 1.0, + "content": "pair classes; (2) optimization difficulty on noisy video-text datasets by training from scratch.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 571, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "Noise-contrastive estimation. In training stage, we adopt noise-contrastive estimation tech-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "nique (Gutmann & Hyvarinen, 2010) to approximate Equation (3) to solve the computational issues ¨", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "by the huge numbers of pairs. The basic idea is to transform the multi-class classification problem", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 644 + ], + "score": 1.0, + "content": "in Equation (3) into a set of binary classification problem. In the binary classification task, the task", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "is to distinguish between data sample and noise sample. 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If we train both models simultaneously in the beginning,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "the random noise produced by video model will destroy the parameters of language model. 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The first tiny network is efficient for abla-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "tion study and then we transfer its optimal setting to the larger backbone and frame resolution. We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 503, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 492, + 356 + ], + "score": 1.0, + "content": "also add a mapping layer to transform the visual features into 256-dimensional embedding space", + "type": "text" + }, + { + "bbox": [ + 493, + 344, + 503, + 354 + ], + "score": 0.64, + "content": "\\mathbf { f } ^ { v }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 355, + 263, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 202, + 366 + ], + "score": 1.0, + "content": "and this 256-d vector is", + "type": "text" + }, + { + "bbox": [ + 202, + 355, + 212, + 366 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 355, + 263, + 366 + ], + "score": 1.0, + "content": "-normalized.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 504, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 504, + 384 + ], + "score": 1.0, + "content": "Text architecture. Our textual stream subnetwork is based on the off-the-shelf language models.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "score": 1.0, + "content": "We choose Word2vec (Mikolov et al., 2013) and DistilBERT (Devlin et al., 2019; Sanh et al., 2019)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "as our textual encoders. Word2vec is an unsupervised word encoder, pre-trained by reconstructing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "the surrounding words of the continue sentences. We average word vectors which are 300 dimen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "sional as textual encoder. BERT (Devlin et al., 2019) encodes long sentences by predicting the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "missing words given their bidirectional context, and DistilBERT achieves comparable performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "with a faster and lighter model via knowledge distillation (Hinton et al., 2015). We average word", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "embeddings of title generated by DistilBERT and obtain 768 dimensional text feature. Finally, two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "fully connected layers with ReLU and Batch Normalization (Ioffe & Szegedy, 2015) are added to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 292, + 483 + ], + "score": 1.0, + "content": "our textual encoder to obtain textual feature", + "type": "text" + }, + { + "bbox": [ + 292, + 470, + 302, + 480 + ], + "score": 0.83, + "content": "\\mathbf { f } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "in the common embedding space, which is also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 482, + 168, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 116, + 492 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 482, + 168, + 493 + ], + "score": 1.0, + "content": "-normalized.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 510, + 200, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 201, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 201, + 525 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "In this section, we present the experimental results of our proposed CPD framework. First, we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "describe the training and evaluation datasets with implementation details. Then, we conduct ablation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "study on our proposed CPD framework. Finally, we verify the effectiveness of CPD from two", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 424, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 424, + 581 + ], + "score": 1.0, + "content": "aspects: weakly-supervised representation learning and representation transfer.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 107, + 595, + 176, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 177, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 177, + 607 + ], + "score": 1.0, + "content": "4.1 DATASETS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "In our experiment, we pre-train our CPD framework on two video-text datasets: Kinetics-210k (Kay", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "et al., 2017) and Instagram-300k (Duan et al., 2020). Then, we fine-tune the video model on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "three human action datasets: Kinetics400 (Kay et al., 2017), UCF101 (Soomro et al., 2012) and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 238, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 238, + 660 + ], + "score": 1.0, + "content": "HMDB51 (Kuehne et al., 2011).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 417, + 678 + ], + "score": 1.0, + "content": "Kinetics-210k. Following the recent self-supervised methods (Wang et al.,", + "type": "text" + }, + { + "bbox": [ + 418, + 666, + 444, + 676 + ], + "score": 0.56, + "content": "2 0 1 9 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "; Korbar et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "2018; Han et al., 2019), we utilize Kinetics (Kay et al., 2017) dataset for weakly-supervised pre-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "training of CPD. It is often called Kinetics400 since it has 400 action classes, but we count training", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 697, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 713 + ], + "score": 1.0, + "content": "video number as we do not use any class information for weakly-supervised representation learning.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Due to invalid urls and data cleaning, the collected dataset contains around 210k video-text pairs,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and thus we call this dataset as Kinetics-210k. To construct video-text pairs, we equip each clip with", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Curriculum learning. To handle the optimization difficulty of directly training from scratch on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "noisy video-text dataset, we present a curriculum training strategy by resorting to the existing un-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 147 + ], + "score": 1.0, + "content": "supervised pre-trained language models. To relieve the training difficulty, our curriculum learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "strategy divides the training procedure into two stages. In the first stage, we fix the pre-trained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "language model and only update the parameters of visual model and embedding function. The mo-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "tivation is that the language model is pre-trained well using corpus much larger than ours and the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 190 + ], + "score": 1.0, + "content": "video model is totally trained from scratch. If we train both models simultaneously in the beginning,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "the random noise produced by video model will destroy the parameters of language model. In the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "second stage, after the good initialization of video model, we start to jointly train the visual-textual", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 247, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 247, + 221 + ], + "score": 1.0, + "content": "model with a smaller learning rate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 110, + 506, + 221 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 235, + 234, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 235, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 235, + 247 + ], + "score": 1.0, + "content": "3.3 ARCHITECTURE DESIGN", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 256, + 505, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "Video architecture. For video representation, we use the 3D CNNs to extract spatiotemporal fea-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "tures from a video clip. Specifically, we randomly sample 8 frames from each video clip and sam-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "pling stride is 4. Following the implementation of slow stream in the recent SlowFast (Feichtenhofer", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 222, + 302 + ], + "score": 1.0, + "content": "et al., 2018), all filters from", + "type": "text" + }, + { + "bbox": [ + 223, + 291, + 248, + 300 + ], + "score": 0.83, + "content": "c o n v _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 289, + 261, + 302 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 261, + 291, + 281, + 300 + ], + "score": 0.69, + "content": "r e s _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "degenerate temporal convolutions into 2D convolution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 328, + 312 + ], + "score": 1.0, + "content": "kernels and it only reserves 3D convolution kernels in", + "type": "text" + }, + { + "bbox": [ + 329, + 302, + 348, + 311 + ], + "score": 0.87, + "content": "r e s _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 300, + 367, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 367, + 302, + 387, + 311 + ], + "score": 0.88, + "content": "r e s _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "without temporal downsam-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 414, + 323 + ], + "score": 1.0, + "content": "pling. We try two kinds of network architectures: (1) 3D ResNet34 trained on", + "type": "text" + }, + { + "bbox": [ + 414, + 311, + 469, + 322 + ], + "score": 0.92, + "content": "1 1 2 \\times 1 1 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "volumes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 236, + 334 + ], + "score": 1.0, + "content": "and (2) 3D ResNet50 trained on", + "type": "text" + }, + { + "bbox": [ + 236, + 322, + 296, + 333 + ], + "score": 0.92, + "content": "2 2 4 \\times 2 2 4 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "volumes. The first tiny network is efficient for abla-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "tion study and then we transfer its optimal setting to the larger backbone and frame resolution. We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 503, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 492, + 356 + ], + "score": 1.0, + "content": "also add a mapping layer to transform the visual features into 256-dimensional embedding space", + "type": "text" + }, + { + "bbox": [ + 493, + 344, + 503, + 354 + ], + "score": 0.64, + "content": "\\mathbf { f } ^ { v }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 355, + 263, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 202, + 366 + ], + "score": 1.0, + "content": "and this 256-d vector is", + "type": "text" + }, + { + "bbox": [ + 202, + 355, + 212, + 366 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 355, + 263, + 366 + ], + "score": 1.0, + "content": "-normalized.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 256, + 505, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 504, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 504, + 384 + ], + "score": 1.0, + "content": "Text architecture. Our textual stream subnetwork is based on the off-the-shelf language models.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "score": 1.0, + "content": "We choose Word2vec (Mikolov et al., 2013) and DistilBERT (Devlin et al., 2019; Sanh et al., 2019)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "as our textual encoders. Word2vec is an unsupervised word encoder, pre-trained by reconstructing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "the surrounding words of the continue sentences. We average word vectors which are 300 dimen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "sional as textual encoder. BERT (Devlin et al., 2019) encodes long sentences by predicting the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "missing words given their bidirectional context, and DistilBERT achieves comparable performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "with a faster and lighter model via knowledge distillation (Hinton et al., 2015). We average word", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "embeddings of title generated by DistilBERT and obtain 768 dimensional text feature. Finally, two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "fully connected layers with ReLU and Batch Normalization (Ioffe & Szegedy, 2015) are added to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 292, + 483 + ], + "score": 1.0, + "content": "our textual encoder to obtain textual feature", + "type": "text" + }, + { + "bbox": [ + 292, + 470, + 302, + 480 + ], + "score": 0.83, + "content": "\\mathbf { f } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "in the common embedding space, which is also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 482, + 168, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 116, + 492 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 482, + 168, + 493 + ], + "score": 1.0, + "content": "-normalized.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 372, + 506, + 493 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 510, + 200, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 201, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 201, + 525 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "In this section, we present the experimental results of our proposed CPD framework. First, we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "describe the training and evaluation datasets with implementation details. Then, we conduct ablation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "study on our proposed CPD framework. Finally, we verify the effectiveness of CPD from two", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 424, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 424, + 581 + ], + "score": 1.0, + "content": "aspects: weakly-supervised representation learning and representation transfer.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 536, + 506, + 581 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 595, + 176, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 177, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 177, + 607 + ], + "score": 1.0, + "content": "4.1 DATASETS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "In our experiment, we pre-train our CPD framework on two video-text datasets: Kinetics-210k (Kay", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "et al., 2017) and Instagram-300k (Duan et al., 2020). Then, we fine-tune the video model on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "three human action datasets: Kinetics400 (Kay et al., 2017), UCF101 (Soomro et al., 2012) and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 238, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 238, + 660 + ], + "score": 1.0, + "content": "HMDB51 (Kuehne et al., 2011).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 615, + 506, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 417, + 678 + ], + "score": 1.0, + "content": "Kinetics-210k. Following the recent self-supervised methods (Wang et al.,", + "type": "text" + }, + { + "bbox": [ + 418, + 666, + 444, + 676 + ], + "score": 0.56, + "content": "2 0 1 9 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "; Korbar et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "2018; Han et al., 2019), we utilize Kinetics (Kay et al., 2017) dataset for weakly-supervised pre-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "training of CPD. It is often called Kinetics400 since it has 400 action classes, but we count training", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 697, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 713 + ], + "score": 1.0, + "content": "video number as we do not use any class information for weakly-supervised representation learning.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Due to invalid urls and data cleaning, the collected dataset contains around 210k video-text pairs,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and thus we call this dataset as Kinetics-210k. To construct video-text pairs, we equip each clip with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "the video title directly crawled from YouTube, termed as Kinetics-title. As the original title may be", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "very noisy, we pre-process the text information in two ways. First, we delete special symbols and", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "characters such as non-English words and emoji, termed as Kinetics-title-clean. Second, we use", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "StanfordNLP (Qi et al., 2018) to obtain the dependency tree of sentences in titles and only reserve", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 284, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 284, + 139 + ], + "score": 1.0, + "content": "verbs and nouns, named Kinetics-title-tree.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "the video title directly crawled from YouTube, termed as Kinetics-title. As the original title may be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "very noisy, we pre-process the text information in two ways. First, we delete special symbols and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "characters such as non-English words and emoji, termed as Kinetics-title-clean. Second, we use", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "StanfordNLP (Qi et al., 2018) to obtain the dependency tree of sentences in titles and only reserve", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 284, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 284, + 139 + ], + "score": 1.0, + "content": "verbs and nouns, named Kinetics-title-tree.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "Instagram-300k. To avoid data bias in Kinetics caused by human annotation (i.e., trimmed videos", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "with an action), we further verify the effectiveness our CPD model on an uncurated web video", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "dataset (Duan et al., 2020). This new dataset is constructed from Instagram by searching action", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "label of Kinetics-400 but without any manual filtering. Due to limited computation resource and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "also for fair comparison with pretraining on Kinetics-210k, we randomly sample 300k from the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 330, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 330, + 210 + ], + "score": 1.0, + "content": "original web video dataset, termed as Instagram-300k.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 504, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 504, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 504, + 227 + ], + "score": 1.0, + "content": "An important difference is that the these videos are with User Generated Content (UGC) and accom-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "panied by captions uploaded by users. Therefore, its video content distribution is much different with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 238, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 248 + ], + "score": 1.0, + "content": "those in Profession Generated Content (PGC) in UCF101 and HMDB51, and the text noise is also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 480, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 480, + 260 + ], + "score": 1.0, + "content": "much higher. So, it is more challenging to train a pre-trained CPD model on Instagram-300k.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "UCF101 and HMDB51. We evaluate the generalization of our pre-trained models by fine-tuning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "on two small human action datasets: UCF101 (Soomro et al., 2012) and HMDB51 (Kuehne et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 194, + 299 + ], + "score": 1.0, + "content": "2011), which contain", + "type": "text" + }, + { + "bbox": [ + 195, + 287, + 211, + 297 + ], + "score": 0.28, + "content": "1 3 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "videos of 101 classes and 7k video of 51 classes respectively. We report", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "ablation study on the first split and report average performance over three splits for fair comparison.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 247, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 249, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 249, + 344 + ], + "score": 1.0, + "content": "4.2 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "Weakly supervised learning of CPD. We train our CPD model on video-text datasets and use", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "video-text retrieval on 1k unseen video-text pairs as validation set duration training. Specifically, 8", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "frames are sampled from each video clip and the sampling stride is 4. We use SGD to optimize our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "objective and the training parameters include a momentum of 0.9 and 1e-4 for weight decay. We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 214, + 412 + ], + "score": 1.0, + "content": "set temperature parameter", + "type": "text" + }, + { + "bbox": [ + 214, + 399, + 254, + 410 + ], + "score": 0.89, + "content": "\\tau = 0 . 0 7", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 399, + 340, + 412 + ], + "score": 1.0, + "content": "and noise frequency", + "type": "text" + }, + { + "bbox": [ + 340, + 401, + 350, + 409 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "to 4096. In the beginning, we fix the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "pre-trained language model and the learning rate is set as 0.2. When the retrieval performance on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "validation set saturates (170 epochs for 3D ResNet34 and 110 epochs for 3D ResNet50), we start to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "update the language model with learning rate of 3e-5 and decrease the rest learning rate to 0.02. The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 341, + 455 + ], + "score": 1.0, + "content": "maximize training number is 250 epochs. For input size of", + "type": "text" + }, + { + "bbox": [ + 342, + 443, + 401, + 453 + ], + "score": 0.91, + "content": "1 1 2 \\times 1 1 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 443, + 506, + 455 + ], + "score": 1.0, + "content": ", the mini-batch size is 64", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 326, + 468 + ], + "score": 1.0, + "content": "clips per GPUs and 16 clips per GPUs for input size of", + "type": "text" + }, + { + "bbox": [ + 327, + 454, + 386, + 465 + ], + "score": 0.91, + "content": "2 2 4 \\times 2 2 4 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 453, + 505, + 468 + ], + "score": 1.0, + "content": ". We use 8 GPUs for training.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "Evaluation on representation learning. We first verify our CPD learned representation by em-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 504, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 393, + 494 + ], + "score": 1.0, + "content": "ploying a shallow classifier on frozen features. Specifically, we utilize", + "type": "text" + }, + { + "bbox": [ + 393, + 482, + 400, + 492 + ], + "score": 0.51, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 482, + 504, + 494 + ], + "score": 1.0, + "content": "-Nearest Neighbor (kNN)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "and linear classifier based on extracted features for classification. For video feature extraction, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "sample 10 clips from each video and each clip contains 8 frames with 4 sampling stride. The 256-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "dimensional embedding feature and the output of global average pooling are extracted as features.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "The extracted features over 10 clips in a video are averaged as a video-level representation. 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We adopt Adam with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 465, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 465, + 572 + ], + "score": 1.0, + "content": "learning rate of 1e-3 and reduce by a factor of 10 every 10 epochs, stopping at 30 epochs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "Evaluation on representation transfer. A main goal of representation learning is to transfer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "them to downstream tasks. We fine-tune the learned spatiotemporal representation on the UCF101,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "HMDB51 and a small fraction of Kinetics400. During fine-tuning, 16 frames with stride 4 are sam-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "pled as input. We simply replace the embedding layer of video model with a new fully-connected", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "layer and multi-way softmax for action recognition. The classifier is trained using the SGD opti-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 642 + ], + "score": 1.0, + "content": "mizer with an initial learning rate 1e-2 and weight decay 5e-4. Learning rate is decreased twice by a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "factor of 10 when the validation loss saturates. 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To avoid data bias in Kinetics caused by human annotation (i.e., trimmed videos", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "with an action), we further verify the effectiveness our CPD model on an uncurated web video", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "dataset (Duan et al., 2020). This new dataset is constructed from Instagram by searching action", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "label of Kinetics-400 but without any manual filtering. Due to limited computation resource and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "also for fair comparison with pretraining on Kinetics-210k, we randomly sample 300k from the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 330, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 330, + 210 + ], + "score": 1.0, + "content": "original web video dataset, termed as Instagram-300k.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 144, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 504, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 504, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 504, + 227 + ], + "score": 1.0, + "content": "An important difference is that the these videos are with User Generated Content (UGC) and accom-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "panied by captions uploaded by users. Therefore, its video content distribution is much different with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 238, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 248 + ], + "score": 1.0, + "content": "those in Profession Generated Content (PGC) in UCF101 and HMDB51, and the text noise is also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 480, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 480, + 260 + ], + "score": 1.0, + "content": "much higher. So, it is more challenging to train a pre-trained CPD model on Instagram-300k.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 215, + 505, + 260 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "UCF101 and HMDB51. 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We report", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "ablation study on the first split and report average performance over three splits for fair comparison.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 264, + 506, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 247, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 249, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 249, + 344 + ], + "score": 1.0, + "content": "4.2 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "Weakly supervised learning of CPD. We train our CPD model on video-text datasets and use", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "video-text retrieval on 1k unseen video-text pairs as validation set duration training. Specifically, 8", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "frames are sampled from each video clip and the sampling stride is 4. We use SGD to optimize our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "objective and the training parameters include a momentum of 0.9 and 1e-4 for weight decay. We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 214, + 412 + ], + "score": 1.0, + "content": "set temperature parameter", + "type": "text" + }, + { + "bbox": [ + 214, + 399, + 254, + 410 + ], + "score": 0.89, + "content": "\\tau = 0 . 0 7", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 399, + 340, + 412 + ], + "score": 1.0, + "content": "and noise frequency", + "type": "text" + }, + { + "bbox": [ + 340, + 401, + 350, + 409 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "to 4096. In the beginning, we fix the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "pre-trained language model and the learning rate is set as 0.2. When the retrieval performance on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "validation set saturates (170 epochs for 3D ResNet34 and 110 epochs for 3D ResNet50), we start to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "update the language model with learning rate of 3e-5 and decrease the rest learning rate to 0.02. The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 341, + 455 + ], + "score": 1.0, + "content": "maximize training number is 250 epochs. For input size of", + "type": "text" + }, + { + "bbox": [ + 342, + 443, + 401, + 453 + ], + "score": 0.91, + "content": "1 1 2 \\times 1 1 2 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 443, + 506, + 455 + ], + "score": 1.0, + "content": ", the mini-batch size is 64", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 326, + 468 + ], + "score": 1.0, + "content": "clips per GPUs and 16 clips per GPUs for input size of", + "type": "text" + }, + { + "bbox": [ + 327, + 454, + 386, + 465 + ], + "score": 0.91, + "content": "2 2 4 \\times 2 2 4 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 453, + 505, + 468 + ], + "score": 1.0, + "content": ". We use 8 GPUs for training.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 354, + 506, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "Evaluation on representation learning. We first verify our CPD learned representation by em-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 504, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 393, + 494 + ], + "score": 1.0, + "content": "ploying a shallow classifier on frozen features. Specifically, we utilize", + "type": "text" + }, + { + "bbox": [ + 393, + 482, + 400, + 492 + ], + "score": 0.51, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 482, + 504, + 494 + ], + "score": 1.0, + "content": "-Nearest Neighbor (kNN)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "and linear classifier based on extracted features for classification. For video feature extraction, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "sample 10 clips from each video and each clip contains 8 frames with 4 sampling stride. The 256-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "dimensional embedding feature and the output of global average pooling are extracted as features.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "The extracted features over 10 clips in a video are averaged as a video-level representation. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 342, + 549 + ], + "score": 1.0, + "content": "choose cosine distance as distance metric in kNN and set", + "type": "text" + }, + { + "bbox": [ + 342, + 537, + 374, + 547 + ], + "score": 0.9, + "content": "k = 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 536, + 505, + 549 + ], + "score": 1.0, + "content": ". As for linear classifier, a fully", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "connected layer after Batch Normalization is added with cross-entropy loss. We adopt Adam with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 465, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 465, + 572 + ], + "score": 1.0, + "content": "learning rate of 1e-3 and reduce by a factor of 10 every 10 epochs, stopping at 30 epochs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 470, + 506, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "Evaluation on representation transfer. A main goal of representation learning is to transfer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "them to downstream tasks. We fine-tune the learned spatiotemporal representation on the UCF101,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "HMDB51 and a small fraction of Kinetics400. During fine-tuning, 16 frames with stride 4 are sam-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "pled as input. We simply replace the embedding layer of video model with a new fully-connected", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "layer and multi-way softmax for action recognition. The classifier is trained using the SGD opti-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 642 + ], + "score": 1.0, + "content": "mizer with an initial learning rate 1e-2 and weight decay 5e-4. Learning rate is decreased twice by a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "factor of 10 when the validation loss saturates. During testing, for each video, we uniformly sample", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "10 clips and each clip contains 3 crops, following the common practice (Feichtenhofer et al., 2018).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 575, + 506, + 665 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 686, + 207, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 209, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 209, + 698 + ], + "score": 1.0, + "content": "4.3 ABLATION STUDY", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 708, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 504, + 723 + ], + "score": 1.0, + "content": "In this study, we pre-train our CPD models on Kinetics-210k dataset and choose the task of repre-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 372, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 372, + 733 + ], + "score": 1.0, + "content": "sentation transfer by fine tuning on UCF101 split 1 for evaluation.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 104, + 708, + 504, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 241, + 93, + 354, + 137 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 241, + 93, + 354, + 137 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 241, + 93, + 354, + 137 + ], + "spans": [ + { + "bbox": [ + 241, + 93, + 354, + 137 + ], + "score": 0.946, + "html": "
TrainingstrategyAccuracy(%)
Random init.50.0
Direct fine-tuning81.3
Curr. learning182.2
Curr.learning284.2
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Textual encoderDataAccuracy(%)
Random Init.-50.0
Word2vecTree83.1
DistilBERTTree82.1
Word2vecClean82.5
DistilBERTClean84.2
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Objective functionAccuracy(%)
Random init.50.0
Ranking loss79.9
Self-instance Dis.51.1
Cross-pair Dis.82.2
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BackbonePre-trained Sup.Layer (Dim)KNNLC
3D-ConvNet (Kay et al.,2017)Kinetics-400Label--56.1
3D ResNet34 (Hara et al., 2018) 3D ResNet50 (ours)Kinetics-400 Label Kinetics-400 Label--60.1 73.2
ResNet50ImageNet Label-- 42.856.1
3DResNet34res5 (2048)
Instagram-300k Captionemb (256)34.537.3
3D ResNet34Instagram-300k Captionres5 (512)36.144.6
3D ResNet50Instagram-300k Captionemb (256)51.151.7
3DResNet50Instagram-300k Captionres5 (2048)51.155.4
3DResNet34Kinetics-210k Titleemb (256)49.950.8
3D ResNet34Kinetics-210k Titleres5 (512)50.153.3
3DResNet50Kinetics-210k Titleemb (256)58.059.6
3DResNet50Kinetics-210k Titleres5 (2048)58.263.8
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Top-1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 313, + 368, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 368, + 326 + ], + "score": 1.0, + "content": "classification accuracy is reported on Kinetics-400 validation set.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + } + ], + "index": 12.25 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 344 + ], + "score": 1.0, + "content": "Objective function. We compare three objective functions for cross-modal pair discrimination de-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "scribed in Section 3.1. We pre-train models by utilizing DistilBERT as textual encoder without fine-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "tuning and the experimental results are reported in Table 1a. Self-instance discrimination almost", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "has no contribution to learn effective representation as there is no cross-modal correlation modeling.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "Cross-pair discrimination gives a better performance than ranking loss as cross-pair discrimination", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "can construct negative video-text pairs from entire dataset while ranking loss is only optimized by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "negative pairs from current batch. More theoretical analysis can be found in Section. A.1 of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 409, + 151, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 151, + 423 + ], + "score": 1.0, + "content": "Appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "Curriculum learning. We design different training strategies from noisy video-text datasets. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "first strategy is to fine-tune the pre-trained textual encoder directly at the beginning. Then we com-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "pare with stage I and stage II of curriculum learning proposed in Section 3.2. All these strategies", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 457, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 506, + 473 + ], + "score": 1.0, + "content": "are pre-trained on Kinetics-title-clean. The numerical results are summarized in Table 1b. Fixing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 506, + 484 + ], + "score": 1.0, + "content": "the pre-trained language model gives better performance than direct fine-tuning at the beginning", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 481, + 142, + 492 + ], + "score": 0.9, + "content": "( + 0 . 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ". We ascribe this to the fact that the random noise produced by video model destroy the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "well pre-trained textual encoder at the beginning. Also, fine-tuning the language model after the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 502, + 374, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 348, + 515 + ], + "score": 1.0, + "content": "video model is well initialized further boost the accuracy by", + "type": "text" + }, + { + "bbox": [ + 348, + 503, + 370, + 514 + ], + "score": 0.85, + "content": "2 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 502, + 374, + 515 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Different textual information. In this experiment, we choose video-text pairs from Kinetics-title-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "score": 1.0, + "content": "tree, Kinetics-title-clean datasets and utilize Word2vec and DistilBERT as a textual extractor. The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "experimental results are reported in Table 1c. For textual encoder, abundant and video-specific text", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "information benefits to train our CPD model with stronger language model such as DistilBERT", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 564, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 475, + 576 + ], + "score": 1.0, + "content": "according to the performance difference between Kinetics-title-tree and Kinetics-title-clean", + "type": "text" + }, + { + "bbox": [ + 475, + 564, + 504, + 575 + ], + "score": 0.83, + "content": "( 8 2 . 1 \\%", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 120, + 587 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 121, + 575, + 148, + 586 + ], + "score": 0.84, + "content": "8 4 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "). As for shallow textual encoder (e.g., Word2vec), simple text information from Kinetics-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 402, + 597 + ], + "score": 1.0, + "content": "title-tree dataset gives better performance than abundant text information", + "type": "text" + }, + { + "bbox": [ + 402, + 586, + 430, + 597 + ], + "score": 0.82, + "content": "( 8 3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 586, + 446, + 597 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 446, + 586, + 474, + 597 + ], + "score": 0.84, + "content": "8 2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "). From", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "above observation, it can be concluded that Word2vec is more good at concise and accurate text", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "while DistilBERT can handle more complex and noisy sentences which is close to realistic setting.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 632 + ], + "score": 1.0, + "content": "Also, it is affordable to utilize strong language models due to our curriculum learning strategy and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 235, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 235, + 642 + ], + "score": 1.0, + "content": "lightweight DistilBERT model.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 656, + 327, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 328, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 328, + 668 + ], + "score": 1.0, + "content": "4.4 EVALUATION ON REPRESENTATION LEARNING", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "To evaluate our learned representation, we report the classification performance on validation set", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of Kinetics via training shallow classifiers on frozen features as shown in Table 4.3. We perform", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "kNN classifiers and linear classifiers (LC) on the embedding features or visual features from global", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "average pooling after res5. In this shallow learning setting, we also compare with ImageNet pre-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "training representation (ResNet50) by using the same classifier. First, the representation learnt from", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 241, + 93, + 354, + 137 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 241, + 93, + 354, + 137 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 241, + 93, + 354, + 137 + ], + "spans": [ + { + "bbox": [ + 241, + 93, + 354, + 137 + ], + "score": 0.946, + "html": "
TrainingstrategyAccuracy(%)
Random init.50.0
Direct fine-tuning81.3
Curr. learning182.2
Curr.learning284.2
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Textual encoderDataAccuracy(%)
Random Init.-50.0
Word2vecTree83.1
DistilBERTTree82.1
Word2vecClean82.5
DistilBERTClean84.2
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Objective functionAccuracy(%)
Random init.50.0
Ranking loss79.9
Self-instance Dis.51.1
Cross-pair Dis.82.2
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BackbonePre-trained Sup.Layer (Dim)KNNLC
3D-ConvNet (Kay et al.,2017)Kinetics-400Label--56.1
3D ResNet34 (Hara et al., 2018) 3D ResNet50 (ours)Kinetics-400 Label Kinetics-400 Label--60.1 73.2
ResNet50ImageNet Label-- 42.856.1
3DResNet34res5 (2048)
Instagram-300k Captionemb (256)34.537.3
3D ResNet34Instagram-300k Captionres5 (512)36.144.6
3D ResNet50Instagram-300k Captionemb (256)51.151.7
3DResNet50Instagram-300k Captionres5 (2048)51.155.4
3DResNet34Kinetics-210k Titleemb (256)49.950.8
3D ResNet34Kinetics-210k Titleres5 (512)50.153.3
3DResNet50Kinetics-210k Titleemb (256)58.059.6
3DResNet50Kinetics-210k Titleres5 (2048)58.263.8
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Top-1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 313, + 368, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 368, + 326 + ], + "score": 1.0, + "content": "classification accuracy is reported on Kinetics-400 validation set.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + } + ], + "index": 12.25 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 344 + ], + "score": 1.0, + "content": "Objective function. We compare three objective functions for cross-modal pair discrimination de-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "scribed in Section 3.1. We pre-train models by utilizing DistilBERT as textual encoder without fine-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "tuning and the experimental results are reported in Table 1a. Self-instance discrimination almost", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "has no contribution to learn effective representation as there is no cross-modal correlation modeling.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "Cross-pair discrimination gives a better performance than ranking loss as cross-pair discrimination", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "can construct negative video-text pairs from entire dataset while ranking loss is only optimized by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "negative pairs from current batch. More theoretical analysis can be found in Section. A.1 of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 409, + 151, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 151, + 423 + ], + "score": 1.0, + "content": "Appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 334, + 505, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "Curriculum learning. We design different training strategies from noisy video-text datasets. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "first strategy is to fine-tune the pre-trained textual encoder directly at the beginning. Then we com-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "pare with stage I and stage II of curriculum learning proposed in Section 3.2. All these strategies", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 457, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 506, + 473 + ], + "score": 1.0, + "content": "are pre-trained on Kinetics-title-clean. The numerical results are summarized in Table 1b. Fixing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 506, + 484 + ], + "score": 1.0, + "content": "the pre-trained language model gives better performance than direct fine-tuning at the beginning", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 481, + 142, + 492 + ], + "score": 0.9, + "content": "( + 0 . 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ". We ascribe this to the fact that the random noise produced by video model destroy the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "well pre-trained textual encoder at the beginning. Also, fine-tuning the language model after the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 502, + 374, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 348, + 515 + ], + "score": 1.0, + "content": "video model is well initialized further boost the accuracy by", + "type": "text" + }, + { + "bbox": [ + 348, + 503, + 370, + 514 + ], + "score": 0.85, + "content": "2 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 502, + 374, + 515 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 426, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Different textual information. In this experiment, we choose video-text pairs from Kinetics-title-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 504, + 542 + ], + "score": 1.0, + "content": "tree, Kinetics-title-clean datasets and utilize Word2vec and DistilBERT as a textual extractor. The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "experimental results are reported in Table 1c. For textual encoder, abundant and video-specific text", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "information benefits to train our CPD model with stronger language model such as DistilBERT", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 564, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 475, + 576 + ], + "score": 1.0, + "content": "according to the performance difference between Kinetics-title-tree and Kinetics-title-clean", + "type": "text" + }, + { + "bbox": [ + 475, + 564, + 504, + 575 + ], + "score": 0.83, + "content": "( 8 2 . 1 \\%", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 120, + 587 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 121, + 575, + 148, + 586 + ], + "score": 0.84, + "content": "8 4 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "). As for shallow textual encoder (e.g., Word2vec), simple text information from Kinetics-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 402, + 597 + ], + "score": 1.0, + "content": "title-tree dataset gives better performance than abundant text information", + "type": "text" + }, + { + "bbox": [ + 402, + 586, + 430, + 597 + ], + "score": 0.82, + "content": "( 8 3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 586, + 446, + 597 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 446, + 586, + 474, + 597 + ], + "score": 0.84, + "content": "8 2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "). From", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "above observation, it can be concluded that Word2vec is more good at concise and accurate text", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "while DistilBERT can handle more complex and noisy sentences which is close to realistic setting.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 632 + ], + "score": 1.0, + "content": "Also, it is affordable to utilize strong language models due to our curriculum learning strategy and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 235, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 235, + 642 + ], + "score": 1.0, + "content": "lightweight DistilBERT model.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 519, + 506, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 656, + 327, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 328, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 328, + 668 + ], + "score": 1.0, + "content": "4.4 EVALUATION ON REPRESENTATION LEARNING", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "To evaluate our learned representation, we report the classification performance on validation set", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of Kinetics via training shallow classifiers on frozen features as shown in Table 4.3. We perform", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "kNN classifiers and linear classifiers (LC) on the embedding features or visual features from global", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "average pooling after res5. In this shallow learning setting, we also compare with ImageNet pre-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "training representation (ResNet50) by using the same classifier. First, the representation learnt from", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 677, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 79, + 498, + 246 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 79, + 498, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 79, + 498, + 246 + ], + "spans": [ + { + "bbox": [ + 109, + 79, + 498, + 246 + ], + "score": 0.984, + "html": "
MethodSupervisionBackbonePre-trained DatasetUCF101HMDB51
RandomInit. (Hara etal.,2018)3DResNet1842.417.1
Kinetics Pre-trained (Hara etal., 2018)Action label3DResNet50Kinetics89.361.0
Supervised SOTA (Xie et al.,2018)Action labelS3DKinetics96.875.9
Shuffle& Learn (Misra et al.,2016)Order verificationCaffeNetUCF101/HMDB5150.218.1
OPN (Lee et al., 2017)Sequence orderVGGNetUCF101/HMDB5159.823.8
CMC (Tian et al., 2019)Optical flowCaffeNetUCF10155.3-
O3N (Fernando et al.,2017)Odd-one-outAlexNetUCF10160.332.5
MASN (Wang et al.,2019a)MotionC3DKinetics-40061.233.4
COP (Xu et al.,2019b)Clip order3D ResNet10UCF10164.929.5
DPC (Han et al.,2019)Prediction3DResNet34Kinetics-40075.735.7
CBT(Sun et al.,2019a)Audio(Text)/ContextS3DKinetics-60079.544.6
AVTS (Korbar et al.,2018)AudioI3DKinetics-60083.753.0
AVTS (Korbar et al., 2018)AudioMC3Audioset-1.8M89.061.6
XDC (Alwassel et al.,2019)AudioR(2+1)DKinetics-40084.247.1
XDC (Alwassel et al.,2019)AudioR(2+1)DIG-65M91.563.1
MIL-NCE (Miech et al., 2020)Audio(Text)S3DHT-100M91.361.0
TWS (Stroud et al.,2020)Text (Title,Des,Tag etc.)S3D-GWVT-70M90.365.3
CPD (Ours)Caption3DResNet50Instagram300k89.963.8
CPD (Ours)Title3DResNet50Kinetics210k90.563.6
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The reason could be ascribed to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "video distribution gap between UGC (Instagram) and PGC (Youtube), and also much noisier textual", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "information in Instagram-300k. Second, we compare with ImageNet pretrained features, and our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 322, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 504, + 334 + ], + "score": 1.0, + "content": "CPD representation is better under the same backbone. 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MethodSupervisionBackbonePre-trained DatasetUCF101HMDB51
RandomInit. (Hara etal.,2018)3DResNet1842.417.1
Kinetics Pre-trained (Hara etal., 2018)Action label3DResNet50Kinetics89.361.0
Supervised SOTA (Xie et al.,2018)Action labelS3DKinetics96.875.9
Shuffle& Learn (Misra et al.,2016)Order verificationCaffeNetUCF101/HMDB5150.218.1
OPN (Lee et al., 2017)Sequence orderVGGNetUCF101/HMDB5159.823.8
CMC (Tian et al., 2019)Optical flowCaffeNetUCF10155.3-
O3N (Fernando et al.,2017)Odd-one-outAlexNetUCF10160.332.5
MASN (Wang et al.,2019a)MotionC3DKinetics-40061.233.4
COP (Xu et al.,2019b)Clip order3D ResNet10UCF10164.929.5
DPC (Han et al.,2019)Prediction3DResNet34Kinetics-40075.735.7
CBT(Sun et al.,2019a)Audio(Text)/ContextS3DKinetics-60079.544.6
AVTS (Korbar et al.,2018)AudioI3DKinetics-60083.753.0
AVTS (Korbar et al., 2018)AudioMC3Audioset-1.8M89.061.6
XDC (Alwassel et al.,2019)AudioR(2+1)DKinetics-40084.247.1
XDC (Alwassel et al.,2019)AudioR(2+1)DIG-65M91.563.1
MIL-NCE (Miech et al., 2020)Audio(Text)S3DHT-100M91.361.0
TWS (Stroud et al.,2020)Text (Title,Des,Tag etc.)S3D-GWVT-70M90.365.3
CPD (Ours)Caption3DResNet50Instagram300k89.963.8
CPD (Ours)Title3DResNet50Kinetics210k90.563.6
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We compare our CPD model with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 265, + 326, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 326, + 277 + ], + "score": 1.0, + "content": "other methods trained on different type of supervision.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 330, + 301 + ], + "score": 1.0, + "content": "Kinetics-210k generally outperforms that of Instagram-", + "type": "text" + }, + { + "bbox": [ + 331, + 289, + 352, + 300 + ], + "score": 0.34, + "content": "3 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 289, + 505, + 301 + ], + "score": 1.0, + "content": ". 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We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "compare our CPD model pre-trained on Instagram-300k and Kinetics-210k with a randomly ini-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "tialized network, self-supervised methods solely based on visual information, including Shuffle &", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "Learn (Misra et al., 2016), CMC (Tian et al., 2019), MASN (Wang et al., 2019a), COP (Xu et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "score": 1.0, + "content": "2019b), DPC (Han et al., 2019) and so on, and representation learning methods based on multi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "modal information (e.g., audio, text), including CBT (Sun et al., 2019a), AVTS (Korbar et al., 2018),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 476, + 479, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 479, + 488 + ], + "score": 1.0, + "content": "XDC (Alwassel et al., 2019), MIL-NCE (Stroud et al., 2020), and TWS (Stroud et al., 2020).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 398, + 506, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "As shown in Table 3, our CPD models generally outperform those self-supervised learning ap-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 280, + 516 + ], + "score": 1.0, + "content": "proaches of only using visual information", + "type": "text" + }, + { + "bbox": [ + 280, + 504, + 314, + 515 + ], + "score": 0.88, + "content": "( \\geq 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 504, + 382, + 516 + ], + "score": 1.0, + "content": "on UCF101 and", + "type": "text" + }, + { + "bbox": [ + 383, + 504, + 415, + 515 + ], + "score": 0.91, + "content": "\\geq 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "on HMDB51), which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "indicates that cross-modal information is useful cue for visual representation learning. 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The code of CPD will be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 145, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 145, + 206 + ], + "score": 1.0, + "content": "released.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 219, + 318, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 319, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 319, + 231 + ], + "score": 1.0, + "content": "A.1 ANALYSIS ON DIFFERENT LOSS FUNCTIONS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 240, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 504, + 251 + ], + "score": 1.0, + "content": "More insight about why our loss is better than ranking loss could be found from gradient back-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 248, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 176, + 264 + ], + "score": 1.0, + "content": "propagation. 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MethodThe Amount ofLabeled Data
1%10%20%
Fromscratch0.310.733.3
ImageNet Inflation12.836.845.7
Ours (Instagram-300k)18.741.347.4
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The purpose of NCE is to transform the multi-class classification problem", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "score": 1.0, + "content": "into a set of binary classification problems by comparing data distribution against noise distribution.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 137, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 119, + 155 + ], + "score": 1.0, + "content": "So", + "type": "text" + }, + { + "bbox": [ + 119, + 140, + 132, + 151 + ], + "score": 0.68, + "content": "p _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 137, + 401, + 155 + ], + "score": 1.0, + "content": "is noise distribution and we formalize it as a uniform distribution:", + "type": "text" + }, + { + "bbox": [ + 402, + 138, + 437, + 152 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { p } _ { n } = \\frac { 1 } { N } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 137, + 469, + 155 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 469, + 139, + 479, + 149 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 137, + 506, + 155 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 218, + 163 + ], + "score": 1.0, + "content": "number of video-text pairs.", + "type": "text" + }, + { + "bbox": [ + 219, + 150, + 250, + 162 + ], + "score": 0.93, + "content": "h ( i _ { t } , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "is the posterior probability of feature from the data distribution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 504, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 276, + 173 + ], + "score": 1.0, + "content": "which means video and text are matched.", + "type": "text" + }, + { + "bbox": [ + 277, + 163, + 287, + 171 + ], + "score": 0.76, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 160, + 504, + 173 + ], + "score": 1.0, + "content": "is the number of negative pairs and we set it as 4096.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 198, + 185 + ], + "score": 1.0, + "content": "For each video feature", + "type": "text" + }, + { + "bbox": [ + 198, + 172, + 208, + 182 + ], + "score": 0.83, + "content": "\\mathbf { f } ^ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 171, + 333, + 185 + ], + "score": 1.0, + "content": ", we take its related text feature", + "type": "text" + }, + { + "bbox": [ + 334, + 172, + 343, + 184 + ], + "score": 0.89, + "content": "\\mathbf { f } _ { i } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "and sample 4096 unrelated text features", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 181, + 507, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 116, + 196 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { j } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 181, + 507, + 197 + ], + "score": 1.0, + "content": "which are all from memory bank. The FPS of training videos are 30. The code of CPD will be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 145, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 145, + 206 + ], + "score": 1.0, + "content": "released.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 106, + 507, + 206 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 219, + 318, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 319, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 319, + 231 + ], + "score": 1.0, + "content": "A.1 ANALYSIS ON DIFFERENT LOSS FUNCTIONS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 240, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 504, + 251 + ], + "score": 1.0, + "content": "More insight about why our loss is better than ranking loss could be found from gradient back-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 248, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 176, + 264 + ], + "score": 1.0, + "content": "propagation. 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MethodThe Amount ofLabeled Data
1%10%20%
Fromscratch0.310.733.3
ImageNet Inflation12.836.845.7
Ours (Instagram-300k)18.741.347.4
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In", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 335, + 723 + ], + "score": 1.0, + "content": "addition, top-1 accuracy of 20 random classes reaches to", + "type": "text" + }, + { + "bbox": [ + 335, + 710, + 362, + 720 + ], + "score": 0.87, + "content": "7 4 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 708, + 505, + 723 + ], + "score": 1.0, + "content": ", which shows the strong capability", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 239, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 239, + 733 + ], + "score": 1.0, + "content": "of our visual-textual embedding.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 622, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 122, + 80, + 486, + 194 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 122, + 80, + 486, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 80, + 486, + 194 + ], + "spans": [ + { + "bbox": [ + 122, + 80, + 486, + 194 + ], + "score": 0.983, + "html": "
MethodsUCF-101Kinetics-400
TrainTestSplitAcc.TrainTestSplitAcc.
Mettes (Mettes& Snoek,2017)1101332.8--11
Ours(3D ResNet34)=101340.6=400138.2
Ours(3D ResNet50)-101339.9-400143.7
Mettes(Mettes& Snoek,2017)-501040.4-111
Ours(3D ResNet34)501047.2=1001055.3
Ours(3D ResNet50)501044.8=1001057.4
Mettes (Mettes& Snoek,2017)201051.21--=
Ours(3D ResNet34)201054.4=201073.1
Ours(3D ResNet50)201058.1=201074.4
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DatasetsAt Least One(%)A11(%)Rel(%)
Kinetics-title-tree90.544.346.3
Kinetics-title-clean91.638.426.0
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MethodsUCF-101Kinetics-400
TrainTestSplitAcc.TrainTestSplitAcc.
Mettes (Mettes& Snoek,2017)1101332.8--11
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Ours(3D ResNet50)501044.8=1001057.4
Mettes (Mettes& Snoek,2017)201051.21--=
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Ours(3D ResNet50)201058.1=201074.4
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DatasetsAt Least One(%)A11(%)Rel(%)
Kinetics-title-tree90.544.346.3
Kinetics-title-clean91.638.426.0
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BackbonePre-trained Sup.Layer (Dim)KNNLC
3D-ConvNet (Kay et al.,2017)Kinetics-400Label--56.1
3D ResNet34 (Hara et al., 2018) 3D ResNet50 (ours)Kinetics-400 Label Kinetics-400 Label--60.1 73.2
ResNet50ImageNet Label-- 42.856.1
3DResNet34res5 (2048)
Instagram-300k Captionemb (256)34.537.3
3D ResNet34Instagram-300k Captionres5 (512)36.144.6
3D ResNet50Instagram-300k Captionemb (256)51.151.7
3DResNet50Instagram-300k Captionres5 (2048)51.155.4
3DResNet34Kinetics-210k Titleemb (256)49.950.8
3D ResNet34Kinetics-210k Titleres5 (512)50.153.3
3DResNet50Kinetics-210k Titleemb (256)58.059.6
3DResNet50Kinetics-210k Titleres5 (2048)58.263.8
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Objective functionAccuracy(%)
Random init.50.0
Ranking loss79.9
Self-instance Dis.51.1
Cross-pair Dis.82.2
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TrainingstrategyAccuracy(%)
Random init.50.0
Direct fine-tuning81.3
Curr. learning182.2
Curr.learning284.2
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Textual encoderDataAccuracy(%)
Random Init.-50.0
Word2vecTree83.1
DistilBERTTree82.1
Word2vecClean82.5
DistilBERTClean84.2
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MethodSupervisionBackbonePre-trained DatasetUCF101HMDB51
RandomInit. (Hara etal.,2018)3DResNet1842.417.1
Kinetics Pre-trained (Hara etal., 2018)Action label3DResNet50Kinetics89.361.0
Supervised SOTA (Xie et al.,2018)Action labelS3DKinetics96.875.9
Shuffle& Learn (Misra et al.,2016)Order verificationCaffeNetUCF101/HMDB5150.218.1
OPN (Lee et al., 2017)Sequence orderVGGNetUCF101/HMDB5159.823.8
CMC (Tian et al., 2019)Optical flowCaffeNetUCF10155.3-
O3N (Fernando et al.,2017)Odd-one-outAlexNetUCF10160.332.5
MASN (Wang et al.,2019a)MotionC3DKinetics-40061.233.4
COP (Xu et al.,2019b)Clip order3D ResNet10UCF10164.929.5
DPC (Han et al.,2019)Prediction3DResNet34Kinetics-40075.735.7
CBT(Sun et al.,2019a)Audio(Text)/ContextS3DKinetics-60079.544.6
AVTS (Korbar et al.,2018)AudioI3DKinetics-60083.753.0
AVTS (Korbar et al., 2018)AudioMC3Audioset-1.8M89.061.6
XDC (Alwassel et al.,2019)AudioR(2+1)DKinetics-40084.247.1
XDC (Alwassel et al.,2019)AudioR(2+1)DIG-65M91.563.1
MIL-NCE (Miech et al., 2020)Audio(Text)S3DHT-100M91.361.0
TWS (Stroud et al.,2020)Text (Title,Des,Tag etc.)S3D-GWVT-70M90.365.3
CPD (Ours)Caption3DResNet50Instagram300k89.963.8
CPD (Ours)Title3DResNet50Kinetics210k90.563.6
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MethodsUCF-101Kinetics-400
TrainTestSplitAcc.TrainTestSplitAcc.
Mettes (Mettes& Snoek,2017)1101332.8--11
Ours(3D ResNet34)=101340.6=400138.2
Ours(3D ResNet50)-101339.9-400143.7
Mettes(Mettes& Snoek,2017)-501040.4-111
Ours(3D ResNet34)501047.2=1001055.3
Ours(3D ResNet50)501044.8=1001057.4
Mettes (Mettes& Snoek,2017)201051.21--=
Ours(3D ResNet34)201054.4=201073.1
Ours(3D ResNet50)201058.1=201074.4
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DatasetsAt Least One(%)A11(%)Rel(%)
Kinetics-title-tree90.544.346.3
Kinetics-title-clean91.638.426.0
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15, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/XQQA6-So14/XQQA6-So14.md b/parse/train/XQQA6-So14/XQQA6-So14.md new file mode 100644 index 0000000000000000000000000000000000000000..954f7605167a6437d0e98d188997df457a214ede --- /dev/null +++ b/parse/train/XQQA6-So14/XQQA6-So14.md @@ -0,0 +1,530 @@ +# NEURAL SPATIO-TEMPORAL POINT PROCESSES + +Ricky T. Q. Chen∗ University of Toronto; Vector Institute rtqichen@cs.toronto.edu + +Brandon Amos, Maximilian Nickel +Facebook AI Research +{bda,maxn}@fb.com + +# ABSTRACT + +We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, highfidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of continuous-time neural networks with two novel neural architectures, i.e., Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience. + +# 1 INTRODUCTION + +Modeling discrete events that are localized in continuous time and space is an important task in many scientific fields and applications. Spatio-temporal point processes (STPPs) are a versatile and principled framework for modeling such event data and have, consequently, found many applications in a diverse range of fields. This includes, for instance, modeling earthquakes and aftershocks (Ogata, 1988; 1998), the occurrence and propagation of wildfires (Hering et al., 2009), epidemics and infectious diseases (Meyer et al., 2012; Schoenberg et al., 2019), urban mobility (Du et al., 2016), the spread of invasive species (Balderama et al., 2012), and brain activity (Tagliazucchi et al., 2012). + +It is of great interest in all of these areas to learn high-fidelity models which can jointly capture spatial and temporal dependencies and their propagation effects. However, existing parameterizations of STPPs are strongly restricted in this regard due to computational considerations: In its general form, STPPs require solving multivariate integrals for computing likelihood values and thus have primarily been studied within the context of different approximations and model restrictions. This includes, for instance, restricting the model class to parameterizations with known closed-form solutions (e.g., exponential Hawkes processes (Ozaki, 1979)), to restrict dependencies between the spatial and temporal domain (e.g., independent and unpredictable marks (Daley & Vere-Jones, 2003)), or to discretize continuous time and space (Ogata, 1998). These restrictions and approximations—which can lead to mis-specified models and loss of information—motivated the development of neural temporal point processes such as Neural Hawkes Processes (Mei & Eisner, 2017) and Neural Jump SDEs (Jia & Benson, 2019). While these methods are more flexible, they can still require approximations such as Monte-Carlo sampling of the likelihood (Mei & Eisner, 2017; Nickel & Le, 2020) and, most importantly, only model restricted spatial distributions (Jia & Benson, 2019). + +![](images/fa57820cf1c346f628551e171ee58cac549d8c865680579795012e82f52dbaf4.jpg) +Figure 1: Color is used to denote $p ( x | t )$ , which can be evaluated for Neural STPPs. After observing an event in one mode, the model is instantaneously updated as it strongly expects an event in the next mode. After a period of no observations, the model smoothly reverts back to the marginal distribution. + +To overcome these issues, we propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and allows us to define flexible, high-fidelity models for spatio-temporal event data. We build upon ideas of Neural Jump SDEs (Jia & Benson, 2019) and Continuous-time Normalizing Flows (CNFs; Chen et al. 2018; Grathwohl et al. 2019; Mathieu & Nickel 2020) to learn parametric models of spatial (or mark1) distributions that are defined continuously in time. Normalizing flows are known to be flexible universal density estimators (e.g. Huang et al. 2018; 2020; Teshima et al. 2020; Kong & Chaudhuri 2020) while retaining computational tractability. As such, our approach allows the computation of exact likelihood values even for highly complex spatio-temporal distributions, and our models create smoothly changing spatial distributions that naturally benefits spatio-temporal modeling. Central to our approach, are two novel neural architectures based on CNFs—using either discontinuous jumps in distribution or self-attention—to condition spatial distributions on the event history. To the best of our knowledge, this is the first method that combines the flexibility of neural TPPs with the ability to learn high-fidelity models of continuous marks that can have complex dependencies on the event history. In addition to our modeling contributions, we also construct five new pre-processed data sets for benchmarking spatio-temporal event models. + +# 2 BACKGROUND + +In the following, we give a brief overview of two core frameworks which our method builds upon, i.e., spatio-temporal point processes and continuous-time normalizing flows. + +Event Modeling with Point Processes Spatio-temporal point processes are concerned with modeling sequences of random events in continuous space and time (Moller & Waagepetersen, 2003; Baddeley et al., 2007). Let $\mathcal { H } = \{ ( t _ { i } , \pmb { x } _ { i } ) \} _ { i = 1 } ^ { n }$ denote the sequence of event times $t _ { i } \in \mathbb { R }$ and their associated locations $\pmb { x } _ { i } \in \mathbb { R } ^ { d }$ , the number of events $n$ being also random. Additionally, let $\mathcal { H } _ { t } = \{ ( t _ { i } , \pmb { x } _ { i } ) ~ | ~ t _ { i } < t , t _ { i } \in \mathcal { H } \}$ denote the history of events predating time $t$ . A spatio-temporal point process is then fully characterized by its conditional intensity function + +$$ +\lambda ( t , \pmb { x } \mid \mathcal { H } _ { t } ) \triangleq \operatorname* { l i m } _ { \Delta t \downarrow 0 , \Delta \pmb { x } \downarrow 0 } \frac { \mathbb { P } \left( t _ { i } \in [ t , t + \Delta t ] , \pmb { x } _ { i } \in B ( \pmb { x } , \Delta \pmb { x } ) \mid \mathcal { H } _ { t } \right) } { | B ( \pmb { x } , \Delta \pmb { x } ) | \Delta t } . +$$ + +where $B ( { \pmb x } , \Delta { \pmb x } )$ denotes a ball centered at $\pmb { x } \in \mathbb { R } ^ { d }$ and with radius $\Delta \mathbfit { x }$ . The only condition is that $\lambda ( t , \pmb { x } \mid \mathcal { H } _ { t } ) \geq 0$ and need not be normalized. Given $i - 1$ previous events, the conditional intensity function describes therefore the instantaneous probability of the $i$ -th event occurring at $t$ and location $_ { \textbf { \em x } }$ . In the following, we will use the common star superscript shorthand $\lambda ^ { * } ( t , \pmb { x } ) = \lambda ( t , \pmb { x } \mid \mathcal { H } _ { t } )$ to denote conditional dependence on the history. The joint log-likelihood of observing $\mathcal { H }$ within a time interval of $[ 0 , T ]$ is then given by (Daley $\&$ Vere-Jones, 2003, Proposition 7.3.III) + +$$ +\log p \left( \mathcal { H } \right) = \sum _ { i = 1 } ^ { n } \log \lambda ^ { * } ( t _ { i } , \pmb { x } _ { i } ) - \int _ { 0 } ^ { T } \int _ { \mathbb { R } ^ { d } } \lambda ^ { * } ( \tau , \pmb { x } ) d \pmb { x } d \tau . +$$ + +Training general STPPs with maximum likelihood is difficult as eq. (2) requires solving a multivariate integral. This need to compute integrals has driven research to focus around the use of kernel density estimators (KDE) with exponential kernels that have known anti-derivatives (Reinhart et al., 2018). + +Continuous-time Normalizing Flows Normalizing flows (Dinh et al., 2014; 2016; Rezende & Mohamed, 2015) is a class of density models that describe flexible distributions by parameterizing an invertible transformation from a simpler base distribution, which enables exact computation of the probability of the transformed distribution, without any unknown normalization constants. + +Given a random variable $\scriptstyle { \pmb x } _ { 0 }$ with known distribution $p ( \pmb { x } _ { 0 } )$ and an invertible transformation $F ( x )$ , the transformed variable $F ( \pmb { x } _ { 0 } )$ is a random variable with a probability distribution function that satisfies + +$$ +\log p ( F ( { \pmb x } _ { 0 } ) ) = \log p ( { \pmb x } _ { 0 } ) - \log \left| \operatorname* { d e t } \frac { \partial F } { \partial { \pmb x } } ( { \pmb x } _ { 0 } ) \right| . +$$ + +There have been many advances in parameterizing $F$ with flexible neural networks that also allow for cheap evaluations of eq. (3). We focus our attention on Continuous-time Normalizing Flows (CNFs), which parameterizes this transformation with a Neural ODE (Chen et al., 2018). CNFs define an infinite set of distributions on the real line that vary smoothly across time, and will be our core component for modeling events in the spatial domain. + +Let $p ( \pmb { x } _ { 0 } )$ be the base distribution2. We then parameterize an instantaneous change in the form of an ordinary differential equation (ODE), $\begin{array} { r } { \frac { d { \pmb x } _ { t } } { d t } = f ( t , { \pmb x } _ { t } ) } \end{array}$ , where the subscript denotes dependence on $t$ . This function can be parameterized using any Lipschitz-continuous neural network. Conditioned on a sample $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ from the base distribution, let $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ be the solution of the initial value problem 3 at time $t$ , i.e. it is from a trajectory that passes through $\scriptstyle { \pmb x } _ { 0 }$ at time 0 and satisfies the ODE $d { \pmb x } _ { t } / d t = f$ . We can express the value of the solution at time $t$ as + +$$ +{ \pmb x } _ { t } = { \pmb x } _ { 0 } + \int _ { 0 } ^ { t } f ( t , { \pmb x } _ { \tau } ) d \tau . +$$ + +The distribution of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ then also continuously changes in $t$ through the following equation, + +$$ +\log p ( { \pmb x } _ { t } | t ) = \log p ( { \pmb x } _ { 0 } ) - \int _ { 0 } ^ { t } \mathrm { t r } \left( \frac { \partial f } { \partial x } ( \tau , { \pmb x } _ { \tau } ) \right) ~ d \tau . +$$ + +In practice, eq. (4) and eq. (5) are solved together from 0 to $t$ , as eq. (5) alone is not an ordinary differential equation but the combination of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ and $\log p ( { \pmb x } _ { t } )$ is. The trace of the Jacobian $\frac { \partial f } { \partial \boldsymbol { x } } ( \tau , \pmb { x } _ { \tau } )$ can be estimated using a Monte Carlo estimate of the identity (Skilling, 1989; Hutchinson, 1990), $\mathrm { t r } ( A ) = \mathbb { E } _ { v \sim \mathcal { N } ( 0 , 1 ) } [ v ^ { \top } A v ]$ . This estimator relies only on a vector-Jacobian product, which can be efficiently computed in modern automatic differentiation and deep learning frameworks. This has been used (Grathwohl et al., 2019) to scale CNFs to higher dimensions using a Monte Carlo estimate of the log likelihood objective, + +$$ +\log p ( { \pmb x } _ { t } | t ) = \log p ( { \pmb x } _ { 0 } ) - \mathbb { E } _ { \pmb { v } \sim \mathcal { N } ( 0 , 1 ) } \left[ \int _ { 0 } ^ { t } v ^ { \top } \frac { \partial f } { \partial x } ( \tau , { \pmb x } _ { \tau } ) v \ d \tau \right] , +$$ + +which, even if only one sample of $v$ is used, is still amenable to training with stochastic gradient descent. Gradients with respect to any parameters in $f$ can be computed with constant memory by solving an adjoint ODE in reverse-time as described in Chen et al. (2018). + +# 3 NEURAL SPATIO-TEMPORAL POINT PROCESSES + +We are interested in modeling high-fidelity distributions in continuous time and space that can be updated based on new event information. For this purpose, we use the Neural ODE framework to parameterize a STPP by combining ideas from Neural Jump SDEs and Continuous Normalizing Flows to create highly flexible models that still allow exact likelihood computation. + +We first (re-)introduce necessary notation. Let $\mathcal { H } = \{ ( t _ { i } , \pmb { x } _ { t _ { i } } ^ { ( i ) } ) \}$ denote a sequence of event times $t _ { i } \in [ 0 , T ]$ and locations $\pmb { x } _ { t _ { i } } ^ { ( i ) } \in \mathbb { R } ^ { d }$ i. The superscript indicates an association with the $i$ -th event, and the use of subscripting with $t _ { i }$ will be useful later in the continuous-time modeling framework. Following Daley & Vere-Jones (2003), we decompose the conditional intensity function as + +$$ +\lambda ^ { * } ( t , { \pmb x } ) = \lambda ^ { * } ( t ) p ^ { * } ( { \pmb x } \mid t ) +$$ + +where $\lambda ^ { * } ( t )$ is the ground intensity of the temporal process and where $\boldsymbol { p } ^ { * } ( \boldsymbol { x } \mid t )$ is the conditional density of a mark $_ { \textbf { \em x } }$ at $t$ given $\mathcal { H } _ { t }$ . The star superscript is used as again shorthand to denote dependence on the history. Since $\textstyle \int _ { \mathbb { R } ^ { d } } p ^ { * } ( { \pmb { x } } \mid t ) = 1$ , eq. (7) allows us now to simplify the log-likelihood function of the joint process from eq. (2), such that + +$$ +\log p ( \mathcal { H } ) = \underbrace { \sum _ { i = 1 } ^ { n } \log \lambda ^ { * } ( t _ { i } ) - \int _ { 0 } ^ { T } \lambda ^ { * } ( \tau ) ~ d \tau } _ { \mathrm { t e m p o r a l ~ l o g - l i k e l i h o o d } } + \underbrace { \sum _ { i = 1 } ^ { n } \log p ^ { * } ( \pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) } _ { \mathrm { s p a t i a l ~ l o g - l i k e l i h o o d } } +$$ + +Furthermore, based on eq. (7), we can derive separate models for the ground intensity and conditional mark density which will be jointly conditioned on a continuous-time hidden state with jumps. In the following, we will first describe how we construct a latent dynamics model, which we use to compute the ground intensity $\lambda ^ { * } ( t )$ . We will then propose three novel CNF-based approaches for modeling the conditional mark density $p ^ { * } ( { \pmb x } | t )$ . We will first describe an unconditional model, which is already a strong baseline when spatial event distributions only follow temporal patterns and there is little to no correlation between the spatial observations. We then devise two new methods of conditioning on the event history $\mathcal { H }$ : one explicitly modeling instantaneous changes in distribution, and another that uses an attention mechanism which is more amenable to parallelism. + +Latent Dynamics and Ground Intensity For the temporal variables $\{ t _ { i } \}$ , parameterize the intensity function using hidden state dynamics with jumps, similar to the work of Jia & Benson (2019). Specifically, we evolve a continuous-time hidden state $^ { h }$ and set + +$$ +\lambda ^ { * } ( t ) = g _ { \lambda } ( h _ { t } ) \qquad \mathrm { ( G r o u n d i n t e n s i t y ) } +$$ + +where $g _ { \lambda }$ is a neural network with a softplus nonlinearity applied to the output, to ensure the intensity is positive. We then capture conditional dependencies through the use of a continuously changing state $\boldsymbol { h } _ { t }$ with instantaneous updates when conditioned on an event. + +The architecture is analogous to a recurrent neural network with a continuous-time hidden state (Mei & Eisner, 2017; Che et al., 2018; Rubanova et al., 2019) modeled by a Neural ODE. This provides us with a vector representation $\boldsymbol { h } _ { t }$ at every time value $t$ that acts as both a summary of the history of events and as a predictor of future behavior. Instantaneous updates to $\boldsymbol { h } _ { t }$ allow to incorporate abrupt changes to the hidden state that are triggered by observed events. This mechanism is important for modeling point processes and allows past events to influence future dynamics in a discontinuous way (e.g., modeling immediate shocks to a system). + +We use $f _ { h }$ to model the continuous change in the form of an ODE and $g _ { h }$ to model instantaneous changes based on an observed event. + +$$ +{ \begin{array} { r l r l } & { h _ { t _ { 0 } } = h _ { 0 } } & & { ( { \mathrm { A n ~ i n i t i a l ~ h i d d e n ~ s t a t e } } ) } \\ & { { \frac { d h _ { t } } { d t } } = f _ { h } ( t , h _ { t } ) } & & { { \mathrm { b e t w e e n ~ e v e n t ~ t i m e s } } } & & { ( { \mathrm { C o n t i n u o u s ~ e v o l u t i o n } } ) } \\ & { \operatorname* { l i m } _ { \varepsilon \to 0 } h _ { t _ { i } + \varepsilon } = g _ { h } \left( t _ { i } , h _ { t _ { i } } , x _ { t _ { i } } ^ { ( i ) } \right) } & & { { \mathrm { a t ~ e v e n t ~ t i m e s ~ } } t _ { i } } & & { ( { \mathrm { I n s t a n t a n e o u s ~ u p d a t e s } } ) } \end{array} } +$$ + +The use of $\varepsilon$ is to portray that $h _ { t }$ is a cagl \` ad\` function, i.e. left-continuous with right limits, with a discontinuous jump modeled by $g _ { h }$ . + +The parameterization of continuous-time hidden states in the form of eqs. (10) to (12) has been used for time series modeling (Rubanova et al., 2019; De Brouwer et al., 2019) as well as TPPs (Jia & Benson, 2019). We parameterize $f _ { h }$ as a standard multi-layer fully connected neural network, and use the GRU update (Cho et al., 2014) to parameterize $g _ { h }$ , as was done in Rubanova et al. (2019). + +Time-varying CNF The first model we consider is a straightforward application of the CNF to time-variable observations. Assuming that the spatial distribution is independent of prior events, + +$$ +\log p ^ { * } ( \pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) = \log p ( \pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) = \log p ( \pmb { x } _ { 0 } ^ { ( i ) } ) - \int _ { 0 } ^ { t _ { i } } \mathrm { t r } \left( \frac { \partial f } { \partial x } ( \tau , \pmb { x } _ { \tau } ^ { ( i ) } ) \right) d \tau +$$ + +where $\pmb { x } _ { \tau } ^ { ( i ) }$ ) is the solution of the ODE f with initial value x(i)t , the observed event location, at $\tau = t _ { i }$ the observed event time. The spatial distribution of an event modeled by a Time-varying CNF changes with respect to the time it occurs. Some spatio-temporal data sets exhibit mostly temporal patterns and little to no dependence on previous events in the spatial domain, which would make a time-varying CNF a good fit. Nevertheless, this model lacks the ability to capture spatial propagation effects, as it does not condition on previous event observations. + +A major benefit of this model is the ability to evaluate the joint log-likelihood fully in parallel across events, since there are no dependencies between events. Most modern ODE solvers that we are aware of only allow a scalar terminal time. Thus, to solve all $n$ integrals in eq. (13) with a single call to an ODE solver, we can simply reparameterize all integrals with a consistent dummy variable and track the terminal time in the state (see Appendix $\mathrm { F }$ for detailed explanation). Intuitively, the idea is that we can reparameterize ODEs that are on $t \in [ 0 , t _ { i } ]$ into an ODE on $s \in [ 0 , 1 ]$ using the change of variables $\bar { s } = t / { { t } _ { i } }$ (or $t = s t _ { i }$ ) and scaling the output of $f$ by $t _ { i }$ . The joint ODE is then + +$$ +\frac { d } { d s } \underbrace { \left[ \begin{array} { c } { x _ { s } ^ { ( 1 ) } } \\ { \vdots } \\ { x _ { s } ^ { ( n ) } } \end{array} \right] } _ { A _ { s } } = \underbrace { \left[ t _ { 1 } f ( s t _ { 1 } , x _ { s } ^ { ( 1 ) } ) \right] } _ { f ( s t _ { n } , z _ { s } ) } \quad \mathrm { ~ w h i c h ~ g i v e s ~ } \quad \underbrace { \left[ \begin{array} { c } { x _ { 0 } ^ { ( 1 ) } } \\ { \vdots } \\ { x _ { 0 } ^ { ( n ) } } \end{array} \right] } _ { A _ { 0 } } + \int _ { 0 } ^ { 1 } f ( s , A _ { s } ) d s = \underbrace { \left[ \begin{array} { c } { x _ { t _ { 1 } } ^ { ( 1 ) } } \\ { \vdots } \\ { x _ { t _ { n } } ^ { ( n ) } } \end{array} \right] } _ { A _ { 1 } } . +$$ + +Thus the full trajectories between 0 to $t _ { i }$ for all events can be computed in parallel using this augmented ODE by simply integrating once from $s = 0$ to $s = 1$ . + +$\mathbf { J u m p C N F }$ For the second model, we condition the dynamics defining the continuous normalizing flow on the hidden state $^ { h }$ , allowing the normalizing flow to update its distribution based on changes in $\mathcal { H }$ . For this purpose, we define continuous-time spatial distributions by making again use of two components: (i) a continuous-time normalizing flow that evolves the distribution continuously, and (ii) a standard (discrete-time) flow model that changes the distribution instantaneously after conditioning on new events. As normalizing flows parameterize distributions through transformations of the samples, these continuous- and discrete-time transformations are composable in a straightforward manner and are end-to-end differentiable. + +The generative process of a single event in a Jump CNF is given by: + +$$ +\begin{array} { r l r l } { x _ { 0 } \sim p ( x _ { 0 } ) } & { } & & { \mathrm { ( A n i n i t i a l ~ d i s t r i b u t i o n ) } } \\ { \displaystyle \frac { d x _ { t } } { d t } = f _ { x } ( t , x _ { t } , h _ { t } ) } & { \mathrm { ~ b e t w e e n ~ e v e n t ~ t i m e s ~ } } & & { \mathrm { ( C o n t i n u o u s ~ e v o l u t i o n ) } } \\ { \displaystyle \operatorname* { l i m } _ { \varepsilon 0 } x _ { t _ { i } + \varepsilon } = g _ { x } ( t _ { i } , x _ { t _ { i } } , h _ { t _ { i } } ) } & { \quad \mathrm { a t ~ e v e n t ~ t i m e s ~ } t _ { i } } & & { \mathrm { ( I n s t a n t a n e o u s ~ u p d a t e s ) } } \end{array} +$$ + +The initial distribution can be parameterized by a normalizing flow. In practice, we set a base distribution at a negative time value and model $p ( \pmb { x } _ { 0 } )$ using the same CNF parameterized by $f _ { x }$ . The instantaneous updates (or jumps) describe conditional updates in distribution after each new event has been observed. This conditioning on $h _ { t _ { i } }$ is required for the continuous and instantaneous updates to depend on the history of observations. Otherwise, a Jump CNF would only be able to model the marginal distribution and behave similarly to a time-varying CNF. We solve for $h _ { t }$ alongside $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ . + +The final probability of an event $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ at some $t > t _ { n }$ after observing $n$ events is given by the sum of changes according to the continuous- and discrete-time normalizing flows. + +$$ +\begin{array} { r l } & { \log p ^ { * } ( { \boldsymbol x } _ { t } | t ) = \log p ( { \boldsymbol x } _ { 0 } ) } \\ & { \quad \quad + \underbrace { \displaystyle \sum _ { t _ { i } \in \mathcal { H } _ { t } } \left( - \int _ { t _ { i - 1 } } ^ { t _ { i } } { \mathrm { t r } \left( \frac { \partial f ( \tau , { \boldsymbol x } _ { \tau } , h _ { \tau } ) } { \partial { \boldsymbol x } } \right) d \tau } - \log \left| \operatorname* { d e t } \frac { \partial g _ { { \boldsymbol x } } ( t _ { i } , { \boldsymbol x } _ { t _ { i } } , h _ { t _ { i } } ) } { \partial { \boldsymbol x } } \right| \right) } _ { \mathrm { C h a p e ~ i n g ~ u p ~ t o ~ l a s t ~ v e n t } } } \\ & { \quad \quad + \underbrace { \displaystyle \int _ { t _ { n } } ^ { t } - { \mathrm { t r } \left( \frac { \partial f ( \tau , { \boldsymbol x } _ { \tau } , h _ { \tau } ) } { \partial { \boldsymbol x } } \right) d \tau } } _ { \mathrm { C h a p e ~ i n g e n ~ l a s t ~ v e n t ~ t o ~ } } } \end{array} +$$ + +As the instantaneous updates must be applied sequentially in a Jump CNF, we can only compute the integrals in eq. (18) one at a time. As such, the number of initial value problems scales linearly with the number of events in the history because the ODE solver must be restarted between each instantaneous update to account for the discontinuous change to state. This incurs a substantial cost when the number of events is large. + +Attentive CNF To design a spatial model with conditional dependencies that alleviates the computational issues of Jump CNFs and can be computed in parallel, we make use of efficient attention mechanisms based on the Transformer architecture (Vaswani et al., 2017). Denoting only the spatial variables for simplicity, each conditional distribution $\log p ( \pmb { x } _ { t _ { i } } \ | \ \mathcal { H } _ { t _ { i } } )$ can be modeled by a CNF that depends on the sample path of prior events. Specifically, we take the dummy-variable reparameterization of eq. (14) and modify it so that the $i$ -th event depends on all previous events using a Transformer architecture for $f$ , + +![](images/63a02fc2549e8b7f69997b935dc6ed8a77774c252453939a052db56ee3e48cab.jpg) +Figure 2: Visualization of the sampling paths of Neural STPP models for a 1-D spatio-temporal data set where $\{ t _ { i } \} _ { i = 1 } ^ { 4 }$ are event times. The Jump CNF uses instantaneous jumps to update its distribution based on newly observed events while the Attentive CNF depends continuously on the sampling paths of prior events. We additionally visualize a second sequence for the Attentive CNF where the random base samples $\{ x _ { 0 } ^ { ( i ) } \} _ { i = 2 } ^ { 4 }$ are the same as in sequence 1. Even so, the sampling paths are different because the first event is different, effectively leading to different conditional spatial distributions. See Figure 9 for visualizations of the learned density. + +$$ +\frac { d } { d s } \left[ \begin{array} { c } { x _ { s } ^ { ( 1 ) } } \\ { x _ { s } ^ { ( 2 ) } } \\ { \vdots } \\ { x _ { s } ^ { ( n ) } } \end{array} \right] = \left[ \begin{array} { c } { t _ { 1 } f \left( s t _ { 1 } , \{ x _ { s } ^ { ( i ) } \} _ { i = 1 } ^ { 1 } , \{ h _ { t _ { i } } \} _ { i = 1 } ^ { 1 } \right) } \\ { t _ { 2 } f \left( s t _ { 2 } , \{ x _ { s } ^ { ( i ) } \} _ { i = 1 } ^ { 2 } , \{ h _ { t _ { i } } \} _ { i = 1 } ^ { 2 } \right) } \\ { \vdots } \\ { t _ { n } f \left( s t _ { n } , \{ x _ { s } ^ { ( i ) } \} _ { i = 1 } ^ { n } , \{ h _ { t _ { i } } \} _ { i = 1 } ^ { n } \right) } \end{array} \right] : = f _ { \mathrm { A t t } } \mathrm { n } . +$$ + +With this formulation, the trajectory of $\pmb { x } _ { \tau } ^ { ( i ) }$ depends continuously on the trajectory of $\pmb { x } _ { \tau } ^ { ( j ) }$ for all $j < i$ and the hidden states $^ { h }$ prior to the $i$ -th event. Similar to eq. (14), an Attention CNF can now solve for the trajectories of all events in parallel while simultaneously depending non-trivially on $\mathcal { H }$ . + +To parameterize $f _ { \mathrm { A t t } \mathrm { n } }$ , we use an embedding layer followed by two multihead attention (MHA) blocks and an output layer to map back into the input space. We use the Lipschitz-continuous multihead attention from Kim et al. (2020) as they recently showed that the dot product multihead attention (Vaswani et al., 2017) is not Lipschitz-continuous and thus may be ill-suited for parameterizing ODEs. + +Low-variance Log-likelihood Estimation The variance of the Hutchinson stochastic trace estimator in eq. (6) grows with the squared Frobenius norm of the Jacobian, $\sum _ { i j } \left[ \partial f / \partial x \right] _ { i j } ^ { 2 }$ (Hutchinson, 1990). For attentive CNFs, we can remove some of the non-diagonal elements of the Jacobian and achieve a lower variance estimator. The attention mechanism creates a blocktriangular Jacobian, where each block corresponds to one event, but the elements outside of the blockdiagonal are solely due to the multihead attention. By detaching the gradient connections between different events in the MHA blocks, we can create a surrogate + +![](images/f5dbfbc0e7accab2cb0542d09d2deb53f0e2cb9c2f4fa3bd16b33ead1f927e20.jpg) +Figure 3: Lower variance estimates of the loglikelihood allows training better Attentive CNFs. + +Jacobian matrix that do not contain cross-event partial derivatives. This effectively allows us to apply Hutchinson’s estimator on a matrix that has the same diagonal elements as the Jacobian $\partial f / \partial x$ —and thus has the same expected value—but has zeros outside of the block-diagonal, leading to a lower variance trace estimator. The procedure consists of selectively removing partial derivatives and is straightforward but notationally cumbersome; the interested reader can find the details in Appendix E. + +This is similar in spirit to Chen & Duvenaud (2019) but instead of constructing a neural network that specifically allows cheap removal of partial derivatives, we make use of the fact that multihead attention already allows cheap removal of (cross-event) partial derivatives. + +An ablation experiment is shown in Figure 3 for training on the PINWHEEL data set, where the lower variance estimates (and gradients) ultimate led to faster convergence and better converged models. + +# 4 RELATED WORK + +Neural Temporal Point Processes Modeling real-world data using restricted models such as Exponential Hawkes Processes (Ozaki, 1979) may lead to poor results due to model mis-specification. While this has led to many works on improving the Hawkes process (e.g. Linderman & Adams 2014; Li & Zha 2014; Zhao et al. 2015; Farajtabar et al. 2017; Li & Ke 2020; Nickel & Le 2020), recent works have begun to explore neural network parameterizations of TPPs. A common approach is to use recurrent neural networks to accumulate the event history in a latent state from which the intensity value can then be derived. Models of this form include, for instance, Recurrent Marked Temporal Point Processes (RMTPPs; Du et al. 2016) and Neural Hawkes Processes (NHPs; Mei & Eisner 2017). In contrast to our approach, these methods can not compute the exact likelihood of the model and have to resort to Monte-Carlo sampling for its approximation. However, this approach is especially problematic for commonly occurring clustered and bursty event sequences as it either requires a very high sampling rate or ignores important temporal dependencies (Nickel & Le, 2020). To overcome this issue, Jia & Benson (2019) proposed Neural Jump SDEs which extend the Neural ODE framework and allow to compute the exact likelihood for neural TPPs, up to numerical errors. This method is closely related to our approach and we build on its ideas to compute the ground intensity of the STPP. However, current Neural Jump SDEs —as well as NHPs and RMTPPs—are not well-suited for modeling complex continuous mark distributions as they are restricted to methods such as Gaussian mixture models in the spatial domain. Finally, Shchur et al. (2019) and Mehrasa et al. (2019) considered combining TPPs with flexible likelihood-based models, however for different purposes as in our case, i.e., for intensity-free learning of only temporal point processes. + +Continuous Normalizing Flows The ability to describe an infinite number of distributions with a Continuous Normalizing Flow has been used by a few recent works. Some works in computer graphics have used the interpolation effect of CNFs to model transformations of point clouds (Yang et al., 2019; Rempe et al., 2020; Li et al., 2020). CNFs have also been used in sequential latent variable models (Deng et al., 2020; Rempe et al., 2020). However, such works do not align the “time” axis of the CNF with the temporal axis of observations, and do not train on observations at more than one value of “time” in the CNF. In contrast, we align the time axis of the CNF with the time of the observations, directly using its ability to model distributions on a real-valued axis. A closely related application of CNFs to spatio-temporal data was done by Tong et al. (2020), who modeled the distribution of cells in a developing human embryo system at five fixed time values. In contrast to this, we extend to applications where observations are made at arbitrary time values, jointly modeling space and time within the spatio-temporal point process framework. Furthermore, Mathieu & Nickel (2020); Lou et al. (2020) recently proposed extensions of CNFs to Riemannian manifolds. For our proposed approach, this is especially interesting in the context of earth and climate science, as it allows us to model STPPs on the sphere simply by replacing the CNF with its Riemannian equivalent. + +# 5 EXPERIMENTS + +Data Sets Many collected data can be represented within the framework of spatio-temporal events. We pre-process data from open sources and make them suitable for spatio-temporal event modeling. Each sequence in these data sets can contain up to thousands of variables, all the while having a large variance in sequence lengths. Varying across a wide range of domains, the data sets we consider are: earthquakes, pandemic spread, consumer demand for a bike sharing app, and high-amplitude brain signals from fMRI scans. We briefly describe these data sets here; further details, pre-processing steps, and data set diagnostics can be found in Appendix C. Code for preprocessing and training are open sourced at https://github.com/facebookresearch/neural_stpp. + +PINWHEEL This is a synthetic data set with multimodal and non-Gaussian spatial distributions designed to test the ability to capture drastic changes due to event history (see fig. 5). The data set consists of 10 clusters which form a pinwheel structure. Events are sampled from a multivariate Hawkes process such that events from one cluster will increase the probability of observing events in the next cluster in a clock-wise rotation. Number of events per sequences ranges between 4 to 108. + +EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological Survey (2020). Number of events per sequences ranges between 18 to 543. + +![](images/64968869c1f670bd0d6d1f78cc771ab88dd0fa3dbd68dc7e1efb980f726fcdbe.jpg) +Figure 5: Evolution of spatial densities on PINWHEEL data. top: Attentive CNF. bottom: Jump CNF. (a) Before observing any events at $\scriptstyle t = 0$ , the distribution is even across all clusters. (b-f) Each event increases the probability of observing a future event from the subsequent cluster in clock-wise ordering. (g-h) After a period of no new events, the distribution smoothly returns back to the initial distribution (a). + +![](images/c2f9aeea16989017a062ac3824365a59e03ce754df2dafff536fa59e95729512.jpg) +Figure 7: Snapshots of conditional spatial distributions modeled by the Jump CNF (top) and a conditional kernel density estimator (KDE; bottom). (a) Distribution before any events at $\scriptstyle t = 0$ . (b-d) The Jump CNF’s distributions concentrate around tectonic plate boundaries where earthquakes and aftershocks gather, whereas the KDE must use a large variance in order to capture propagation of aftershocks in multiple directions. + +COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19 cases in New Jersey state. The data is aggregated at the county level, which we dequantize uniformly across the county. Number of events per sequences ranges between 3 to 323. + +BOLD5000 This consists of fMRI scans as participants are given visual stimuli (Chang et al., 2019). We convert brain responses into spatio-temporal events following the z-score thresholding approach in Tagliazucchi et al. (2012; 2016). Number of events per sequences ranges between 6 to 1741. + +In addition to these datasets, we also report in Appendix A results for CITIBIKE, a data set consisting of rental events from a bike sharing service in New York City. + +Baselines To evaluate the capability of our proposed models, we compare against commonly-used baselines and state-of-the-art models. In some settings, ground intensity and conditional mark density are independent of each other and we can freely combine different baselines for the temporal and spatial domains. As temporal baselines, we use a homogeneous Poisson process, a self-correction process, a Hawkes process, and the Neural Hawkes Process, which were trained using their officially released code. As spatial baselines, we use a conditional kernel density estimator (KDE) with learned parameters, where $p ( { \pmb x } | t )$ is essentially modeled as a history-dependent Gaussian mixture model (see Appendix B), as well as the Time-varying CNF. In addition, we also compare to our implementation of Neural Jump SDEs (Jia & Benson, 2019) where the spatial distribution is a Gaussian mixture model. We use the same architecture as our GRU-based continuous-time hidden states for fair comparison, as we found the simpler parameterization in Jia & Benson (2019) to be numerically unstable for large number of events. Range of hyperparameter values are outlined in Appendix D. + +
PinwheelEarthquakes JPCOVID-19 NJBOLD5000
ModelTemporalSpatialTemporalSpatialTemporalSpatialTemporalSpatial
Poisson Process-0.784±0.0011-0.111±0.00110.878±0.01610.862±0.0181
Self-correcting Process-2.117±0.222-7.051±0.780-10.053±1.150-6.470±0.827
Hawkes Process-0.276±0.0330.114±0.0052.092±0.0232.860±0.050
Neural Hawkes Process-0.023±0.0010.198±0.0012.229±0.0133.080±0.019
Conditional KDE-2.958±0.000-2.259±0.001-2.583±0.000-3.467±0.000
Time-varying CNF-2.185±0.003-1.459±0.016-2.002±0.0021-1.846±0.019
Neural Jump SDE (GRU)-0.006±0.042-2.077±0.0260.186±0.005 -1.652±0.0122.251±0.004 -2.214±0.0055.675±0.0030.743±0.089
Jump CNF0.027±0.002-1.562±0.0150.166±0.001-1.007±0.0502.242±0.002 -1.904±0.0045.536±0.0161.246±0.185
Attentive CNF0.034±0.001 -1.572±0.0020.204±0.001-1.237±0.0752.258±0.002 -1.864±0.0015.842±0.0051.252±0.026
+ +Table 1: Log-likelihood per event on held-out test data (higher is better). Standard devs. estimated over 3 runs. + +Results & Analyses The results of our evaluation are shown in table 1. We highlight all results where the intervals containing one standard deviation away from the mean overlap. + +Across all data sets, the Time-varying CNF outperforms the conditional KDE baseline despite not being conditional on history. This suggests that the overall spatial distribution is rather complex. We also see from Figure 7 that Gaussian clusters tend to compensate for far-reaching events by learning a larger band-width whereas a flexible CNF can easily model multi-modal event propagation. + +The Jump and Attentive CNF models achieve better log-likelihoods than the Time-varying CNF, suggesting prediction in these data sets benefit from modeling dependence on event history. + +For COVID-19, the self-exciting Hawkes process is a strong baseline which aligns with similar results for other infectious diseases (Park et al., 2019), but Neural STPPs can achieve substantially better spatial likelihoods. Overall, NHP is competitive with the Neural Jump SDE; however, it tends to fall short of the Attentive CNF which jointly models spatial and temporal variables. + +In a closer comparison to the temporal likelihood of Neural Jump SDEs (Jia & Benson, 2019), we find that overly-restricted spatial models can negatively affect the temporal model since both domains are tightly coupled. Since our realization of Neural Jump SDEs and our STPPs use the same underlying architecture to model the temporal domain, the temporal likelihood values are often close. However, there is still a statistically significant difference between our Neural STPP models and Neural Jump SDEs even for the temporal log-likelihood on all data sets. + +Finally, we note that the results of the Jump and Attentive CNFs are typically close. The attentive model generally achieves better temporal log-likelihoods while maintaining competitive spatial log-likelihoods. This difference is likely due to the Attentive CNF’s ability to attend to all previous events, while the Jump CNF has to compress all history information inside the hidden state at the time of event. The Attentive CNF also enjoys substantially faster computations (see Appendix A). + +# 6 CONCLUSION + +To learn high-fidelity models of stochastic events occurring in continuous space and time, we have proposed a new class of parameterizations for spatio-temporal point processes. Our approach combines ideas of Neural Jump SDEs with Continuous Normalizing Flows and allows to retain the flexibility of neural temporal point processes while enabling highly expressive models of continuous marks. We leverage Neural ODEs as a computational method that allows computing, up to negligible numerical error, the likelihood of the joint model, and we show that our approach achieves state-ofthe-art performance on spatio-temporal datasets collected from a wide range of domains. + +A promising area for future work are applications of our method in earth and climate science which often are concerned with modeling highly complex spatio-temporal data. In this context, the use of Riemannian CNFs (Mathieu & Nickel, 2020; Lou et al., 2020; Falorsi & Forre´, 2020) is especially interesting as it allows us to model Neural STPPs on manifolds (e.g. the earth’s surface) by simply replacing the CNF in our models with a Riemannian counterpart. + +# ACKNOWLEDGMENTS + +We acknowledge the Python community (Van Rossum & Drake Jr, 1995; Oliphant, 2007) for developing the core set of tools that enabled this work, including PyTorch (Paszke et al., 2019), torchdiffeq (Chen, 2018), fairseq (Ott et al., 2019), Jupyter (Kluyver et al., 2016), Matplotlib (Hunter, 2007), seaborn (Waskom et al., 2018), Cython (Behnel et al., 2011), numpy (Oliphant, 2006; Van Der Walt et al., 2011), pandas (McKinney, 2012), and SciPy (Jones et al., 2014). + +# REFERENCES + +Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. + +Adrian Baddeley, Imre Bar´ any, and Rolf Schneider. 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Citibike NY
ModelTemporalSpatial
Poisson Process0.609±0.012
Self-correcting Process-5.649±1.433
Hawkes Process1.062±0.000
Neural Hawkes Process 1.030±0.015
Conditional KDE-2.856±0.000
Time-varying CNF-2.132±0.012
Neural Jump SDE1.092±0.002-2.731±0.001
Jump CNF1.105±0.002-2.155±0.015
Attentive CNF1.112±0.002-2.095±0.006
+ +![](images/3b3f8ed141ab2f08bd1ebcfa696bac421996048d70ab7a1c592ac5145da7e4d0.jpg) +Figure 8: Runtime comparison of Jump and Attentive CNF. +Table 2: Log-likelihood values on held-out test data for an urban mobility data set. + +![](images/39a14c8009ca62fbe1c850ddd66d48d4ee3b174b7ebe6492ce5e6ec129b8812d.jpg) +Figure 9: Both the Jump CNF and Attentive CNF are capable of modeling different the spatial distributions based on event history, so the appearance of a new event effectively shifts the distribution instantaneously. Shown on a synthetic 1-D data set similar to PINWHEEL, except we use a mixture of three Gaussians. Each event increases the likelihood of events for the cluster to the right. + +![](images/b85d5ea6427ba727b8c5be454da079d0dca9eaaee7f8cdf688b11c3f384509fc.jpg) +Figure 10: Learned attention weights for random event sequences. + +# B BASELINE + +Our self-excitation baseline uses a Hawkes process to model the temporal variable, then uses a Gaussian mixture model to describe the spatial distribution conditioned on history of events. This corresponds to the following likelihood decomposition + +$$ +\log p ( t _ { 1 } , \dots , t _ { n } , x _ { 1 } , \dots , x _ { n } ) = \sum _ { i = 1 } ^ { n } \log p ( x _ { i } | t _ { i } , t _ { 1 } , \dots , t _ { i - 1 } , x _ { 1 } , \dots , x _ { i - 1 } ) + \sum _ { i = 1 } ^ { n } \log p ( t _ { i } | t _ { 1 } , \dots , t _ { i - 1 } ) +$$ + +Note that $t _ { i }$ does not depend on the spatial variables associated with previous events. This dependence structure allows the usage of simple temporal point processes to model $t _ { i }$ , e.g. a Hawkes process, since temporal variables do not depend on the spatial information. The spatial distribution conditions all past events as well as the current time of occurance. + +Our baseline model assumes a simple Gaussian conditional model, that new events are likely to appear near previous events. + +$$ +\log p ( x _ { i } | t _ { i } , t _ { 1 } , \dots , t _ { i - 1 } , x _ { 1 } , \dots , x _ { i - 1 } ) = \sum _ { j = 1 } ^ { i - 1 } \alpha _ { j } \mathcal { N } ( x _ { j } | \sigma ^ { 2 } ) , \quad \alpha _ { j } = \frac { \exp \{ ( t _ { j } - t _ { i } ) / \tau \} } { \sum _ { j ^ { \prime } = 1 } ^ { i - 1 } \exp \{ ( t _ { j ^ { \prime } } - t _ { i } ) / \tau \} } +$$ + +This parameteric model has two learnable parameters: $\sigma ^ { 2 }$ and $\tau$ , which control the rate of decay in the spatial and temporal domains, respectively. + +However, this Gaussian spatial model assumes events are propagated in all directions equally and can only model local self-excitation behavior. These assumptions are often used for simplifications but are generally incorrect for many spatio-temporal data. To name a few, earthquakes occur more frequently along boundaries of tectonic plates, epidemics propagate along traffic routes, taxi demands saturate locally and change as customers move around. + +# C PRE-PROCESSING STEPS FOR EACH DATA SET + +PINWHEEL We sample from a multivariate Hawkes process with 10 dimensions. We turn this into continuous spatial variables by assigning each dimension to a cluster from a “pinwheel” distribution, and sample from the corresponding cluster for each event. Number of events per sequences ranges between 4 to 108. + +EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological Survey (2020). Starting from January 01, 1990, we created sequences with a gap of 7 days. Each sequence was of length 30 days. We ensured there was no contamination between train/val/test sets by removing intermediate sequences. We removed earthquakes from 2010 November to 2011 December, as these sequences were too long and only served as outliers in the data. This resulted in 950 training sequences, 50 validation sequences, and 50 test sequences. Number of events per sequence ranges between 18 to 543. + +COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19 cases in the New Jersey state, from March to July of 2020. The data is aggregated at the county level, which we dequantize uniformly across the county. We also dequantize the temporal axis by assigning new cases uniformly within the day. Starting at March 15, and every 3 days, we took a 7 day length sequence. For each sequence, we sampled each event with a probability of 0.01. This was done 50 times per sequence. We ensured there was no contamination between train/val/test sets by removing intermediate sequences. This resulted in 1450 training sequences, 100 validation sequences, and 100 test sequences. Number of events per sequence ranges between 3 to 323. + +CITIBIKE Citibike is a bike sharing service in New York City. We treat the start of each trip as an event, and use the data from April to August of 2019. We split into sequences of length 1 day starting at $5 { : } 0 0 { \mathrm { a m } }$ of each day. For each sequence, we subsampled with a probability of 0.005 per event, 20 times. This resulted in 2440 training sequences, 300 validation sequences, and 320 test sequences. Number of events per sequence ranges between 9 to 231. + +![](images/749703a2fc1655f14b5816bcd8926bc3db67af8d8051f198cc235bae973ff268.jpg) +Figure 11: Histograms of the number of events per sequence in each processed data set. + +BOLD5000 This consists of fMRI scans of four participants as they are given visual stimuli (Chang et al., 2019). We use the sessions of a single patient and for each run, we split into 3 sequences, treated individually. We converted brain responses into spatio-temporal events following the $\mathbf { Z }$ -score thresholding approach in Tagliazucchi et al. 2016, Equation (2). We used a threshold of $\gamma = 6 . 0$ . We split the data into 1050 training sequences, 150 val sequences, 220 test sequences. Number of events per sequence ranges between 6 to 1741. + +Each data set contains sequences with highly variable number of events, with varying degrees of dependence between events, making them difficult to model with traditional point process models. We plot histograms showing the number of events per sequence in Figure 11. + +# D HYPERPARAMETERS CHOSEN AND TESTED + +For the time-varying, jump, and attentive CNF models, we parameterized the CNF drift as a multilayer perceptron (MLP) with dimensions $[ d - 6 4 - 6 4 - 6 4 - d ]$ , where $d$ is the number of spatial variables. We swept over activation functions between using softplus or a time-dependent Swish (Ramachandran et al., 2017). + +$$ +\mathrm { T i m e D e p e n d e n t S w i s h } ( t , z ) = h \sigma ( \beta ( t ) \odot z ) +$$ + +where $\sigma$ is the logistic sigmoid function, $\odot$ is the Hadamard (element-wise) product, and $\beta : \mathbb { R } \mathbb { R } _ { d _ { z } }$ is a MLP with widths $[ \bar { 1 } - 6 4 - d _ { z } ]$ where $d _ { z }$ is the dimension of $z$ , using the softplus activation function. We ultimately decided on using the time-dependent Swish for all experiments. + +We swept over the MLP for defining $f _ { h }$ for the continuous-time hidden state in eq. (11) using hidden widths of $[ 8 - 2 0 ]$ , $[ 3 2 - 3 2 ]$ , $[ 6 4 - 6 4 ]$ , $\left[ 3 2 - 3 2 - 3 2 \right]$ , and $[ 6 4 - 6 4 - 6 4 ]$ . The majority of models used $3 2 - 3 2$ as it provided enough flexibility while remaining easy to solve. We used the softplus activation function. We tried MLP for parameterizing the instantaneous change in eq. (12); however, it was too unstable for long sequences. We therefore switched to the GRU parameterization, which takes an input (new event), the hidden state at the time of event, and outputs a new hidden state. + +We regularized the $L _ { 2 }$ norm of the hidden state drift with a strength of 1e-4, chosen from $\{ 0$ , 1e-4, 1e-3, 1e- $\langle 2 \}$ . We optionally used optimal transport-inspired regularization from Finlay et al. (2020), which adds a Frobenius norm regularization to the gradient of the drift in addition to the $L _ { 2 }$ norm regularization, to the CNF models with a strength of $\{ 0 \}$ , 1e-4, 1e-3, 1e- $\cdot 2 \}$ . The Time-varying and Attentive CNF models did not require regularization and were mostly kept at 0, but the Jump CNF models benefited from some amount of regularization to avoid numerical instability. + +To model a non-trivial spatial distribution for the entire data interval, we shift the data interval to start at $t = 2$ for all CNF models. Thus the interval used for parameterizing the CNF is $[ 2 , T + 2 ]$ . Generally, the “time” variable is a dummy one; we can place the base distribution at any time, and we can choose any interval on the real line to be the data interval; this does not limit the model in any way. + +For the Jump CNF, we used a composition of 4 radial flows (Rezende & Mohamed, 2015) to parameterize the instantaneous updates in eq. (17). All parameters of the radial flows were parameterized to be the output of a MLP that takes as input the hidden state at the time of the event (before the hidden state is updated based on the current event). The radial flows were initialized in such a way that the log determinant is near zero. + +For the Attentive CNF, the drift function consists of + +Time-dependent $\mathrm { M L P } ( d - 6 4 - 6 4 ) 2 \times ]$ MultiheadAttention Time-dependent MLP(64 − 64 − d) + +where the Time-dependent MLPs make use of the TimeDependentSwish. As was done in Vaswani et al. (2017), the multihead attention is used within a residual branch, except we swapped LayerNorm (Ba et al., 2016) with ActNorm (Kingma & Dhariwal, 2018) as LayerNorm has an unbounded Lipschitz and can be ill-suited for use in ODEs. We tested both standard multihead attention (Vaswani et al., 2017) and the Lipschitz multihead attention (Kim et al., 2020). The Lipschitz multihead attention typically produced similar validation NLL as the standard multihead attention but were more stable on multiple occassions. We therefore kept the Lipschitz multihead attention for all experiments. We additionally, use an auxiliary (non-attentive, simply with the two multihead attention layers removed) CNF to map from 0 (i.e. the time of base distribution) to the beginning of the data interval. + +We initialized all Neural ODEs (for the hidden state and CNFs) with zero drift by initializing the weights and biases of the final layer to zero. + +The log-likelihood values reported are after the spatial variables have been standardized using the empirical mean and standard deviation from the training set. + +We train and test on log-likelihood (in nats) per event, which normalizes eq. (8) of each sequence by the number of events. + +All integrals were solved using Chen (2018) to within a relative and absolute tolerance of 1E-4 or 1E-6, chosen based on preliminary testing for convergence and stability. + +Our implementation of the Neural Jump SDE shares the same continuous-time hidden state parameterization but uses a mixture of Gaussians as the spatial model. We used 5 mixtures, and a MLP that maps from the hidden state to the parameters of this mixture of Gaussians (the means, log standard deviations, and mixture coefficients). + +# E REMOVING CROSS-EVENT PARTIAL DERIVATIVES + +This results in a lower-variance gradient estimator for training, and allows parallel computation of conditional log probabilities at test time. + +We first summarily describe the attention mechanism. For an input $\boldsymbol { X } \in \mathbb { R } ^ { n \times d }$ representing the f $n$ variabled values $\{ x _ { s } ^ { ( 0 ) } , \ldots , x _ { s } ^ { ( n ) } \}$ . . . , x(n)s } at some valudependent on f , $s$ , this attention mechanism creates logitsch that the output is $P \in \mathbb { R } ^ { n \times n }$ $V \in R ^ { n \times d }$ $X$ + +$$ +O = \underbrace { \operatorname { s o f t m a x } ( P ) } _ { : = S } V . +$$ + +where the softmax is taken over each row of $P$ . The output is then added to $X$ as a residual connection. The multihead attention computes $P$ in a way such that $P _ { i j }$ depends on $X _ { i }$ and $X _ { j }$ , and $V _ { i }$ depends on $X _ { i }$ . This is true for both the vanilla MHA (Vaswani et al., 2017) and the L2 MHA (Kim et al., 2020). For our use case, $P _ { i j }$ is set to −inf for $j > i$ as we don’t want to attend to future events. + +We retain only the block-diagonal gradients where each block contains variables corresponding to one event. This is equivalent to removing all the cross-event dependencies. + +$$ +{ \frac { \partial { \cal O } _ { i } } { \partial X _ { i } } } = S _ { : , i } { \frac { \partial V _ { i } } { \partial X _ { i } } } + V ^ { \mathsf { T } } { \frac { \partial S } { \partial P _ { i } } } { \frac { \partial P _ { i } } { \partial X _ { i } } } + V ^ { \mathsf { T } } { \frac { \partial S } { \partial P _ { : , i } } } { \frac { \partial P _ { : , i } } { \partial X _ { i } } } +$$ + +# F PARALLEL SOLVING OF MULTIPLE ODES WITH VARYING INTERVALS + +Our numerical ODE solvers integrate a single ODE system $\begin{array} { r } { \frac { d x } { d t } = f ( t , x ) } \end{array}$ , where $x \in \mathbb { R } ^ { d }$ and $f : \mathbb { R } ^ { 1 + d } \mathbb { R } ^ { d }$ , on a single fixed interval $[ t _ { s t a r t } , t _ { e n d } ]$ . We can express the inputs and outputs of an ODE solver with + +$$ +\odot \mathtt { D E S o l v e } ( x _ { 0 } , f , t _ { s t a r t } , t _ { e n d } ) \triangleq x _ { 0 } + \int _ { t _ { s t a r t } } ^ { t _ { e n d } } f ( t , x ( t ) ) d t = x ( t _ { e n d } ) . +$$ + +where $x _ { 0 }$ is a vector containing the initial state at the initial time $t _ { 0 }$ . + +Multiple ODEs Now suppose we have a set of that we would like to solve. If all systems had the $M$ systems (i.e. e initial time $\begin{array} { r } { { \frac { d x _ { m } } { d t } } = f _ { m } } \end{array}$ for outp $m = 1 , \ldots , M )$ $t _ { s t a r t }$ and $t _ { e n d }$ , we can readily create a joint system + +$$ +\boldsymbol { x } _ { j o i n t } = \left[ \begin{array} { c } { x _ { 1 } } \\ { \vdots } \\ { x _ { M } } \end{array} \right] \qquad \mathrm { t h a t ~ f o l l o w s } \quad \frac { d x _ { j o i n t } } { d t } = \left[ \begin{array} { c } { f _ { 1 } ( t , x _ { 1 } ) } \\ { \vdots } \\ { f _ { M } ( t , x _ { M } ) } \end{array} \right] +$$ + +Solving this joint system can be done in parallel with a single call to ODESolve: + +$$ +x _ { j o i n t } ( t _ { 1 } ) = \mathsf { O D E S o l v e } ( x _ { 0 j o i n t } , f _ { j o i n t } , t _ { 0 } , t _ { 1 } ) +$$ + +which computes $x _ { m } ( t _ { 1 } )$ for all $m = 1 , \ldots , M$ . This is the standard method used for solving a batch of Neural ODEs. + +Adding dependencies is straightforward Note that extending this further, the ODE systems do not need to be independent. They can depend on other variables at the same concurrent time value because the joint system is still an ordinary differential equation. For instance, we can have + +$$ +\frac { d x _ { m } } { d t } = f _ { m } ( t , x _ { 1 } , \ldots , x _ { m } ) +$$ + +where $f _ { m } : \mathbb { R } ^ { 1 + M d } \mathbb { R } ^ { d }$ . Each system is now a partial differential equation, but the joint system $x _ { j o i n t }$ is still an ODE and can be solved with one call to ODESolve. + +Varied time intervals Now suppose each system has a different time interval that we want to solve. Different initial times and different end times. Let’s denote the start and end time for the $m$ -th system as t start and $t _ { e n d } ^ { ( m ) }$ respectively. We can construct a dummy variable that always integrates from 0 to 1, and perform a change of variables (reparameterization) to transform every system to use this dummy variable. + +As a concrete example of this reparameterzation procedure, consider just one system $x ( t )$ with drift function $f ( t , x )$ that we want to integrate from $t _ { s t a r t }$ to $t _ { e n d }$ with the initial value $x _ { 0 }$ . We can transform $x ( t )$ using the relation $\begin{array} { r } { s = \frac { t - \overline { { t } } _ { s t a r t } } { t _ { e n d } - t _ { s t a r t } } } \end{array}$ , or equivalently + +$$ +t = s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } , +$$ + +into a solution $\tilde { { \boldsymbol { x } } } ( s )$ on the unit interval $[ 0 , 1 ]$ such that + +$$ +\tilde { x } ( s ) = x ( s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } ) +$$ + +The drift function for $\tilde { x }$ then follows as + +$$ +\begin{array} { l } { \displaystyle \tilde { f } ( s , \tilde { x } ( s ) ) \triangleq \frac { d \tilde { x } ( s ) } { d s } = \frac { d x ( t ) } { d t } \bigg \rvert _ { t = s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } } \frac { d t } { d s } } \\ { \displaystyle = f ( t , x ( t ) ) \bigg \rvert _ { t = s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } } ( t _ { e n d } - t _ { s t a r t } ) } \\ { \displaystyle = f \big ( s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } , \tilde { x } ( s ) \big ) \big ( t _ { e n d } - t _ { s t a r t } \big ) } \end{array} +$$ + +Now since $\tilde { x } ( 0 ) = x ( t _ { s t a r t } )$ and $\tilde { x } ( 1 ) = x ( t _ { e n d } )$ , the following are equivalent + +$$ +\boldsymbol { x } ( t _ { e n d } ) = \boldsymbol { \mathrm { O D E S O 1 v e } } ( x _ { 0 } , \tilde { f } , 0 , 1 ) = \boldsymbol { \mathrm { O D E S O 1 v e } } ( x _ { 0 } , f , t _ { s t a r t } , t _ { e n d } ) +$$ + +Putting it all together Let $\widetilde { x } _ { m } ( s )$ be the reparameterized solution for $x _ { m } ( t )$ such that + +$$ +\tilde { x } _ { m } ( s ) = x _ { m } \left( s \left( t _ { e n d } ^ { ( m ) } - t _ { s t a r t } ^ { ( m ) } \right) + t _ { s t a r t } \right) +$$ + +We can then solve for all $M$ systems, with different varying time intervals, using + +$$ +\tilde { x } _ { j o i n t } = \left[ \begin{array} { l } { \tilde { x } _ { 1 } } \\ { \vdots } \\ { \tilde { x } _ { M } } \end{array} \right] \qquad \quad \mathrm { t h a t ~ f o l l o w s } \quad \frac { d \tilde { x } _ { j o i n t } } { d s } = \left[ \begin{array} { l } { \tilde { f } _ { 1 } ( s , \tilde { x } _ { 1 } ) } \\ { \vdots } \\ { \tilde { f } _ { M } ( s , \tilde { x } _ { M } ) } \end{array} \right] . +$$ + +Solving this system to $s = 1$ yields $\tilde { x } _ { m } ( 1 ) = x _ { m } ( t _ { e n d } ^ { ( m ) } )$ + +Assuming $t _ { s t a r t } ^ { ( m ) } = 0$ for all $m = 1 , \ldots , M$ in order to reduce notational complexity, we can write this joint system in terms of the original systems as + +$$ +\frac { d \tilde { x } _ { j o i n t } } { d s } = \left[ \begin{array} { c } { f _ { 1 } \left( s t _ { e n d } ^ { ( 1 ) } , \tilde { x } ( s ) \right) \ t _ { e n d } } \\ { \vdots } \\ { f _ { 1 } \left( s t _ { e n d } ^ { ( 1 ) } , \tilde { x } ( s ) \right) \ t _ { e n d } } \end{array} \right] . +$$ + +This is the joint system written in equation 14. The joint system in equation 19 adds dependence between the $M$ systems but can still be solved with a single ODESolve. \ No newline at end of file diff --git a/parse/train/XQQA6-So14/XQQA6-So14_content_list.json b/parse/train/XQQA6-So14/XQQA6-So14_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..bb7d92aa52700c923dbebb074ab6c0625dc306f4 --- /dev/null +++ b/parse/train/XQQA6-So14/XQQA6-So14_content_list.json @@ -0,0 +1,2578 @@ +[ + { + "type": "text", + "text": "NEURAL SPATIO-TEMPORAL POINT PROCESSES ", + "text_level": 1, + "bbox": [ + 173, + 98, + 748, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ricky T. Q. Chen∗ University of Toronto; Vector Institute rtqichen@cs.toronto.edu ", + "bbox": [ + 183, + 145, + 436, + 186 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Brandon Amos, Maximilian Nickel \nFacebook AI Research \n{bda,maxn}@fb.com ", + "bbox": [ + 511, + 145, + 756, + 188 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 224, + 544, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, highfidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of continuous-time neural networks with two novel neural architectures, i.e., Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience. ", + "bbox": [ + 232, + 256, + 766, + 382 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 409, + 336, + 425 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modeling discrete events that are localized in continuous time and space is an important task in many scientific fields and applications. Spatio-temporal point processes (STPPs) are a versatile and principled framework for modeling such event data and have, consequently, found many applications in a diverse range of fields. This includes, for instance, modeling earthquakes and aftershocks (Ogata, 1988; 1998), the occurrence and propagation of wildfires (Hering et al., 2009), epidemics and infectious diseases (Meyer et al., 2012; Schoenberg et al., 2019), urban mobility (Du et al., 2016), the spread of invasive species (Balderama et al., 2012), and brain activity (Tagliazucchi et al., 2012). ", + "bbox": [ + 174, + 441, + 825, + 539 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "It is of great interest in all of these areas to learn high-fidelity models which can jointly capture spatial and temporal dependencies and their propagation effects. However, existing parameterizations of STPPs are strongly restricted in this regard due to computational considerations: In its general form, STPPs require solving multivariate integrals for computing likelihood values and thus have primarily been studied within the context of different approximations and model restrictions. This includes, for instance, restricting the model class to parameterizations with known closed-form solutions (e.g., exponential Hawkes processes (Ozaki, 1979)), to restrict dependencies between the spatial and temporal domain (e.g., independent and unpredictable marks (Daley & Vere-Jones, 2003)), or to discretize continuous time and space (Ogata, 1998). These restrictions and approximations—which can lead to mis-specified models and loss of information—motivated the development of neural temporal point processes such as Neural Hawkes Processes (Mei & Eisner, 2017) and Neural Jump SDEs (Jia & Benson, 2019). While these methods are more flexible, they can still require approximations such as Monte-Carlo sampling of the likelihood (Mei & Eisner, 2017; Nickel & Le, 2020) and, most importantly, only model restricted spatial distributions (Jia & Benson, 2019). ", + "bbox": [ + 176, + 545, + 576, + 863 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/fa57820cf1c346f628551e171ee58cac549d8c865680579795012e82f52dbaf4.jpg", + "image_caption": [ + "Figure 1: Color is used to denote $p ( x | t )$ , which can be evaluated for Neural STPPs. After observing an event in one mode, the model is instantaneously updated as it strongly expects an event in the next mode. After a period of no observations, the model smoothly reverts back to the marginal distribution. " + ], + "image_footnote": [], + "bbox": [ + 596, + 559, + 821, + 734 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To overcome these issues, we propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and allows us to define flexible, high-fidelity models for spatio-temporal event data. We build upon ideas of Neural Jump SDEs (Jia & Benson, 2019) and Continuous-time Normalizing Flows (CNFs; Chen et al. 2018; Grathwohl et al. 2019; Mathieu & Nickel 2020) to learn parametric models of spatial (or mark1) distributions that are defined continuously in time. Normalizing flows are known to be flexible universal density estimators (e.g. Huang et al. 2018; 2020; Teshima et al. 2020; Kong & Chaudhuri 2020) while retaining computational tractability. As such, our approach allows the computation of exact likelihood values even for highly complex spatio-temporal distributions, and our models create smoothly changing spatial distributions that naturally benefits spatio-temporal modeling. Central to our approach, are two novel neural architectures based on CNFs—using either discontinuous jumps in distribution or self-attention—to condition spatial distributions on the event history. To the best of our knowledge, this is the first method that combines the flexibility of neural TPPs with the ability to learn high-fidelity models of continuous marks that can have complex dependencies on the event history. In addition to our modeling contributions, we also construct five new pre-processed data sets for benchmarking spatio-temporal event models. ", + "bbox": [ + 178, + 872, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 299 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 174, + 318, + 326, + 334 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the following, we give a brief overview of two core frameworks which our method builds upon, i.e., spatio-temporal point processes and continuous-time normalizing flows. ", + "bbox": [ + 173, + 349, + 825, + 378 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Event Modeling with Point Processes Spatio-temporal point processes are concerned with modeling sequences of random events in continuous space and time (Moller & Waagepetersen, 2003; Baddeley et al., 2007). Let $\\mathcal { H } = \\{ ( t _ { i } , \\pmb { x } _ { i } ) \\} _ { i = 1 } ^ { n }$ denote the sequence of event times $t _ { i } \\in \\mathbb { R }$ and their associated locations $\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d }$ , the number of events $n$ being also random. Additionally, let $\\mathcal { H } _ { t } = \\{ ( t _ { i } , \\pmb { x } _ { i } ) ~ | ~ t _ { i } < t , t _ { i } \\in \\mathcal { H } \\}$ denote the history of events predating time $t$ . A spatio-temporal point process is then fully characterized by its conditional intensity function ", + "bbox": [ + 173, + 392, + 825, + 479 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4f7e7f224143536a52f67e4eda4fd42a48fba60bb49d43f6f27c707585ae7f18.jpg", + "text": "$$\n\\lambda ( t , \\pmb { x } \\mid \\mathcal { H } _ { t } ) \\triangleq \\operatorname* { l i m } _ { \\Delta t \\downarrow 0 , \\Delta \\pmb { x } \\downarrow 0 } \\frac { \\mathbb { P } \\left( t _ { i } \\in [ t , t + \\Delta t ] , \\pmb { x } _ { i } \\in B ( \\pmb { x } , \\Delta \\pmb { x } ) \\mid \\mathcal { H } _ { t } \\right) } { | B ( \\pmb { x } , \\Delta \\pmb { x } ) | \\Delta t } .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 484, + 725, + 518 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $B ( { \\pmb x } , \\Delta { \\pmb x } )$ denotes a ball centered at $\\pmb { x } \\in \\mathbb { R } ^ { d }$ and with radius $\\Delta \\mathbfit { x }$ . The only condition is that $\\lambda ( t , \\pmb { x } \\mid \\mathcal { H } _ { t } ) \\geq 0$ and need not be normalized. Given $i - 1$ previous events, the conditional intensity function describes therefore the instantaneous probability of the $i$ -th event occurring at $t$ and location $_ { \\textbf { \\em x } }$ . In the following, we will use the common star superscript shorthand $\\lambda ^ { * } ( t , \\pmb { x } ) = \\lambda ( t , \\pmb { x } \\mid \\mathcal { H } _ { t } )$ to denote conditional dependence on the history. The joint log-likelihood of observing $\\mathcal { H }$ within a time interval of $[ 0 , T ]$ is then given by (Daley $\\&$ Vere-Jones, 2003, Proposition 7.3.III) ", + "bbox": [ + 174, + 525, + 825, + 611 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/154b3cb6fc8df09d85c0e00fb57f6d8ce802154d73478ed675f7b1a432eac975.jpg", + "text": "$$\n\\log p \\left( \\mathcal { H } \\right) = \\sum _ { i = 1 } ^ { n } \\log \\lambda ^ { * } ( t _ { i } , \\pmb { x } _ { i } ) - \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { d } } \\lambda ^ { * } ( \\tau , \\pmb { x } ) d \\pmb { x } d \\tau .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 617, + 691, + 659 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Training general STPPs with maximum likelihood is difficult as eq. (2) requires solving a multivariate integral. This need to compute integrals has driven research to focus around the use of kernel density estimators (KDE) with exponential kernels that have known anti-derivatives (Reinhart et al., 2018). ", + "bbox": [ + 174, + 662, + 825, + 707 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Continuous-time Normalizing Flows Normalizing flows (Dinh et al., 2014; 2016; Rezende & Mohamed, 2015) is a class of density models that describe flexible distributions by parameterizing an invertible transformation from a simpler base distribution, which enables exact computation of the probability of the transformed distribution, without any unknown normalization constants. ", + "bbox": [ + 174, + 719, + 825, + 777 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given a random variable $\\scriptstyle { \\pmb x } _ { 0 }$ with known distribution $p ( \\pmb { x } _ { 0 } )$ and an invertible transformation $F ( x )$ , the transformed variable $F ( \\pmb { x } _ { 0 } )$ is a random variable with a probability distribution function that satisfies ", + "bbox": [ + 174, + 784, + 825, + 824 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/55e1de5b39b256d9b57686ce254219f89b418107dccaa9bdadd216912910814e.jpg", + "text": "$$\n\\log p ( F ( { \\pmb x } _ { 0 } ) ) = \\log p ( { \\pmb x } _ { 0 } ) - \\log \\left| \\operatorname* { d e t } \\frac { \\partial F } { \\partial { \\pmb x } } ( { \\pmb x } _ { 0 } ) \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 820, + 661, + 856 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "There have been many advances in parameterizing $F$ with flexible neural networks that also allow for cheap evaluations of eq. (3). We focus our attention on Continuous-time Normalizing Flows (CNFs), which parameterizes this transformation with a Neural ODE (Chen et al., 2018). CNFs define an infinite set of distributions on the real line that vary smoothly across time, and will be our core component for modeling events in the spatial domain. ", + "bbox": [ + 176, + 857, + 825, + 900 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $p ( \\pmb { x } _ { 0 } )$ be the base distribution2. We then parameterize an instantaneous change in the form of an ordinary differential equation (ODE), $\\begin{array} { r } { \\frac { d { \\pmb x } _ { t } } { d t } = f ( t , { \\pmb x } _ { t } ) } \\end{array}$ , where the subscript denotes dependence on $t$ . This function can be parameterized using any Lipschitz-continuous neural network. Conditioned on a sample $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ from the base distribution, let $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ be the solution of the initial value problem 3 at time $t$ , i.e. it is from a trajectory that passes through $\\scriptstyle { \\pmb x } _ { 0 }$ at time 0 and satisfies the ODE $d { \\pmb x } _ { t } / d t = f$ . We can express the value of the solution at time $t$ as ", + "bbox": [ + 173, + 138, + 826, + 224 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/4a4e130017cfc88e417f41d4a8a43a7769c7cbd6c88fad8b3a15fbe45ddc147d.jpg", + "text": "$$\n{ \\pmb x } _ { t } = { \\pmb x } _ { 0 } + \\int _ { 0 } ^ { t } f ( t , { \\pmb x } _ { \\tau } ) d \\tau .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 231, + 589, + 267 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The distribution of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ then also continuously changes in $t$ through the following equation, ", + "bbox": [ + 171, + 273, + 763, + 289 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ec80c7f65b8daad1b8a3d47caf903a79bc1817108c14738528be55108f3542be.jpg", + "text": "$$\n\\log p ( { \\pmb x } _ { t } | t ) = \\log p ( { \\pmb x } _ { 0 } ) - \\int _ { 0 } ^ { t } \\mathrm { t r } \\left( \\frac { \\partial f } { \\partial x } ( \\tau , { \\pmb x } _ { \\tau } ) \\right) ~ d \\tau .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 295, + 669, + 332 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In practice, eq. (4) and eq. (5) are solved together from 0 to $t$ , as eq. (5) alone is not an ordinary differential equation but the combination of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ and $\\log p ( { \\pmb x } _ { t } )$ is. The trace of the Jacobian $\\frac { \\partial f } { \\partial \\boldsymbol { x } } ( \\tau , \\pmb { x } _ { \\tau } )$ can be estimated using a Monte Carlo estimate of the identity (Skilling, 1989; Hutchinson, 1990), $\\mathrm { t r } ( A ) = \\mathbb { E } _ { v \\sim \\mathcal { N } ( 0 , 1 ) } [ v ^ { \\top } A v ]$ . This estimator relies only on a vector-Jacobian product, which can be efficiently computed in modern automatic differentiation and deep learning frameworks. This has been used (Grathwohl et al., 2019) to scale CNFs to higher dimensions using a Monte Carlo estimate of the log likelihood objective, ", + "bbox": [ + 173, + 337, + 826, + 438 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/27b2119dddad7d5f0b689a5ba12c90e460c76707f20980bc3a7a03168e050f83.jpg", + "text": "$$\n\\log p ( { \\pmb x } _ { t } | t ) = \\log p ( { \\pmb x } _ { 0 } ) - \\mathbb { E } _ { \\pmb { v } \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ \\int _ { 0 } ^ { t } v ^ { \\top } \\frac { \\partial f } { \\partial x } ( \\tau , { \\pmb x } _ { \\tau } ) v \\ d \\tau \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 290, + 444, + 707, + 481 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "which, even if only one sample of $v$ is used, is still amenable to training with stochastic gradient descent. Gradients with respect to any parameters in $f$ can be computed with constant memory by solving an adjoint ODE in reverse-time as described in Chen et al. (2018). ", + "bbox": [ + 174, + 486, + 825, + 530 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 NEURAL SPATIO-TEMPORAL POINT PROCESSES ", + "text_level": 1, + "bbox": [ + 174, + 549, + 604, + 566 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We are interested in modeling high-fidelity distributions in continuous time and space that can be updated based on new event information. For this purpose, we use the Neural ODE framework to parameterize a STPP by combining ideas from Neural Jump SDEs and Continuous Normalizing Flows to create highly flexible models that still allow exact likelihood computation. ", + "bbox": [ + 173, + 582, + 825, + 638 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We first (re-)introduce necessary notation. Let $\\mathcal { H } = \\{ ( t _ { i } , \\pmb { x } _ { t _ { i } } ^ { ( i ) } ) \\}$ denote a sequence of event times $t _ { i } \\in [ 0 , T ]$ and locations $\\pmb { x } _ { t _ { i } } ^ { ( i ) } \\in \\mathbb { R } ^ { d }$ i. The superscript indicates an association with the $i$ -th event, and the use of subscripting with $t _ { i }$ will be useful later in the continuous-time modeling framework. Following Daley & Vere-Jones (2003), we decompose the conditional intensity function as ", + "bbox": [ + 173, + 646, + 826, + 704 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cb917c645357ea3aff34b0b5723d772a8a0d236998b75ef0b259ec9993df83bd.jpg", + "text": "$$\n\\lambda ^ { * } ( t , { \\pmb x } ) = \\lambda ^ { * } ( t ) p ^ { * } ( { \\pmb x } \\mid t )\n$$", + "text_format": "latex", + "bbox": [ + 410, + 712, + 588, + 729 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\lambda ^ { * } ( t )$ is the ground intensity of the temporal process and where $\\boldsymbol { p } ^ { * } ( \\boldsymbol { x } \\mid t )$ is the conditional density of a mark $_ { \\textbf { \\em x } }$ at $t$ given $\\mathcal { H } _ { t }$ . The star superscript is used as again shorthand to denote dependence on the history. Since $\\textstyle \\int _ { \\mathbb { R } ^ { d } } p ^ { * } ( { \\pmb { x } } \\mid t ) = 1$ , eq. (7) allows us now to simplify the log-likelihood function of the joint process from eq. (2), such that ", + "bbox": [ + 174, + 736, + 825, + 792 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b7c5e6c3cbb5ae884c7ccd715abaebd6f7b84025acd004cd6a822d82eec87478.jpg", + "text": "$$\n\\log p ( \\mathcal { H } ) = \\underbrace { \\sum _ { i = 1 } ^ { n } \\log \\lambda ^ { * } ( t _ { i } ) - \\int _ { 0 } ^ { T } \\lambda ^ { * } ( \\tau ) ~ d \\tau } _ { \\mathrm { t e m p o r a l ~ l o g - l i k e l i h o o d } } + \\underbrace { \\sum _ { i = 1 } ^ { n } \\log p ^ { * } ( \\pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) } _ { \\mathrm { s p a t i a l ~ l o g - l i k e l i h o o d } }\n$$", + "text_format": "latex", + "bbox": [ + 287, + 799, + 714, + 857 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Furthermore, based on eq. (7), we can derive separate models for the ground intensity and conditional mark density which will be jointly conditioned on a continuous-time hidden state with jumps. In the following, we will first describe how we construct a latent dynamics model, which we use to compute the ground intensity $\\lambda ^ { * } ( t )$ . We will then propose three novel CNF-based approaches for modeling the conditional mark density $p ^ { * } ( { \\pmb x } | t )$ . We will first describe an unconditional model, which is already a strong baseline when spatial event distributions only follow temporal patterns and there is little to no correlation between the spatial observations. We then devise two new methods of conditioning on the event history $\\mathcal { H }$ : one explicitly modeling instantaneous changes in distribution, and another that uses an attention mechanism which is more amenable to parallelism. ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Latent Dynamics and Ground Intensity For the temporal variables $\\{ t _ { i } \\}$ , parameterize the intensity function using hidden state dynamics with jumps, similar to the work of Jia & Benson (2019). Specifically, we evolve a continuous-time hidden state $^ { h }$ and set ", + "bbox": [ + 174, + 243, + 826, + 286 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/03da588f0e82dd714dad317c7dbd620fb94df538199bdca04d0ce0f5f742eefe.jpg", + "text": "$$\n\\lambda ^ { * } ( t ) = g _ { \\lambda } ( h _ { t } ) \\qquad \\mathrm { ( G r o u n d i n t e n s i t y ) }\n$$", + "text_format": "latex", + "bbox": [ + 369, + 292, + 629, + 310 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $g _ { \\lambda }$ is a neural network with a softplus nonlinearity applied to the output, to ensure the intensity is positive. We then capture conditional dependencies through the use of a continuously changing state $\\boldsymbol { h } _ { t }$ with instantaneous updates when conditioned on an event. ", + "bbox": [ + 174, + 316, + 826, + 359 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The architecture is analogous to a recurrent neural network with a continuous-time hidden state (Mei & Eisner, 2017; Che et al., 2018; Rubanova et al., 2019) modeled by a Neural ODE. This provides us with a vector representation $\\boldsymbol { h } _ { t }$ at every time value $t$ that acts as both a summary of the history of events and as a predictor of future behavior. Instantaneous updates to $\\boldsymbol { h } _ { t }$ allow to incorporate abrupt changes to the hidden state that are triggered by observed events. This mechanism is important for modeling point processes and allows past events to influence future dynamics in a discontinuous way (e.g., modeling immediate shocks to a system). ", + "bbox": [ + 173, + 366, + 825, + 464 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We use $f _ { h }$ to model the continuous change in the form of an ODE and $g _ { h }$ to model instantaneous changes based on an observed event. ", + "bbox": [ + 173, + 469, + 821, + 500 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0394f6b4bbd83939cd06ce26581c4cbed5ade52df0c2d8ae6559ab9f75be55d1.jpg", + "text": "$$\n{ \\begin{array} { r l r l } & { h _ { t _ { 0 } } = h _ { 0 } } & & { ( { \\mathrm { A n ~ i n i t i a l ~ h i d d e n ~ s t a t e } } ) } \\\\ & { { \\frac { d h _ { t } } { d t } } = f _ { h } ( t , h _ { t } ) } & & { { \\mathrm { b e t w e e n ~ e v e n t ~ t i m e s } } } & & { ( { \\mathrm { C o n t i n u o u s ~ e v o l u t i o n } } ) } \\\\ & { \\operatorname* { l i m } _ { \\varepsilon \\to 0 } h _ { t _ { i } + \\varepsilon } = g _ { h } \\left( t _ { i } , h _ { t _ { i } } , x _ { t _ { i } } ^ { ( i ) } \\right) } & & { { \\mathrm { a t ~ e v e n t ~ t i m e s ~ } } t _ { i } } & & { ( { \\mathrm { I n s t a n t a n e o u s ~ u p d a t e s } } ) } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 228, + 503, + 767, + 584 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The use of $\\varepsilon$ is to portray that $h _ { t }$ is a cagl \\` ad\\` function, i.e. left-continuous with right limits, with a discontinuous jump modeled by $g _ { h }$ . ", + "bbox": [ + 176, + 588, + 823, + 618 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The parameterization of continuous-time hidden states in the form of eqs. (10) to (12) has been used for time series modeling (Rubanova et al., 2019; De Brouwer et al., 2019) as well as TPPs (Jia & Benson, 2019). We parameterize $f _ { h }$ as a standard multi-layer fully connected neural network, and use the GRU update (Cho et al., 2014) to parameterize $g _ { h }$ , as was done in Rubanova et al. (2019). ", + "bbox": [ + 173, + 625, + 825, + 681 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Time-varying CNF The first model we consider is a straightforward application of the CNF to time-variable observations. Assuming that the spatial distribution is independent of prior events, ", + "bbox": [ + 173, + 695, + 826, + 724 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2b6fe850c2d807b16e547aeeb5ebe8e91a44a27b5cc4aa04921882ef32802565.jpg", + "text": "$$\n\\log p ^ { * } ( \\pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) = \\log p ( \\pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) = \\log p ( \\pmb { x } _ { 0 } ^ { ( i ) } ) - \\int _ { 0 } ^ { t _ { i } } \\mathrm { t r } \\left( \\frac { \\partial f } { \\partial x } ( \\tau , \\pmb { x } _ { \\tau } ^ { ( i ) } ) \\right) d \\tau\n$$", + "text_format": "latex", + "bbox": [ + 258, + 731, + 743, + 767 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\pmb { x } _ { \\tau } ^ { ( i ) }$ ) is the solution of the ODE f with initial value x(i)t , the observed event location, at $\\tau = t _ { i }$ the observed event time. The spatial distribution of an event modeled by a Time-varying CNF changes with respect to the time it occurs. Some spatio-temporal data sets exhibit mostly temporal patterns and little to no dependence on previous events in the spatial domain, which would make a time-varying CNF a good fit. Nevertheless, this model lacks the ability to capture spatial propagation effects, as it does not condition on previous event observations. ", + "bbox": [ + 173, + 773, + 825, + 861 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A major benefit of this model is the ability to evaluate the joint log-likelihood fully in parallel across events, since there are no dependencies between events. Most modern ODE solvers that we are aware of only allow a scalar terminal time. Thus, to solve all $n$ integrals in eq. (13) with a single call to an ODE solver, we can simply reparameterize all integrals with a consistent dummy variable and track the terminal time in the state (see Appendix $\\mathrm { F }$ for detailed explanation). Intuitively, the idea is that we can reparameterize ODEs that are on $t \\in [ 0 , t _ { i } ]$ into an ODE on $s \\in [ 0 , 1 ]$ using the change of variables $\\bar { s } = t / { { t } _ { i } }$ (or $t = s t _ { i }$ ) and scaling the output of $f$ by $t _ { i }$ . The joint ODE is then ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 826, + 147 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/687725abdc254bc5dd2be0852ede57c07699e16925e0353d2b09bfed823f62c6.jpg", + "text": "$$\n\\frac { d } { d s } \\underbrace { \\left[ \\begin{array} { c } { x _ { s } ^ { ( 1 ) } } \\\\ { \\vdots } \\\\ { x _ { s } ^ { ( n ) } } \\end{array} \\right] } _ { A _ { s } } = \\underbrace { \\left[ t _ { 1 } f ( s t _ { 1 } , x _ { s } ^ { ( 1 ) } ) \\right] } _ { f ( s t _ { n } , z _ { s } ) } \\quad \\mathrm { ~ w h i c h ~ g i v e s ~ } \\quad \\underbrace { \\left[ \\begin{array} { c } { x _ { 0 } ^ { ( 1 ) } } \\\\ { \\vdots } \\\\ { x _ { 0 } ^ { ( n ) } } \\end{array} \\right] } _ { A _ { 0 } } + \\int _ { 0 } ^ { 1 } f ( s , A _ { s } ) d s = \\underbrace { \\left[ \\begin{array} { c } { x _ { t _ { 1 } } ^ { ( 1 ) } } \\\\ { \\vdots } \\\\ { x _ { t _ { n } } ^ { ( n ) } } \\end{array} \\right] } _ { A _ { 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 154, + 782, + 236 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Thus the full trajectories between 0 to $t _ { i }$ for all events can be computed in parallel using this augmented ODE by simply integrating once from $s = 0$ to $s = 1$ . ", + "bbox": [ + 173, + 243, + 823, + 272 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\mathbf { J u m p C N F }$ For the second model, we condition the dynamics defining the continuous normalizing flow on the hidden state $^ { h }$ , allowing the normalizing flow to update its distribution based on changes in $\\mathcal { H }$ . For this purpose, we define continuous-time spatial distributions by making again use of two components: (i) a continuous-time normalizing flow that evolves the distribution continuously, and (ii) a standard (discrete-time) flow model that changes the distribution instantaneously after conditioning on new events. As normalizing flows parameterize distributions through transformations of the samples, these continuous- and discrete-time transformations are composable in a straightforward manner and are end-to-end differentiable. ", + "bbox": [ + 173, + 287, + 825, + 400 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The generative process of a single event in a Jump CNF is given by: ", + "bbox": [ + 173, + 406, + 619, + 421 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e972a7159e4559150a0d904c385f299acffa2c620f52a8dabe4f58eb7411d32a.jpg", + "text": "$$\n\\begin{array} { r l r l } { x _ { 0 } \\sim p ( x _ { 0 } ) } & { } & & { \\mathrm { ( A n i n i t i a l ~ d i s t r i b u t i o n ) } } \\\\ { \\displaystyle \\frac { d x _ { t } } { d t } = f _ { x } ( t , x _ { t } , h _ { t } ) } & { \\mathrm { ~ b e t w e e n ~ e v e n t ~ t i m e s ~ } } & & { \\mathrm { ( C o n t i n u o u s ~ e v o l u t i o n ) } } \\\\ { \\displaystyle \\operatorname* { l i m } _ { \\varepsilon 0 } x _ { t _ { i } + \\varepsilon } = g _ { x } ( t _ { i } , x _ { t _ { i } } , h _ { t _ { i } } ) } & { \\quad \\mathrm { a t ~ e v e n t ~ t i m e s ~ } t _ { i } } & & { \\mathrm { ( I n s t a n t a n e o u s ~ u p d a t e s ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 426, + 763, + 503 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The initial distribution can be parameterized by a normalizing flow. In practice, we set a base distribution at a negative time value and model $p ( \\pmb { x } _ { 0 } )$ using the same CNF parameterized by $f _ { x }$ . The instantaneous updates (or jumps) describe conditional updates in distribution after each new event has been observed. This conditioning on $h _ { t _ { i } }$ is required for the continuous and instantaneous updates to depend on the history of observations. Otherwise, a Jump CNF would only be able to model the marginal distribution and behave similarly to a time-varying CNF. We solve for $h _ { t }$ alongside $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ . ", + "bbox": [ + 173, + 508, + 825, + 593 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The final probability of an event $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ at some $t > t _ { n }$ after observing $n$ events is given by the sum of changes according to the continuous- and discrete-time normalizing flows. ", + "bbox": [ + 173, + 599, + 825, + 628 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/db021b23841b128e5cbe27bb12031d191e2f0eab74394e2b966342b9c5b5bbb9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\log p ^ { * } ( { \\boldsymbol x } _ { t } | t ) = \\log p ( { \\boldsymbol x } _ { 0 } ) } \\\\ & { \\quad \\quad + \\underbrace { \\displaystyle \\sum _ { t _ { i } \\in \\mathcal { H } _ { t } } \\left( - \\int _ { t _ { i - 1 } } ^ { t _ { i } } { \\mathrm { t r } \\left( \\frac { \\partial f ( \\tau , { \\boldsymbol x } _ { \\tau } , h _ { \\tau } ) } { \\partial { \\boldsymbol x } } \\right) d \\tau } - \\log \\left| \\operatorname* { d e t } \\frac { \\partial g _ { { \\boldsymbol x } } ( t _ { i } , { \\boldsymbol x } _ { t _ { i } } , h _ { t _ { i } } ) } { \\partial { \\boldsymbol x } } \\right| \\right) } _ { \\mathrm { C h a p e ~ i n g ~ u p ~ t o ~ l a s t ~ v e n t } } } \\\\ & { \\quad \\quad + \\underbrace { \\displaystyle \\int _ { t _ { n } } ^ { t } - { \\mathrm { t r } \\left( \\frac { \\partial f ( \\tau , { \\boldsymbol x } _ { \\tau } , h _ { \\tau } ) } { \\partial { \\boldsymbol x } } \\right) d \\tau } } _ { \\mathrm { C h a p e ~ i n g e n ~ l a s t ~ v e n t ~ t o ~ } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 633, + 767, + 772 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As the instantaneous updates must be applied sequentially in a Jump CNF, we can only compute the integrals in eq. (18) one at a time. As such, the number of initial value problems scales linearly with the number of events in the history because the ODE solver must be restarted between each instantaneous update to account for the discontinuous change to state. This incurs a substantial cost when the number of events is large. ", + "bbox": [ + 173, + 781, + 825, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Attentive CNF To design a spatial model with conditional dependencies that alleviates the computational issues of Jump CNFs and can be computed in parallel, we make use of efficient attention mechanisms based on the Transformer architecture (Vaswani et al., 2017). Denoting only the spatial variables for simplicity, each conditional distribution $\\log p ( \\pmb { x } _ { t _ { i } } \\ | \\ \\mathcal { H } _ { t _ { i } } )$ can be modeled by a CNF that depends on the sample path of prior events. Specifically, we take the dummy-variable reparameterization of eq. (14) and modify it so that the $i$ -th event depends on all previous events using a Transformer architecture for $f$ , ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/63a02fc2549e8b7f69997b935dc6ed8a77774c252453939a052db56ee3e48cab.jpg", + "image_caption": [ + "Figure 2: Visualization of the sampling paths of Neural STPP models for a 1-D spatio-temporal data set where $\\{ t _ { i } \\} _ { i = 1 } ^ { 4 }$ are event times. The Jump CNF uses instantaneous jumps to update its distribution based on newly observed events while the Attentive CNF depends continuously on the sampling paths of prior events. We additionally visualize a second sequence for the Attentive CNF where the random base samples $\\{ x _ { 0 } ^ { ( i ) } \\} _ { i = 2 } ^ { 4 }$ are the same as in sequence 1. Even so, the sampling paths are different because the first event is different, effectively leading to different conditional spatial distributions. See Figure 9 for visualizations of the learned density. " + ], + "image_footnote": [], + "bbox": [ + 186, + 104, + 810, + 218 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 330, + 825, + 375 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/34906d52461329081c1a6d035e2866af591ff2bc3c83e628b17867efba850154.jpg", + "text": "$$\n\\frac { d } { d s } \\left[ \\begin{array} { c } { x _ { s } ^ { ( 1 ) } } \\\\ { x _ { s } ^ { ( 2 ) } } \\\\ { \\vdots } \\\\ { x _ { s } ^ { ( n ) } } \\end{array} \\right] = \\left[ \\begin{array} { c } { t _ { 1 } f \\left( s t _ { 1 } , \\{ x _ { s } ^ { ( i ) } \\} _ { i = 1 } ^ { 1 } , \\{ h _ { t _ { i } } \\} _ { i = 1 } ^ { 1 } \\right) } \\\\ { t _ { 2 } f \\left( s t _ { 2 } , \\{ x _ { s } ^ { ( i ) } \\} _ { i = 1 } ^ { 2 } , \\{ h _ { t _ { i } } \\} _ { i = 1 } ^ { 2 } \\right) } \\\\ { \\vdots } \\\\ { t _ { n } f \\left( s t _ { n } , \\{ x _ { s } ^ { ( i ) } \\} _ { i = 1 } ^ { n } , \\{ h _ { t _ { i } } \\} _ { i = 1 } ^ { n } \\right) } \\end{array} \\right] : = f _ { \\mathrm { A t t } } \\mathrm { n } .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 377, + 694, + 473 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "With this formulation, the trajectory of $\\pmb { x } _ { \\tau } ^ { ( i ) }$ depends continuously on the trajectory of $\\pmb { x } _ { \\tau } ^ { ( j ) }$ for all $j < i$ and the hidden states $^ { h }$ prior to the $i$ -th event. Similar to eq. (14), an Attention CNF can now solve for the trajectories of all events in parallel while simultaneously depending non-trivially on $\\mathcal { H }$ . ", + "bbox": [ + 173, + 478, + 825, + 522 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To parameterize $f _ { \\mathrm { A t t } \\mathrm { n } }$ , we use an embedding layer followed by two multihead attention (MHA) blocks and an output layer to map back into the input space. We use the Lipschitz-continuous multihead attention from Kim et al. (2020) as they recently showed that the dot product multihead attention (Vaswani et al., 2017) is not Lipschitz-continuous and thus may be ill-suited for parameterizing ODEs. ", + "bbox": [ + 173, + 529, + 825, + 585 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Low-variance Log-likelihood Estimation The variance of the Hutchinson stochastic trace estimator in eq. (6) grows with the squared Frobenius norm of the Jacobian, $\\sum _ { i j } \\left[ \\partial f / \\partial x \\right] _ { i j } ^ { 2 }$ (Hutchinson, 1990). For attentive CNFs, we can remove some of the non-diagonal elements of the Jacobian and achieve a lower variance estimator. The attention mechanism creates a blocktriangular Jacobian, where each block corresponds to one event, but the elements outside of the blockdiagonal are solely due to the multihead attention. By detaching the gradient connections between different events in the MHA blocks, we can create a surrogate ", + "bbox": [ + 174, + 599, + 531, + 770 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/f5dbfbc0e7accab2cb0542d09d2deb53f0e2cb9c2f4fa3bd16b33ead1f927e20.jpg", + "image_caption": [ + "Figure 3: Lower variance estimates of the loglikelihood allows training better Attentive CNFs. " + ], + "image_footnote": [], + "bbox": [ + 550, + 602, + 816, + 720 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Jacobian matrix that do not contain cross-event partial derivatives. This effectively allows us to apply Hutchinson’s estimator on a matrix that has the same diagonal elements as the Jacobian $\\partial f / \\partial x$ —and thus has the same expected value—but has zeros outside of the block-diagonal, leading to a lower variance trace estimator. The procedure consists of selectively removing partial derivatives and is straightforward but notationally cumbersome; the interested reader can find the details in Appendix E. ", + "bbox": [ + 174, + 770, + 825, + 840 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This is similar in spirit to Chen & Duvenaud (2019) but instead of constructing a neural network that specifically allows cheap removal of partial derivatives, we make use of the fact that multihead attention already allows cheap removal of (cross-event) partial derivatives. ", + "bbox": [ + 174, + 847, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "An ablation experiment is shown in Figure 3 for training on the PINWHEEL data set, where the lower variance estimates (and gradients) ultimate led to faster convergence and better converged models. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 102, + 343, + 117 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Neural Temporal Point Processes Modeling real-world data using restricted models such as Exponential Hawkes Processes (Ozaki, 1979) may lead to poor results due to model mis-specification. While this has led to many works on improving the Hawkes process (e.g. Linderman & Adams 2014; Li & Zha 2014; Zhao et al. 2015; Farajtabar et al. 2017; Li & Ke 2020; Nickel & Le 2020), recent works have begun to explore neural network parameterizations of TPPs. A common approach is to use recurrent neural networks to accumulate the event history in a latent state from which the intensity value can then be derived. Models of this form include, for instance, Recurrent Marked Temporal Point Processes (RMTPPs; Du et al. 2016) and Neural Hawkes Processes (NHPs; Mei & Eisner 2017). In contrast to our approach, these methods can not compute the exact likelihood of the model and have to resort to Monte-Carlo sampling for its approximation. However, this approach is especially problematic for commonly occurring clustered and bursty event sequences as it either requires a very high sampling rate or ignores important temporal dependencies (Nickel & Le, 2020). To overcome this issue, Jia & Benson (2019) proposed Neural Jump SDEs which extend the Neural ODE framework and allow to compute the exact likelihood for neural TPPs, up to numerical errors. This method is closely related to our approach and we build on its ideas to compute the ground intensity of the STPP. However, current Neural Jump SDEs —as well as NHPs and RMTPPs—are not well-suited for modeling complex continuous mark distributions as they are restricted to methods such as Gaussian mixture models in the spatial domain. Finally, Shchur et al. (2019) and Mehrasa et al. (2019) considered combining TPPs with flexible likelihood-based models, however for different purposes as in our case, i.e., for intensity-free learning of only temporal point processes. ", + "bbox": [ + 174, + 132, + 825, + 410 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Continuous Normalizing Flows The ability to describe an infinite number of distributions with a Continuous Normalizing Flow has been used by a few recent works. Some works in computer graphics have used the interpolation effect of CNFs to model transformations of point clouds (Yang et al., 2019; Rempe et al., 2020; Li et al., 2020). CNFs have also been used in sequential latent variable models (Deng et al., 2020; Rempe et al., 2020). However, such works do not align the “time” axis of the CNF with the temporal axis of observations, and do not train on observations at more than one value of “time” in the CNF. In contrast, we align the time axis of the CNF with the time of the observations, directly using its ability to model distributions on a real-valued axis. A closely related application of CNFs to spatio-temporal data was done by Tong et al. (2020), who modeled the distribution of cells in a developing human embryo system at five fixed time values. In contrast to this, we extend to applications where observations are made at arbitrary time values, jointly modeling space and time within the spatio-temporal point process framework. Furthermore, Mathieu & Nickel (2020); Lou et al. (2020) recently proposed extensions of CNFs to Riemannian manifolds. For our proposed approach, this is especially interesting in the context of earth and climate science, as it allows us to model STPPs on the sphere simply by replacing the CNF with its Riemannian equivalent. ", + "bbox": [ + 174, + 425, + 825, + 633 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 654, + 326, + 669 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Data Sets Many collected data can be represented within the framework of spatio-temporal events. We pre-process data from open sources and make them suitable for spatio-temporal event modeling. Each sequence in these data sets can contain up to thousands of variables, all the while having a large variance in sequence lengths. Varying across a wide range of domains, the data sets we consider are: earthquakes, pandemic spread, consumer demand for a bike sharing app, and high-amplitude brain signals from fMRI scans. We briefly describe these data sets here; further details, pre-processing steps, and data set diagnostics can be found in Appendix C. Code for preprocessing and training are open sourced at https://github.com/facebookresearch/neural_stpp. ", + "bbox": [ + 174, + 684, + 825, + 796 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "PINWHEEL This is a synthetic data set with multimodal and non-Gaussian spatial distributions designed to test the ability to capture drastic changes due to event history (see fig. 5). The data set consists of 10 clusters which form a pinwheel structure. Events are sampled from a multivariate Hawkes process such that events from one cluster will increase the probability of observing events in the next cluster in a clock-wise rotation. Number of events per sequences ranges between 4 to 108. ", + "bbox": [ + 176, + 808, + 825, + 877 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological Survey (2020). Number of events per sequences ranges between 18 to 543. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/64968869c1f670bd0d6d1f78cc771ab88dd0fa3dbd68dc7e1efb980f726fcdbe.jpg", + "image_caption": [ + "Figure 5: Evolution of spatial densities on PINWHEEL data. top: Attentive CNF. bottom: Jump CNF. (a) Before observing any events at $\\scriptstyle t = 0$ , the distribution is even across all clusters. (b-f) Each event increases the probability of observing a future event from the subsequent cluster in clock-wise ordering. (g-h) After a period of no new events, the distribution smoothly returns back to the initial distribution (a). " + ], + "image_footnote": [], + "bbox": [ + 179, + 102, + 818, + 246 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/c2f9aeea16989017a062ac3824365a59e03ce754df2dafff536fa59e95729512.jpg", + "image_caption": [ + "Figure 7: Snapshots of conditional spatial distributions modeled by the Jump CNF (top) and a conditional kernel density estimator (KDE; bottom). (a) Distribution before any events at $\\scriptstyle t = 0$ . (b-d) The Jump CNF’s distributions concentrate around tectonic plate boundaries where earthquakes and aftershocks gather, whereas the KDE must use a large variance in order to capture propagation of aftershocks in multiple directions. " + ], + "image_footnote": [], + "bbox": [ + 220, + 323, + 772, + 503 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19 cases in New Jersey state. The data is aggregated at the county level, which we dequantize uniformly across the county. Number of events per sequences ranges between 3 to 323. ", + "bbox": [ + 174, + 598, + 825, + 640 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "BOLD5000 This consists of fMRI scans as participants are given visual stimuli (Chang et al., 2019). We convert brain responses into spatio-temporal events following the z-score thresholding approach in Tagliazucchi et al. (2012; 2016). Number of events per sequences ranges between 6 to 1741. ", + "bbox": [ + 174, + 648, + 825, + 691 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In addition to these datasets, we also report in Appendix A results for CITIBIKE, a data set consisting of rental events from a bike sharing service in New York City. ", + "bbox": [ + 174, + 708, + 820, + 736 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Baselines To evaluate the capability of our proposed models, we compare against commonly-used baselines and state-of-the-art models. In some settings, ground intensity and conditional mark density are independent of each other and we can freely combine different baselines for the temporal and spatial domains. As temporal baselines, we use a homogeneous Poisson process, a self-correction process, a Hawkes process, and the Neural Hawkes Process, which were trained using their officially released code. As spatial baselines, we use a conditional kernel density estimator (KDE) with learned parameters, where $p ( { \\pmb x } | t )$ is essentially modeled as a history-dependent Gaussian mixture model (see Appendix B), as well as the Time-varying CNF. In addition, we also compare to our implementation of Neural Jump SDEs (Jia & Benson, 2019) where the spatial distribution is a Gaussian mixture model. We use the same architecture as our GRU-based continuous-time hidden states for fair comparison, as we found the simpler parameterization in Jia & Benson (2019) to be numerically unstable for large number of events. Range of hyperparameter values are outlined in Appendix D. ", + "bbox": [ + 173, + 757, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2111b30cc24d7fe8fbeb2177c2219f0516b14ccee56250f827cbf5622cbbb6bf.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Log-likelihood per event on held-out test data (higher is better). Standard devs. estimated over 3 runs. " + ], + "table_body": "
PinwheelEarthquakes JPCOVID-19 NJBOLD5000
ModelTemporalSpatialTemporalSpatialTemporalSpatialTemporalSpatial
Poisson Process-0.784±0.0011-0.111±0.00110.878±0.01610.862±0.0181
Self-correcting Process-2.117±0.222-7.051±0.780-10.053±1.150-6.470±0.827
Hawkes Process-0.276±0.0330.114±0.0052.092±0.0232.860±0.050
Neural Hawkes Process-0.023±0.0010.198±0.0012.229±0.0133.080±0.019
Conditional KDE-2.958±0.000-2.259±0.001-2.583±0.000-3.467±0.000
Time-varying CNF-2.185±0.003-1.459±0.016-2.002±0.0021-1.846±0.019
Neural Jump SDE (GRU)-0.006±0.042-2.077±0.0260.186±0.005 -1.652±0.0122.251±0.004 -2.214±0.0055.675±0.0030.743±0.089
Jump CNF0.027±0.002-1.562±0.0150.166±0.001-1.007±0.0502.242±0.002 -1.904±0.0045.536±0.0161.246±0.185
Attentive CNF0.034±0.001 -1.572±0.0020.204±0.001-1.237±0.0752.258±0.002 -1.864±0.0015.842±0.0051.252±0.026
", + "bbox": [ + 173, + 101, + 823, + 265 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Results & Analyses The results of our evaluation are shown in table 1. We highlight all results where the intervals containing one standard deviation away from the mean overlap. ", + "bbox": [ + 173, + 325, + 821, + 353 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Across all data sets, the Time-varying CNF outperforms the conditional KDE baseline despite not being conditional on history. This suggests that the overall spatial distribution is rather complex. We also see from Figure 7 that Gaussian clusters tend to compensate for far-reaching events by learning a larger band-width whereas a flexible CNF can easily model multi-modal event propagation. ", + "bbox": [ + 174, + 359, + 825, + 416 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The Jump and Attentive CNF models achieve better log-likelihoods than the Time-varying CNF, suggesting prediction in these data sets benefit from modeling dependence on event history. ", + "bbox": [ + 176, + 422, + 823, + 450 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For COVID-19, the self-exciting Hawkes process is a strong baseline which aligns with similar results for other infectious diseases (Park et al., 2019), but Neural STPPs can achieve substantially better spatial likelihoods. Overall, NHP is competitive with the Neural Jump SDE; however, it tends to fall short of the Attentive CNF which jointly models spatial and temporal variables. ", + "bbox": [ + 174, + 458, + 825, + 513 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In a closer comparison to the temporal likelihood of Neural Jump SDEs (Jia & Benson, 2019), we find that overly-restricted spatial models can negatively affect the temporal model since both domains are tightly coupled. Since our realization of Neural Jump SDEs and our STPPs use the same underlying architecture to model the temporal domain, the temporal likelihood values are often close. However, there is still a statistically significant difference between our Neural STPP models and Neural Jump SDEs even for the temporal log-likelihood on all data sets. ", + "bbox": [ + 174, + 521, + 825, + 604 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we note that the results of the Jump and Attentive CNFs are typically close. The attentive model generally achieves better temporal log-likelihoods while maintaining competitive spatial log-likelihoods. This difference is likely due to the Attentive CNF’s ability to attend to all previous events, while the Jump CNF has to compress all history information inside the hidden state at the time of event. The Attentive CNF also enjoys substantially faster computations (see Appendix A). ", + "bbox": [ + 174, + 612, + 825, + 681 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 712, + 318, + 728 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To learn high-fidelity models of stochastic events occurring in continuous space and time, we have proposed a new class of parameterizations for spatio-temporal point processes. Our approach combines ideas of Neural Jump SDEs with Continuous Normalizing Flows and allows to retain the flexibility of neural temporal point processes while enabling highly expressive models of continuous marks. We leverage Neural ODEs as a computational method that allows computing, up to negligible numerical error, the likelihood of the joint model, and we show that our approach achieves state-ofthe-art performance on spatio-temporal datasets collected from a wide range of domains. ", + "bbox": [ + 174, + 750, + 825, + 847 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A promising area for future work are applications of our method in earth and climate science which often are concerned with modeling highly complex spatio-temporal data. In this context, the use of Riemannian CNFs (Mathieu & Nickel, 2020; Lou et al., 2020; Falorsi & Forre´, 2020) is especially interesting as it allows us to model Neural STPPs on manifolds (e.g. the earth’s surface) by simply replacing the CNF in our models with a Riemannian counterpart. 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", + "bbox": [ + 169, + 74, + 828, + 934 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A ADDITIONAL RESULTS AND FIGURES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 522, + 118 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/3ab6ab434661b66f64176a2bbb4b71c8cef89d2503beb5adddf489b0b29e2dd6.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Citibike NY
ModelTemporalSpatial
Poisson Process0.609±0.012
Self-correcting Process-5.649±1.433
Hawkes Process1.062±0.000
Neural Hawkes Process 1.030±0.015
Conditional KDE-2.856±0.000
Time-varying CNF-2.132±0.012
Neural Jump SDE1.092±0.002-2.731±0.001
Jump CNF1.105±0.002-2.155±0.015
Attentive CNF1.112±0.002-2.095±0.006
", + "bbox": [ + 560, + 141, + 820, + 314 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/3b3f8ed141ab2f08bd1ebcfa696bac421996048d70ab7a1c592ac5145da7e4d0.jpg", + "image_caption": [ + "Figure 8: Runtime comparison of Jump and Attentive CNF. ", + "Table 2: Log-likelihood values on held-out test data for an urban mobility data set. " + ], + "image_footnote": [], + "bbox": [ + 192, + 156, + 540, + 309 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/39a14c8009ca62fbe1c850ddd66d48d4ee3b174b7ebe6492ce5e6ec129b8812d.jpg", + "image_caption": [ + "Figure 9: Both the Jump CNF and Attentive CNF are capable of modeling different the spatial distributions based on event history, so the appearance of a new event effectively shifts the distribution instantaneously. Shown on a synthetic 1-D data set similar to PINWHEEL, except we use a mixture of three Gaussians. Each event increases the likelihood of events for the cluster to the right. " + ], + "image_footnote": [], + "bbox": [ + 241, + 382, + 758, + 601 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/b85d5ea6427ba727b8c5be454da079d0dca9eaaee7f8cdf688b11c3f384509fc.jpg", + "image_caption": [ + "Figure 10: Learned attention weights for random event sequences. " + ], + "image_footnote": [], + "bbox": [ + 173, + 722, + 820, + 892 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B BASELINE", + "text_level": 1, + "bbox": [ + 174, + 102, + 295, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Our self-excitation baseline uses a Hawkes process to model the temporal variable, then uses a Gaussian mixture model to describe the spatial distribution conditioned on history of events. This corresponds to the following likelihood decomposition ", + "bbox": [ + 173, + 135, + 826, + 178 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/136e263f2ecf8be4ea53b2b1b176a5d7a92806a711093741476359ad3b0b2c71.jpg", + "text": "$$\n\\log p ( t _ { 1 } , \\dots , t _ { n } , x _ { 1 } , \\dots , x _ { n } ) = \\sum _ { i = 1 } ^ { n } \\log p ( x _ { i } | t _ { i } , t _ { 1 } , \\dots , t _ { i - 1 } , x _ { 1 } , \\dots , x _ { i - 1 } ) + \\sum _ { i = 1 } ^ { n } \\log p ( t _ { i } | t _ { 1 } , \\dots , t _ { i - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 185, + 839, + 227 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Note that $t _ { i }$ does not depend on the spatial variables associated with previous events. This dependence structure allows the usage of simple temporal point processes to model $t _ { i }$ , e.g. a Hawkes process, since temporal variables do not depend on the spatial information. The spatial distribution conditions all past events as well as the current time of occurance. ", + "bbox": [ + 174, + 238, + 828, + 295 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Our baseline model assumes a simple Gaussian conditional model, that new events are likely to appear near previous events. ", + "bbox": [ + 171, + 301, + 823, + 329 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/feabdb2a183d8d4a59042c16056b2eb557b46cbd77f6b90b5a949efa9d7563e5.jpg", + "text": "$$\n\\log p ( x _ { i } | t _ { i } , t _ { 1 } , \\dots , t _ { i - 1 } , x _ { 1 } , \\dots , x _ { i - 1 } ) = \\sum _ { j = 1 } ^ { i - 1 } \\alpha _ { j } \\mathcal { N } ( x _ { j } | \\sigma ^ { 2 } ) , \\quad \\alpha _ { j } = \\frac { \\exp \\{ ( t _ { j } - t _ { i } ) / \\tau \\} } { \\sum _ { j ^ { \\prime } = 1 } ^ { i - 1 } \\exp \\{ ( t _ { j ^ { \\prime } } - t _ { i } ) / \\tau \\} }\n$$", + "text_format": "latex", + "bbox": [ + 191, + 338, + 810, + 383 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This parameteric model has two learnable parameters: $\\sigma ^ { 2 }$ and $\\tau$ , which control the rate of decay in the spatial and temporal domains, respectively. ", + "bbox": [ + 176, + 393, + 823, + 422 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "However, this Gaussian spatial model assumes events are propagated in all directions equally and can only model local self-excitation behavior. These assumptions are often used for simplifications but are generally incorrect for many spatio-temporal data. To name a few, earthquakes occur more frequently along boundaries of tectonic plates, epidemics propagate along traffic routes, taxi demands saturate locally and change as customers move around. ", + "bbox": [ + 174, + 429, + 825, + 500 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C PRE-PROCESSING STEPS FOR EACH DATA SET ", + "text_level": 1, + "bbox": [ + 173, + 523, + 583, + 539 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "PINWHEEL We sample from a multivariate Hawkes process with 10 dimensions. We turn this into continuous spatial variables by assigning each dimension to a cluster from a “pinwheel” distribution, and sample from the corresponding cluster for each event. Number of events per sequences ranges between 4 to 108. ", + "bbox": [ + 174, + 555, + 825, + 611 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological Survey (2020). Starting from January 01, 1990, we created sequences with a gap of 7 days. Each sequence was of length 30 days. We ensured there was no contamination between train/val/test sets by removing intermediate sequences. We removed earthquakes from 2010 November to 2011 December, as these sequences were too long and only served as outliers in the data. This resulted in 950 training sequences, 50 validation sequences, and 50 test sequences. Number of events per sequence ranges between 18 to 543. ", + "bbox": [ + 173, + 617, + 825, + 729 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19 cases in the New Jersey state, from March to July of 2020. The data is aggregated at the county level, which we dequantize uniformly across the county. We also dequantize the temporal axis by assigning new cases uniformly within the day. Starting at March 15, and every 3 days, we took a 7 day length sequence. For each sequence, we sampled each event with a probability of 0.01. This was done 50 times per sequence. We ensured there was no contamination between train/val/test sets by removing intermediate sequences. This resulted in 1450 training sequences, 100 validation sequences, and 100 test sequences. Number of events per sequence ranges between 3 to 323. ", + "bbox": [ + 173, + 736, + 825, + 848 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "CITIBIKE Citibike is a bike sharing service in New York City. We treat the start of each trip as an event, and use the data from April to August of 2019. We split into sequences of length 1 day starting at $5 { : } 0 0 { \\mathrm { a m } }$ of each day. For each sequence, we subsampled with a probability of 0.005 per event, 20 times. This resulted in 2440 training sequences, 300 validation sequences, and 320 test sequences. Number of events per sequence ranges between 9 to 231. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/749703a2fc1655f14b5816bcd8926bc3db67af8d8051f198cc235bae973ff268.jpg", + "image_caption": [ + "Figure 11: Histograms of the number of events per sequence in each processed data set. " + ], + "image_footnote": [], + "bbox": [ + 210, + 104, + 787, + 390 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "BOLD5000 This consists of fMRI scans of four participants as they are given visual stimuli (Chang et al., 2019). We use the sessions of a single patient and for each run, we split into 3 sequences, treated individually. We converted brain responses into spatio-temporal events following the $\\mathbf { Z }$ -score thresholding approach in Tagliazucchi et al. 2016, Equation (2). We used a threshold of $\\gamma = 6 . 0$ . We split the data into 1050 training sequences, 150 val sequences, 220 test sequences. Number of events per sequence ranges between 6 to 1741. ", + "bbox": [ + 173, + 439, + 826, + 523 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Each data set contains sequences with highly variable number of events, with varying degrees of dependence between events, making them difficult to model with traditional point process models. We plot histograms showing the number of events per sequence in Figure 11. ", + "bbox": [ + 174, + 535, + 825, + 578 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D HYPERPARAMETERS CHOSEN AND TESTED ", + "text_level": 1, + "bbox": [ + 173, + 597, + 570, + 613 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For the time-varying, jump, and attentive CNF models, we parameterized the CNF drift as a multilayer perceptron (MLP) with dimensions $[ d - 6 4 - 6 4 - 6 4 - d ]$ , where $d$ is the number of spatial variables. We swept over activation functions between using softplus or a time-dependent Swish (Ramachandran et al., 2017). ", + "bbox": [ + 173, + 627, + 825, + 683 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/25a40ede27a0e706f5451b6c4a8c470341d5d0406ac302803cf29405c1acdd85.jpg", + "text": "$$\n\\mathrm { T i m e D e p e n d e n t S w i s h } ( t , z ) = h \\sigma ( \\beta ( t ) \\odot z )\n$$", + "text_format": "latex", + "bbox": [ + 352, + 683, + 645, + 699 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\sigma$ is the logistic sigmoid function, $\\odot$ is the Hadamard (element-wise) product, and $\\beta : \\mathbb { R } \\mathbb { R } _ { d _ { z } }$ is a MLP with widths $[ \\bar { 1 } - 6 4 - d _ { z } ]$ where $d _ { z }$ is the dimension of $z$ , using the softplus activation function. We ultimately decided on using the time-dependent Swish for all experiments. ", + "bbox": [ + 174, + 700, + 823, + 742 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We swept over the MLP for defining $f _ { h }$ for the continuous-time hidden state in eq. (11) using hidden widths of $[ 8 - 2 0 ]$ , $[ 3 2 - 3 2 ]$ , $[ 6 4 - 6 4 ]$ , $\\left[ 3 2 - 3 2 - 3 2 \\right]$ , and $[ 6 4 - 6 4 - 6 4 ]$ . The majority of models used $3 2 - 3 2$ as it provided enough flexibility while remaining easy to solve. We used the softplus activation function. We tried MLP for parameterizing the instantaneous change in eq. (12); however, it was too unstable for long sequences. We therefore switched to the GRU parameterization, which takes an input (new event), the hidden state at the time of event, and outputs a new hidden state. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We regularized the $L _ { 2 }$ norm of the hidden state drift with a strength of 1e-4, chosen from $\\{ 0$ , 1e-4, 1e-3, 1e- $\\langle 2 \\}$ . We optionally used optimal transport-inspired regularization from Finlay et al. (2020), which adds a Frobenius norm regularization to the gradient of the drift in addition to the $L _ { 2 }$ norm regularization, to the CNF models with a strength of $\\{ 0 \\}$ , 1e-4, 1e-3, 1e- $\\cdot 2 \\}$ . The Time-varying and Attentive CNF models did not require regularization and were mostly kept at 0, but the Jump CNF models benefited from some amount of regularization to avoid numerical instability. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "To model a non-trivial spatial distribution for the entire data interval, we shift the data interval to start at $t = 2$ for all CNF models. Thus the interval used for parameterizing the CNF is $[ 2 , T + 2 ]$ . Generally, the “time” variable is a dummy one; we can place the base distribution at any time, and we can choose any interval on the real line to be the data interval; this does not limit the model in any way. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For the Jump CNF, we used a composition of 4 radial flows (Rezende & Mohamed, 2015) to parameterize the instantaneous updates in eq. (17). All parameters of the radial flows were parameterized to be the output of a MLP that takes as input the hidden state at the time of the event (before the hidden state is updated based on the current event). The radial flows were initialized in such a way that the log determinant is near zero. ", + "bbox": [ + 174, + 180, + 825, + 251 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For the Attentive CNF, the drift function consists of ", + "bbox": [ + 174, + 257, + 514, + 271 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Time-dependent $\\mathrm { M L P } ( d - 6 4 - 6 4 ) 2 \\times ]$ MultiheadAttention Time-dependent MLP(64 − 64 − d) ", + "bbox": [ + 178, + 281, + 823, + 296 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the Time-dependent MLPs make use of the TimeDependentSwish. As was done in Vaswani et al. (2017), the multihead attention is used within a residual branch, except we swapped LayerNorm (Ba et al., 2016) with ActNorm (Kingma & Dhariwal, 2018) as LayerNorm has an unbounded Lipschitz and can be ill-suited for use in ODEs. We tested both standard multihead attention (Vaswani et al., 2017) and the Lipschitz multihead attention (Kim et al., 2020). The Lipschitz multihead attention typically produced similar validation NLL as the standard multihead attention but were more stable on multiple occassions. We therefore kept the Lipschitz multihead attention for all experiments. We additionally, use an auxiliary (non-attentive, simply with the two multihead attention layers removed) CNF to map from 0 (i.e. the time of base distribution) to the beginning of the data interval. ", + "bbox": [ + 173, + 305, + 825, + 431 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We initialized all Neural ODEs (for the hidden state and CNFs) with zero drift by initializing the weights and biases of the final layer to zero. ", + "bbox": [ + 174, + 438, + 821, + 465 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The log-likelihood values reported are after the spatial variables have been standardized using the empirical mean and standard deviation from the training set. ", + "bbox": [ + 174, + 473, + 821, + 501 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We train and test on log-likelihood (in nats) per event, which normalizes eq. (8) of each sequence by the number of events. ", + "bbox": [ + 174, + 507, + 820, + 536 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "All integrals were solved using Chen (2018) to within a relative and absolute tolerance of 1E-4 or 1E-6, chosen based on preliminary testing for convergence and stability. ", + "bbox": [ + 173, + 542, + 823, + 571 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Our implementation of the Neural Jump SDE shares the same continuous-time hidden state parameterization but uses a mixture of Gaussians as the spatial model. We used 5 mixtures, and a MLP that maps from the hidden state to the parameters of this mixture of Gaussians (the means, log standard deviations, and mixture coefficients). ", + "bbox": [ + 174, + 578, + 825, + 635 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E REMOVING CROSS-EVENT PARTIAL DERIVATIVES ", + "text_level": 1, + "bbox": [ + 174, + 655, + 614, + 670 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "This results in a lower-variance gradient estimator for training, and allows parallel computation of conditional log probabilities at test time. ", + "bbox": [ + 174, + 684, + 825, + 713 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We first summarily describe the attention mechanism. For an input $\\boldsymbol { X } \\in \\mathbb { R } ^ { n \\times d }$ representing the f $n$ variabled values $\\{ x _ { s } ^ { ( 0 ) } , \\ldots , x _ { s } ^ { ( n ) } \\}$ . . . , x(n)s } at some valudependent on f , $s$ , this attention mechanism creates logitsch that the output is $P \\in \\mathbb { R } ^ { n \\times n }$ $V \\in R ^ { n \\times d }$ $X$ ", + "bbox": [ + 173, + 719, + 825, + 765 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/62810ae4cfae7a92235c44b6047361745bc5eaf04e5c17c918f62d9620b35e1b.jpg", + "text": "$$\nO = \\underbrace { \\operatorname { s o f t m a x } ( P ) } _ { : = S } V .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 767, + 565, + 801 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the softmax is taken over each row of $P$ . The output is then added to $X$ as a residual connection. The multihead attention computes $P$ in a way such that $P _ { i j }$ depends on $X _ { i }$ and $X _ { j }$ , and $V _ { i }$ depends on $X _ { i }$ . This is true for both the vanilla MHA (Vaswani et al., 2017) and the L2 MHA (Kim et al., 2020). For our use case, $P _ { i j }$ is set to −inf for $j > i$ as we don’t want to attend to future events. ", + "bbox": [ + 173, + 801, + 825, + 859 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We retain only the block-diagonal gradients where each block contains variables corresponding to one event. This is equivalent to removing all the cross-event dependencies. ", + "bbox": [ + 173, + 864, + 821, + 893 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0c00dad3ae001cb5e66a5f532d2bb9061ba820ecd3682c1dc08f7833a3779150.jpg", + "text": "$$\n{ \\frac { \\partial { \\cal O } _ { i } } { \\partial X _ { i } } } = S _ { : , i } { \\frac { \\partial V _ { i } } { \\partial X _ { i } } } + V ^ { \\mathsf { T } } { \\frac { \\partial S } { \\partial P _ { i } } } { \\frac { \\partial P _ { i } } { \\partial X _ { i } } } + V ^ { \\mathsf { T } } { \\frac { \\partial S } { \\partial P _ { : , i } } } { \\frac { \\partial P _ { : , i } } { \\partial X _ { i } } }\n$$", + "text_format": "latex", + "bbox": [ + 333, + 895, + 665, + 928 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F PARALLEL SOLVING OF MULTIPLE ODES WITH VARYING INTERVALS ", + "text_level": 1, + "bbox": [ + 169, + 101, + 785, + 119 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Our numerical ODE solvers integrate a single ODE system $\\begin{array} { r } { \\frac { d x } { d t } = f ( t , x ) } \\end{array}$ , where $x \\in \\mathbb { R } ^ { d }$ and $f : \\mathbb { R } ^ { 1 + d } \\mathbb { R } ^ { d }$ , on a single fixed interval $[ t _ { s t a r t } , t _ { e n d } ]$ . We can express the inputs and outputs of an ODE solver with ", + "bbox": [ + 173, + 131, + 825, + 178 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/470e7c5d983116d82347219875c8b533069cb8f848555f7f4040d9e97d6ff996.jpg", + "text": "$$\n\\odot \\mathtt { D E S o l v e } ( x _ { 0 } , f , t _ { s t a r t } , t _ { e n d } ) \\triangleq x _ { 0 } + \\int _ { t _ { s t a r t } } ^ { t _ { e n d } } f ( t , x ( t ) ) d t = x ( t _ { e n d } ) .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 180, + 733, + 217 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $x _ { 0 }$ is a vector containing the initial state at the initial time $t _ { 0 }$ . ", + "bbox": [ + 174, + 222, + 616, + 237 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Multiple ODEs Now suppose we have a set of that we would like to solve. If all systems had the $M$ systems (i.e. e initial time $\\begin{array} { r } { { \\frac { d x _ { m } } { d t } } = f _ { m } } \\end{array}$ for outp $m = 1 , \\ldots , M )$ $t _ { s t a r t }$ and $t _ { e n d }$ , we can readily create a joint system ", + "bbox": [ + 173, + 251, + 828, + 295 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/b1875201c01747b468454623718f8f7e09fe86d90571c40cc7ee57513bb065e3.jpg", + "text": "$$\n\\boldsymbol { x } _ { j o i n t } = \\left[ \\begin{array} { c } { x _ { 1 } } \\\\ { \\vdots } \\\\ { x _ { M } } \\end{array} \\right] \\qquad \\mathrm { t h a t ~ f o l l o w s } \\quad \\frac { d x _ { j o i n t } } { d t } = \\left[ \\begin{array} { c } { f _ { 1 } ( t , x _ { 1 } ) } \\\\ { \\vdots } \\\\ { f _ { M } ( t , x _ { M } ) } \\end{array} \\right]\n$$", + "text_format": "latex", + "bbox": [ + 282, + 299, + 714, + 357 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Solving this joint system can be done in parallel with a single call to ODESolve: ", + "bbox": [ + 173, + 361, + 709, + 376 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/54fb8c05724d30688e3a2ce0d3a6e49351fc229d66b9b8354729e1caa67f8d91.jpg", + "text": "$$\nx _ { j o i n t } ( t _ { 1 } ) = \\mathsf { O D E S o l v e } ( x _ { 0 j o i n t } , f _ { j o i n t } , t _ { 0 } , t _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 339, + 381, + 658, + 398 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "which computes $x _ { m } ( t _ { 1 } )$ for all $m = 1 , \\ldots , M$ . This is the standard method used for solving a batch of Neural ODEs. ", + "bbox": [ + 174, + 402, + 823, + 433 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Adding dependencies is straightforward Note that extending this further, the ODE systems do not need to be independent. They can depend on other variables at the same concurrent time value because the joint system is still an ordinary differential equation. For instance, we can have ", + "bbox": [ + 173, + 446, + 825, + 489 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/85b6b39fff3a4257d1ea4e1de2ce4ba24118e00a15781c1efb67b4fc69a5ad21.jpg", + "text": "$$\n\\frac { d x _ { m } } { d t } = f _ { m } ( t , x _ { 1 } , \\ldots , x _ { m } )\n$$", + "text_format": "latex", + "bbox": [ + 408, + 494, + 589, + 525 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $f _ { m } : \\mathbb { R } ^ { 1 + M d } \\mathbb { R } ^ { d }$ . Each system is now a partial differential equation, but the joint system $x _ { j o i n t }$ is still an ODE and can be solved with one call to ODESolve. ", + "bbox": [ + 173, + 530, + 825, + 560 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Varied time intervals Now suppose each system has a different time interval that we want to solve. Different initial times and different end times. Let’s denote the start and end time for the $m$ -th system as t start and $t _ { e n d } ^ { ( m ) }$ respectively. We can construct a dummy variable that always integrates from 0 to 1, and perform a change of variables (reparameterization) to transform every system to use this dummy variable. ", + "bbox": [ + 173, + 574, + 826, + 647 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As a concrete example of this reparameterzation procedure, consider just one system $x ( t )$ with drift function $f ( t , x )$ that we want to integrate from $t _ { s t a r t }$ to $t _ { e n d }$ with the initial value $x _ { 0 }$ . We can transform $x ( t )$ using the relation $\\begin{array} { r } { s = \\frac { t - \\overline { { t } } _ { s t a r t } } { t _ { e n d } - t _ { s t a r t } } } \\end{array}$ , or equivalently ", + "bbox": [ + 174, + 654, + 825, + 700 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/288011842e72fb83c1a38f396181ed80317aa5ad0cd4d7b9f18995b652cc56b9.jpg", + "text": "$$\nt = s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 704, + 596, + 722 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "into a solution $\\tilde { { \\boldsymbol { x } } } ( s )$ on the unit interval $[ 0 , 1 ]$ such that ", + "bbox": [ + 173, + 726, + 531, + 742 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7b72355d317e45125776fef0ca946a6b1fa3742533c7c67c9f5c2814c258d123.jpg", + "text": "$$\n\\tilde { x } ( s ) = x ( s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } )\n$$", + "text_format": "latex", + "bbox": [ + 380, + 747, + 617, + 765 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The drift function for $\\tilde { x }$ then follows as ", + "bbox": [ + 173, + 768, + 431, + 784 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/b05fa8c06ab75857f7f0129f7fc4c38a7fcfa8fdeb6c53df125094d06896bdd1.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\tilde { f } ( s , \\tilde { x } ( s ) ) \\triangleq \\frac { d \\tilde { x } ( s ) } { d s } = \\frac { d x ( t ) } { d t } \\bigg \\rvert _ { t = s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } } \\frac { d t } { d s } } \\\\ { \\displaystyle = f ( t , x ( t ) ) \\bigg \\rvert _ { t = s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } } ( t _ { e n d } - t _ { s t a r t } ) } \\\\ { \\displaystyle = f \\big ( s ( t _ { e n d } - t _ { s t a r t } ) + t _ { s t a r t } , \\tilde { x } ( s ) \\big ) \\big ( t _ { e n d } - t _ { s t a r t } \\big ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 786, + 736, + 882 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Now since $\\tilde { x } ( 0 ) = x ( t _ { s t a r t } )$ and $\\tilde { x } ( 1 ) = x ( t _ { e n d } )$ , the following are equivalent ", + "bbox": [ + 173, + 885, + 686, + 901 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/64720316e7911927aae73eeabf16741bbf5c9cb79cfc8d293cf591ef8cc358a9.jpg", + "text": "$$\n\\boldsymbol { x } ( t _ { e n d } ) = \\boldsymbol { \\mathrm { O D E S O 1 v e } } ( x _ { 0 } , \\tilde { f } , 0 , 1 ) = \\boldsymbol { \\mathrm { O D E S O 1 v e } } ( x _ { 0 } , f , t _ { s t a r t } , t _ { e n d } )\n$$", + "text_format": "latex", + "bbox": [ + 271, + 906, + 725, + 925 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Putting it all together Let $\\widetilde { x } _ { m } ( s )$ be the reparameterized solution for $x _ { m } ( t )$ such that ", + "bbox": [ + 169, + 102, + 750, + 119 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/7aac81eade4bb7d0a843be908c9a2657480e25cb8a55c0985e64f77b599e4068.jpg", + "text": "$$\n\\tilde { x } _ { m } ( s ) = x _ { m } \\left( s \\left( t _ { e n d } ^ { ( m ) } - t _ { s t a r t } ^ { ( m ) } \\right) + t _ { s t a r t } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 357, + 125, + 640, + 152 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We can then solve for all $M$ systems, with different varying time intervals, using ", + "bbox": [ + 173, + 157, + 702, + 174 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/661def14f29944507a2598bbd2f562f1f8c8ea41d91328ee66e5579f65bbc512.jpg", + "text": "$$\n\\tilde { x } _ { j o i n t } = \\left[ \\begin{array} { l } { \\tilde { x } _ { 1 } } \\\\ { \\vdots } \\\\ { \\tilde { x } _ { M } } \\end{array} \\right] \\qquad \\quad \\mathrm { t h a t ~ f o l l o w s } \\quad \\frac { d \\tilde { x } _ { j o i n t } } { d s } = \\left[ \\begin{array} { l } { \\tilde { f } _ { 1 } ( s , \\tilde { x } _ { 1 } ) } \\\\ { \\vdots } \\\\ { \\tilde { f } _ { M } ( s , \\tilde { x } _ { M } ) } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 179, + 720, + 238 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Solving this system to $s = 1$ yields $\\tilde { x } _ { m } ( 1 ) = x _ { m } ( t _ { e n d } ^ { ( m ) } )$ ", + "bbox": [ + 174, + 246, + 540, + 265 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Assuming $t _ { s t a r t } ^ { ( m ) } = 0$ for all $m = 1 , \\ldots , M$ in order to reduce notational complexity, we can write this joint system in terms of the original systems as ", + "bbox": [ + 174, + 272, + 823, + 304 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/1d9c1c8a9e39030a28b440d03bab3a8bf823cab71b9fb2680b4e9d510e6d6072.jpg", + "text": "$$\n\\frac { d \\tilde { x } _ { j o i n t } } { d s } = \\left[ \\begin{array} { c } { f _ { 1 } \\left( s t _ { e n d } ^ { ( 1 ) } , \\tilde { x } ( s ) \\right) \\ t _ { e n d } } \\\\ { \\vdots } \\\\ { f _ { 1 } \\left( s t _ { e n d } ^ { ( 1 ) } , \\tilde { x } ( s ) \\right) \\ t _ { e n d } } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 309, + 624, + 383 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "This is the joint system written in equation 14. The joint system in equation 19 adds dependence between the $M$ systems but can still be solved with a single ODESolve. ", + "bbox": [ + 173, + 387, + 825, + 417 + ], + "page_idx": 18 + } +] \ No newline at end of file diff --git a/parse/train/XQQA6-So14/XQQA6-So14_middle.json b/parse/train/XQQA6-So14/XQQA6-So14_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..d1e2f109981f53f196d42473a94c8e9071c7c85c --- /dev/null +++ b/parse/train/XQQA6-So14/XQQA6-So14_middle.json @@ -0,0 +1,52585 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 78, + 458, + 96 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 460, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 460, + 99 + ], + "score": 1.0, + "content": "NEURAL SPATIO-TEMPORAL POINT PROCESSES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 115, + 267, + 148 + ], + "lines": [ + { + "bbox": [ + 112, + 114, + 192, + 127 + ], + "spans": [ + { + "bbox": [ + 112, + 114, + 192, + 127 + ], + "score": 1.0, + "content": "Ricky T. Q. 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Spatio-temporal point processes (STPPs) are a versatile and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "principled framework for modeling such event data and have, consequently, found many applications", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 506, + 395 + ], + "score": 1.0, + "content": "in a diverse range of fields. 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However, existing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 354, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 354, + 478 + ], + "score": 1.0, + "content": "parameterizations of STPPs are strongly restricted in this re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 476, + 354, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 354, + 489 + ], + "score": 1.0, + "content": "gard due to computational considerations: In its general form,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 353, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 353, + 501 + ], + "score": 1.0, + "content": "STPPs require solving multivariate integrals for computing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 353, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 353, + 510 + ], + "score": 1.0, + "content": "likelihood values and thus have primarily been studied within", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 510, + 353, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 353, + 521 + ], + "score": 1.0, + "content": "the context of different approximations and model restric-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 520, + 353, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 353, + 532 + ], + "score": 1.0, + "content": "tions. This includes, for instance, restricting the model class", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 530, + 354, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 354, + 545 + ], + "score": 1.0, + "content": "to parameterizations with known closed-form solutions (e.g.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 542, + 354, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 354, + 554 + ], + "score": 1.0, + "content": "exponential Hawkes processes (Ozaki, 1979)), to restrict de-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 552, + 354, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 354, + 566 + ], + "score": 1.0, + "content": "pendencies between the spatial and temporal domain (e.g.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 564, + 353, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 353, + 576 + ], + "score": 1.0, + "content": "independent and unpredictable marks (Daley & Vere-Jones,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 353, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 353, + 587 + ], + "score": 1.0, + "content": "2003)), or to discretize continuous time and space (Ogata,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 586, + 352, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 352, + 597 + ], + "score": 1.0, + "content": "1998). 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We build upon ideas of Neural Jump SDEs (Jia", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "& Benson, 2019) and Continuous-time Normalizing Flows (CNFs; Chen et al. 2018; Grathwohl", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "et al. 2019; Mathieu & Nickel 2020) to learn parametric models of spatial (or mark1) distributions", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "that are defined continuously in time. Normalizing flows are known to be flexible universal density", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "score": 1.0, + "content": "estimators (e.g. Huang et al. 2018; 2020; Teshima et al. 2020; Kong & Chaudhuri 2020) while retaining", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "computational tractability. As such, our approach allows the computation of exact likelihood values", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 163 + ], + "score": 1.0, + "content": "even for highly complex spatio-temporal distributions, and our models create smoothly changing", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "spatial distributions that naturally benefits spatio-temporal modeling. Central to our approach, are", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 182 + ], + "score": 1.0, + "content": "two novel neural architectures based on CNFs—using either discontinuous jumps in distribution or", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "self-attention—to condition spatial distributions on the event history. 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We build upon ideas of Neural Jump SDEs (Jia", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "& Benson, 2019) and Continuous-time Normalizing Flows (CNFs; Chen et al. 2018; Grathwohl", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "et al. 2019; Mathieu & Nickel 2020) to learn parametric models of spatial (or mark1) distributions", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "that are defined continuously in time. Normalizing flows are known to be flexible universal density", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "score": 1.0, + "content": "estimators (e.g. Huang et al. 2018; 2020; Teshima et al. 2020; Kong & Chaudhuri 2020) while retaining", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "computational tractability. As such, our approach allows the computation of exact likelihood values", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 163 + ], + "score": 1.0, + "content": "even for highly complex spatio-temporal distributions, and our models create smoothly changing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "spatial distributions that naturally benefits spatio-temporal modeling. Central to our approach, are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 182 + ], + "score": 1.0, + "content": "two novel neural architectures based on CNFs—using either discontinuous jumps in distribution or", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "self-attention—to condition spatial distributions on the event history. To the best of our knowledge,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "this is the first method that combines the flexibility of neural TPPs with the ability to learn high-fidelity", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "models of continuous marks that can have complex dependencies on the event history. In addition to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "our modeling contributions, we also construct five new pre-processed data sets for benchmarking", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 228, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 228, + 237 + ], + "score": 1.0, + "content": "spatio-temporal event models.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 107, + 252, + 200, + 265 + ], + "lines": [ + { + "bbox": [ + 104, + 251, + 201, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 251, + 201, + 268 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 507, + 291 + ], + "score": 1.0, + "content": "In the following, we give a brief overview of two core frameworks which our method builds upon,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 414, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 414, + 301 + ], + "score": 1.0, + "content": "i.e., spatio-temporal point processes and continuous-time normalizing flows.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "Event Modeling with Point Processes Spatio-temporal point processes are concerned with mod-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "eling sequences of random events in continuous space and time (Moller & Waagepetersen, 2003;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 331, + 507, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 331, + 224, + 349 + ], + "score": 1.0, + "content": "Baddeley et al., 2007). Let", + "type": "text" + }, + { + "bbox": [ + 225, + 334, + 303, + 346 + ], + "score": 0.92, + "content": "\\mathcal { H } = \\{ ( t _ { i } , \\pmb { x } _ { i } ) \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 331, + 454, + 349 + ], + "score": 1.0, + "content": "denote the sequence of event times", + "type": "text" + }, + { + "bbox": [ + 454, + 334, + 486, + 345 + ], + "score": 0.92, + "content": "t _ { i } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 331, + 507, + 349 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 212, + 358 + ], + "score": 1.0, + "content": "their associated locations", + "type": "text" + }, + { + "bbox": [ + 213, + 345, + 249, + 357 + ], + "score": 0.9, + "content": "\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 345, + 343, + 358 + ], + "score": 1.0, + "content": ", the number of events", + "type": "text" + }, + { + "bbox": [ + 344, + 348, + 351, + 356 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "being also random. 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Given", + "type": "text" + }, + { + "bbox": [ + 317, + 429, + 339, + 439 + ], + "score": 0.87, + "content": "i - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 427, + 504, + 440 + ], + "score": 1.0, + "content": "previous events, the conditional intensity", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 358, + 452 + ], + "score": 1.0, + "content": "function describes therefore the instantaneous probability of the", + "type": "text" + }, + { + "bbox": [ + 359, + 440, + 363, + 449 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 438, + 448, + 452 + ], + "score": 1.0, + "content": "-th event occurring at", + "type": "text" + }, + { + "bbox": [ + 448, + 441, + 453, + 449 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "and location", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 107, + 453, + 114, + 460 + ], + "score": 0.61, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 449, + 398, + 463 + ], + "score": 1.0, + "content": ". 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Gradients with respect to any parameters in", + "type": "text" + }, + { + "bbox": [ + 321, + 398, + 328, + 409 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 396, + 505, + 411 + ], + "score": 1.0, + "content": "can be computed with constant memory by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 403, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 403, + 420 + ], + "score": 1.0, + "content": "solving an adjoint ODE in reverse-time as described in Chen et al. (2018).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 435, + 370, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 370, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 370, + 451 + ], + "score": 1.0, + "content": "3 NEURAL SPATIO-TEMPORAL POINT PROCESSES", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "We are interested in modeling high-fidelity distributions in continuous time and space that can be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "updated based on new event information. 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This has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "been used (Grathwohl et al., 2019) to scale CNFs to higher dimensions using a Monte Carlo estimate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 231, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 231, + 348 + ], + "score": 1.0, + "content": "of the log likelihood objective,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 268, + 507, + 348 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 352, + 433, + 381 + ], + "lines": [ + { + "bbox": [ + 178, + 352, + 433, + 381 + ], + "spans": [ + { + "bbox": [ + 178, + 352, + 433, + 381 + ], + "score": 0.94, + "content": "\\log p ( { \\pmb x } _ { t } | t ) = \\log p ( { \\pmb x } _ { 0 } ) - \\mathbb { E } _ { \\pmb { v } \\sim \\mathcal { N } ( 0 , 1 ) } \\left[ \\int _ { 0 } ^ { t } v ^ { \\top } \\frac { \\partial f } { \\partial x } ( \\tau , { \\pmb x } _ { \\tau } ) v \\ d \\tau \\right] ,", + "type": "interline_equation", + "image_path": "27b2119dddad7d5f0b689a5ba12c90e460c76707f20980bc3a7a03168e050f83.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 178, + 352, + 433, + 361.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 178, + 361.6666666666667, + 433, + 371.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 178, + 371.33333333333337, + 433, + 381.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 249, + 398 + ], + "score": 1.0, + "content": "which, even if only one sample of", + "type": "text" + }, + { + "bbox": [ + 249, + 388, + 255, + 396 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "is used, is still amenable to training with stochastic gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 320, + 411 + ], + "score": 1.0, + "content": "descent. Gradients with respect to any parameters in", + "type": "text" + }, + { + "bbox": [ + 321, + 398, + 328, + 409 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 396, + 505, + 411 + ], + "score": 1.0, + "content": "can be computed with constant memory by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 403, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 403, + 420 + ], + "score": 1.0, + "content": "solving an adjoint ODE in reverse-time as described in Chen et al. (2018).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 386, + 505, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 435, + 370, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 370, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 370, + 451 + ], + "score": 1.0, + "content": "3 NEURAL SPATIO-TEMPORAL POINT PROCESSES", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "We are interested in modeling high-fidelity distributions in continuous time and space that can be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "updated based on new event information. For this purpose, we use the Neural ODE framework to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "parameterize a STPP by combining ideas from Neural Jump SDEs and Continuous Normalizing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 495, + 441, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 441, + 507 + ], + "score": 1.0, + "content": "Flows to create highly flexible models that still allow exact likelihood computation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 461, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 512, + 506, + 558 + ], + "lines": [ + { + "bbox": [ + 104, + 511, + 507, + 527 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 298, + 527 + ], + "score": 1.0, + "content": "We first (re-)introduce necessary notation. Let", + "type": "text" + }, + { + "bbox": [ + 298, + 511, + 367, + 527 + ], + "score": 0.93, + "content": "\\mathcal { H } = \\{ ( t _ { i } , \\pmb { x } _ { t _ { i } } ^ { ( i ) } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 511, + 507, + 527 + ], + "score": 1.0, + "content": "denote a sequence of event times", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 522, + 507, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 525, + 151, + 537 + ], + "score": 0.92, + "content": "t _ { i } \\in [ 0 , T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 522, + 210, + 540 + ], + "score": 1.0, + "content": "and locations", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 252, + 537 + ], + "score": 0.92, + "content": "\\pmb { x } _ { t _ { i } } ^ { ( i ) } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 522, + 461, + 540 + ], + "score": 1.0, + "content": "i. The superscript indicates an association with the", + "type": "text" + }, + { + "bbox": [ + 462, + 526, + 466, + 535 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 522, + 507, + 540 + ], + "score": 1.0, + "content": "-th event,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 534, + 507, + 548 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 237, + 548 + ], + "score": 1.0, + "content": "and the use of subscripting with", + "type": "text" + }, + { + "bbox": [ + 237, + 537, + 245, + 547 + ], + "score": 0.83, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 534, + 507, + 548 + ], + "score": 1.0, + "content": "will be useful later in the continuous-time modeling framework.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 546, + 471, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 471, + 560 + ], + "score": 1.0, + "content": "Following Daley & Vere-Jones (2003), we decompose the conditional intensity function as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 511, + 507, + 560 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 564, + 360, + 578 + ], + "lines": [ + { + "bbox": [ + 251, + 564, + 360, + 578 + ], + "spans": [ + { + "bbox": [ + 251, + 564, + 360, + 578 + ], + "score": 0.92, + "content": "\\lambda ^ { * } ( t , { \\pmb x } ) = \\lambda ^ { * } ( t ) p ^ { * } ( { \\pmb x } \\mid t )", + "type": "interline_equation", + "image_path": "cb917c645357ea3aff34b0b5723d772a8a0d236998b75ef0b259ec9993df83bd.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 251, + 564, + 360, + 578 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 133, + 596 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 583, + 156, + 596 + ], + "score": 0.92, + "content": "\\lambda ^ { * } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 583, + 393, + 596 + ], + "score": 1.0, + "content": "is the ground intensity of the temporal process and where", + "type": "text" + }, + { + "bbox": [ + 393, + 583, + 430, + 596 + ], + "score": 0.93, + "content": "\\boldsymbol { p } ^ { * } ( \\boldsymbol { x } \\mid t )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "is the conditional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 174, + 606 + ], + "score": 1.0, + "content": "density of a mark", + "type": "text" + }, + { + "bbox": [ + 175, + 596, + 182, + 605 + ], + "score": 0.81, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 594, + 192, + 606 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 192, + 596, + 197, + 605 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 594, + 221, + 606 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 221, + 595, + 234, + 606 + ], + "score": 0.87, + "content": "\\mathcal { H } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 594, + 504, + 606 + ], + "score": 1.0, + "content": ". 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In the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "following, we will first describe how we construct a latent dynamics model, which we use to compute", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 190, + 129 + ], + "score": 1.0, + "content": "the ground intensity", + "type": "text" + }, + { + "bbox": [ + 190, + 115, + 213, + 127 + ], + "score": 0.91, + "content": "\\lambda ^ { * } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 114, + 505, + 129 + ], + "score": 1.0, + "content": ". 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This provides us", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 222, + 325 + ], + "score": 1.0, + "content": "with a vector representation", + "type": "text" + }, + { + "bbox": [ + 222, + 313, + 234, + 324 + ], + "score": 0.88, + "content": "\\boldsymbol { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 312, + 315, + 325 + ], + "score": 1.0, + "content": "at every time value", + "type": "text" + }, + { + "bbox": [ + 315, + 313, + 320, + 322 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "that acts as both a summary of the history of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 382, + 336 + ], + "score": 1.0, + "content": "events and as a predictor of future behavior. 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(10) to (12) has been used", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "for time series modeling (Rubanova et al., 2019; De Brouwer et al., 2019) as well as TPPs (Jia &", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 242, + 529 + ], + "score": 1.0, + "content": "Benson, 2019). 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(7), we can derive separate models for the ground intensity and conditional", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "mark density which will be jointly conditioned on a continuous-time hidden state with jumps. In the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "following, we will first describe how we construct a latent dynamics model, which we use to compute", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 190, + 129 + ], + "score": 1.0, + "content": "the ground intensity", + "type": "text" + }, + { + "bbox": [ + 190, + 115, + 213, + 127 + ], + "score": 0.91, + "content": "\\lambda ^ { * } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 114, + 505, + 129 + ], + "score": 1.0, + "content": ". 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We will first describe an unconditional model, which is already", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "a strong baseline when spatial event distributions only follow temporal patterns and there is little to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "no correlation between the spatial observations. We then devise two new methods of conditioning on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 175, + 172 + ], + "score": 1.0, + "content": "the event history", + "type": "text" + }, + { + "bbox": [ + 175, + 160, + 185, + 169 + ], + "score": 0.76, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 159, + 506, + 172 + ], + "score": 1.0, + "content": ": one explicitly modeling instantaneous changes in distribution, and another that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 382, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 382, + 182 + ], + "score": 1.0, + "content": "uses an attention mechanism which is more amenable to parallelism.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 506, + 182 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 507, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 391, + 206 + ], + "score": 1.0, + "content": "Latent Dynamics and Ground Intensity For the temporal variables", + "type": "text" + }, + { + "bbox": [ + 391, + 194, + 408, + 206 + ], + "score": 0.91, + "content": "\\{ t _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 192, + 507, + 206 + ], + "score": 1.0, + "content": ", parameterize the inten-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "sity function using hidden state dynamics with jumps, similar to the work of Jia & Benson (2019).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 366, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 325, + 227 + ], + "score": 1.0, + "content": "Specifically, we evolve a continuous-time hidden state", + "type": "text" + }, + { + "bbox": [ + 325, + 216, + 333, + 226 + ], + "score": 0.82, + "content": "^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 216, + 366, + 227 + ], + "score": 1.0, + "content": "and set", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 192, + 507, + 227 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 232, + 385, + 246 + ], + "lines": [ + { + "bbox": [ + 226, + 232, + 385, + 246 + ], + "spans": [ + { + "bbox": [ + 226, + 232, + 385, + 246 + ], + "score": 0.9, + "content": "\\lambda ^ { * } ( t ) = g _ { \\lambda } ( h _ { t } ) \\qquad \\mathrm { ( G r o u n d i n t e n s i t y ) }", + "type": "interline_equation", + "image_path": "03da588f0e82dd714dad317c7dbd620fb94df538199bdca04d0ce0f5f742eefe.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 226, + 232, + 385, + 246 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 506, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 132, + 264 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 254, + 144, + 263 + ], + "score": 0.84, + "content": "g _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 250, + 505, + 264 + ], + "score": 1.0, + "content": "is a neural network with a softplus nonlinearity applied to the output, to ensure the intensity", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "is positive. We then capture conditional dependencies through the use of a continuously changing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 374, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 127, + 286 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 127, + 274, + 138, + 285 + ], + "score": 0.88, + "content": "\\boldsymbol { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 274, + 374, + 286 + ], + "score": 1.0, + "content": "with instantaneous updates when conditioned on an event.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 250, + 506, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "The architecture is analogous to a recurrent neural network with a continuous-time hidden state (Mei", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "& Eisner, 2017; Che et al., 2018; Rubanova et al., 2019) modeled by a Neural ODE. This provides us", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 222, + 325 + ], + "score": 1.0, + "content": "with a vector representation", + "type": "text" + }, + { + "bbox": [ + 222, + 313, + 234, + 324 + ], + "score": 0.88, + "content": "\\boldsymbol { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 312, + 315, + 325 + ], + "score": 1.0, + "content": "at every time value", + "type": "text" + }, + { + "bbox": [ + 315, + 313, + 320, + 322 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "that acts as both a summary of the history of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 382, + 336 + ], + "score": 1.0, + "content": "events and as a predictor of future behavior. Instantaneous updates to", + "type": "text" + }, + { + "bbox": [ + 383, + 324, + 394, + 334 + ], + "score": 0.88, + "content": "\\boldsymbol { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "allow to incorporate abrupt", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "changes to the hidden state that are triggered by observed events. This mechanism is important for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "modeling point processes and allows past events to influence future dynamics in a discontinuous way", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 356, + 296, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 296, + 368 + ], + "score": 1.0, + "content": "(e.g., modeling immediate shocks to a system).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 289, + 506, + 368 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 503, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 138, + 385 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 138, + 373, + 149, + 385 + ], + "score": 0.89, + "content": "f _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 372, + 396, + 385 + ], + "score": 1.0, + "content": "to model the continuous change in the form of an ODE and", + "type": "text" + }, + { + "bbox": [ + 396, + 375, + 407, + 385 + ], + "score": 0.85, + "content": "g _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "to model instantaneous", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 384, + 255, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 255, + 396 + ], + "score": 1.0, + "content": "changes based on an observed event.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 372, + 505, + 396 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 399, + 470, + 463 + ], + "lines": [ + { + "bbox": [ + 140, + 399, + 470, + 463 + ], + "spans": [ + { + "bbox": [ + 140, + 399, + 470, + 463 + ], + "score": 0.8, + "content": "{ \\begin{array} { r l r l } & { h _ { t _ { 0 } } = h _ { 0 } } & & { ( { \\mathrm { A n ~ i n i t i a l ~ h i d d e n ~ s t a t e } } ) } \\\\ & { { \\frac { d h _ { t } } { d t } } = f _ { h } ( t , h _ { t } ) } & & { { \\mathrm { b e t w e e n ~ e v e n t ~ t i m e s } } } & & { ( { \\mathrm { C o n t i n u o u s ~ e v o l u t i o n } } ) } \\\\ & { \\operatorname* { l i m } _ { \\varepsilon \\to 0 } h _ { t _ { i } + \\varepsilon } = g _ { h } \\left( t _ { i } , h _ { t _ { i } } , x _ { t _ { i } } ^ { ( i ) } \\right) } & & { { \\mathrm { a t ~ e v e n t ~ t i m e s ~ } } t _ { i } } & & { ( { \\mathrm { I n s t a n t a n e o u s ~ u p d a t e s } } ) } \\end{array} }", + "type": "interline_equation", + "image_path": "0394f6b4bbd83939cd06ce26581c4cbed5ade52df0c2d8ae6559ab9f75be55d1.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 140, + 399, + 470, + 420.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 140, + 420.3333333333333, + 470, + 441.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 140, + 441.66666666666663, + 470, + 462.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 466, + 504, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 151, + 480 + ], + "score": 1.0, + "content": "The use of", + "type": "text" + }, + { + "bbox": [ + 152, + 470, + 158, + 477 + ], + "score": 0.78, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 467, + 228, + 480 + ], + "score": 1.0, + "content": "is to portray that", + "type": "text" + }, + { + "bbox": [ + 229, + 468, + 240, + 478 + ], + "score": 0.89, + "content": "h _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "is a cagl ` ad` function, i.e. left-continuous with right limits, with a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 477, + 252, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 236, + 492 + ], + "score": 1.0, + "content": "discontinuous jump modeled by", + "type": "text" + }, + { + "bbox": [ + 236, + 480, + 247, + 490 + ], + "score": 0.85, + "content": "g _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 477, + 252, + 492 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 467, + 505, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "The parameterization of continuous-time hidden states in the form of eqs. (10) to (12) has been used", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "for time series modeling (Rubanova et al., 2019; De Brouwer et al., 2019) as well as TPPs (Jia &", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 242, + 529 + ], + "score": 1.0, + "content": "Benson, 2019). We parameterize", + "type": "text" + }, + { + "bbox": [ + 242, + 518, + 253, + 529 + ], + "score": 0.88, + "content": "f _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "as a standard multi-layer fully connected neural network, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 497, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 325, + 541 + ], + "score": 1.0, + "content": "use the GRU update (Cho et al., 2014) to parameterize", + "type": "text" + }, + { + "bbox": [ + 326, + 530, + 336, + 540 + ], + "score": 0.84, + "content": "g _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 528, + 497, + 541 + ], + "score": 1.0, + "content": ", as was done in Rubanova et al. (2019).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 494, + 506, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 506, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "Time-varying CNF The first model we consider is a straightforward application of the CNF to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 561, + 493, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 493, + 576 + ], + "score": 1.0, + "content": "time-variable observations. Assuming that the spatial distribution is independent of prior events,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 551, + 505, + 576 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 579, + 455, + 608 + ], + "lines": [ + { + "bbox": [ + 158, + 579, + 455, + 608 + ], + "spans": [ + { + "bbox": [ + 158, + 579, + 455, + 608 + ], + "score": 0.93, + "content": "\\log p ^ { * } ( \\pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) = \\log p ( \\pmb { x } _ { t _ { i } } ^ { ( i ) } | t _ { i } ) = \\log p ( \\pmb { x } _ { 0 } ^ { ( i ) } ) - \\int _ { 0 } ^ { t _ { i } } \\mathrm { t r } \\left( \\frac { \\partial f } { \\partial x } ( \\tau , \\pmb { x } _ { \\tau } ^ { ( i ) } ) \\right) d \\tau", + "type": "interline_equation", + "image_path": "2b6fe850c2d807b16e547aeeb5ebe8e91a44a27b5cc4aa04921882ef32802565.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 158, + 579, + 455, + 588.6666666666666 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 158, + 588.6666666666666, + 455, + 598.3333333333333 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 158, + 598.3333333333333, + 455, + 607.9999999999999 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 502, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 132, + 629 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 613, + 150, + 627 + ], + "score": 0.92, + "content": "\\pmb { x } _ { \\tau } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 612, + 354, + 630 + ], + "score": 1.0, + "content": ") is the solution of the ODE f with initial value x(i)t ,", + "type": "text" + }, + { + "bbox": [ + 351, + 613, + 475, + 630 + ], + "score": 1.0, + "content": "the observed event location, at", + "type": "text" + }, + { + "bbox": [ + 476, + 617, + 502, + 627 + ], + "score": 0.88, + "content": "\\tau = t _ { i }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "the observed event time. The spatial distribution of an event modeled by a Time-varying CNF changes", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "with respect to the time it occurs. Some spatio-temporal data sets exhibit mostly temporal patterns and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 647, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 663 + ], + "score": 1.0, + "content": "little to no dependence on previous events in the spatial domain, which would make a time-varying", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "CNF a good fit. Nevertheless, this model lacks the ability to capture spatial propagation effects, as it", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 310, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 310, + 683 + ], + "score": 1.0, + "content": "does not condition on previous event observations.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 612, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "A major benefit of this model is the ability to evaluate the joint log-likelihood fully in parallel across", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "events, since there are no dependencies between events. 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This incurs a substantial cost", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 662, + 250, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 250, + 677 + ], + "score": 1.0, + "content": "when the number of events is large.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 507, + 700 + ], + "score": 1.0, + "content": "Attentive CNF To design a spatial model with conditional dependencies that alleviates the compu-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "tational issues of Jump CNFs and can be computed in parallel, we make use of efficient attention", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "mechanisms based on the Transformer architecture (Vaswani et al., 2017). 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For this purpose, we define continuous-time spatial distributions by making again use of two", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 262, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 506, + 274 + ], + "score": 1.0, + "content": "components: (i) a continuous-time normalizing flow that evolves the distribution continuously, and (ii)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 271, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 506, + 286 + ], + "score": 1.0, + "content": "a standard (discrete-time) flow model that changes the distribution instantaneously after conditioning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 295 + ], + "score": 1.0, + "content": "on new events. As normalizing flows parameterize distributions through transformations of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "samples, these continuous- and discrete-time transformations are composable in a straightforward", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 306, + 274, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 274, + 317 + ], + "score": 1.0, + "content": "manner and are end-to-end differentiable.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 228, + 506, + 317 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 322, + 379, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 380, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 380, + 338 + ], + "score": 1.0, + "content": "The generative process of a single event in a Jump CNF is given by:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 320, + 380, + 338 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 338, + 467, + 399 + ], + "lines": [ + { + "bbox": [ + 142, + 338, + 467, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 338, + 467, + 399 + ], + "score": 0.88, + "content": "\\begin{array} { r l r l } { x _ { 0 } \\sim p ( x _ { 0 } ) } & { } & & { \\mathrm { ( A n i n i t i a l ~ d i s t r i b u t i o n ) } } \\\\ { \\displaystyle \\frac { d x _ { t } } { d t } = f _ { x } ( t , x _ { t } , h _ { t } ) } & { \\mathrm { ~ b e t w e e n ~ e v e n t ~ t i m e s ~ } } & & { \\mathrm { ( C o n t i n u o u s ~ e v o l u t i o n ) } } \\\\ { \\displaystyle \\operatorname* { l i m } _ { \\varepsilon 0 } x _ { t _ { i } + \\varepsilon } = g _ { x } ( t _ { i } , x _ { t _ { i } } , h _ { t _ { i } } ) } & { \\quad \\mathrm { a t ~ e v e n t ~ t i m e s ~ } t _ { i } } & & { \\mathrm { ( I n s t a n t a n e o u s ~ u p d a t e s ) } } \\end{array}", + "type": "interline_equation", + "image_path": "e972a7159e4559150a0d904c385f299acffa2c620f52a8dabe4f58eb7411d32a.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 142, + 338, + 467, + 358.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 142, + 358.3333333333333, + 467, + 378.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 142, + 378.66666666666663, + 467, + 398.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "The initial distribution can be parameterized by a normalizing flow. In practice, we set a base", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 292, + 427 + ], + "score": 1.0, + "content": "distribution at a negative time value and model", + "type": "text" + }, + { + "bbox": [ + 293, + 415, + 317, + 426 + ], + "score": 0.92, + "content": "p ( \\pmb { x } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 414, + 472, + 427 + ], + "score": 1.0, + "content": "using the same CNF parameterized by", + "type": "text" + }, + { + "bbox": [ + 472, + 415, + 483, + 426 + ], + "score": 0.88, + "content": "f _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 414, + 505, + 427 + ], + "score": 1.0, + "content": ". 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This conditioning on", + "type": "text" + }, + { + "bbox": [ + 267, + 437, + 281, + 448 + ], + "score": 0.9, + "content": "h _ { t _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "is required for the continuous and instantaneous updates", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 447, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 504, + 459 + ], + "score": 1.0, + "content": "to depend on the history of observations. Otherwise, a Jump CNF would only be able to model the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 458, + 493, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 425, + 471 + ], + "score": 1.0, + "content": "marginal distribution and behave similarly to a time-varying CNF. 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Similar to eq. (14), an Attention CNF can now", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 493, + 415 + ], + "score": 1.0, + "content": "solve for the trajectories of all events in parallel while simultaneously depending non-trivially on", + "type": "text" + }, + { + "bbox": [ + 493, + 403, + 503, + 412 + ], + "score": 0.78, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 402, + 506, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 171, + 433 + ], + "score": 1.0, + "content": "To parameterize", + "type": "text" + }, + { + "bbox": [ + 171, + 420, + 190, + 431 + ], + "score": 0.9, + "content": "f _ { \\mathrm { A t t } \\mathrm { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 419, + 506, + 433 + ], + "score": 1.0, + "content": ", we use an embedding layer followed by two multihead attention (MHA) blocks", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "and an output layer to map back into the input space. We use the Lipschitz-continuous multihead atten-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "tion from Kim et al. (2020) as they recently showed that the dot product multihead attention (Vaswani", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 478, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 478, + 464 + ], + "score": 1.0, + "content": "et al., 2017) is not Lipschitz-continuous and thus may be ill-suited for parameterizing ODEs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 325, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 326, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 326, + 487 + ], + "score": 1.0, + "content": "Low-variance Log-likelihood Estimation The vari-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 486, + 326, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 326, + 498 + ], + "score": 1.0, + "content": "ance of the Hutchinson stochastic trace estimator", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 497, + 325, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 325, + 510 + ], + "score": 1.0, + "content": "in eq. (6) grows with the squared Frobenius norm of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 506, + 327, + 527 + ], + "spans": [ + { + "bbox": [ + 103, + 506, + 160, + 527 + ], + "score": 1.0, + "content": "the Jacobian,", + "type": "text" + }, + { + "bbox": [ + 160, + 508, + 215, + 524 + ], + "score": 0.94, + "content": "\\sum _ { i j } \\left[ \\partial f / \\partial x \\right] _ { i j } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 506, + 327, + 527 + ], + "score": 1.0, + "content": "(Hutchinson, 1990). For at-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 325, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 325, + 534 + ], + "score": 1.0, + "content": "tentive CNFs, we can remove some of the non-diagonal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 326, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 326, + 545 + ], + "score": 1.0, + "content": "elements of the Jacobian and achieve a lower variance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 326, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 326, + 556 + ], + "score": 1.0, + "content": "estimator. 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This effectively allows us to apply", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 456, + 633 + ], + "score": 1.0, + "content": "Hutchinson’s estimator on a matrix that has the same diagonal elements as the Jacobian", + "type": "text" + }, + { + "bbox": [ + 456, + 621, + 479, + 633 + ], + "score": 0.88, + "content": "\\partial f / \\partial x", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "—and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "thus has the same expected value—but has zeros outside of the block-diagonal, leading to a lower", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "variance trace estimator. The procedure consists of selectively removing partial derivatives and is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "straightforward but notationally cumbersome; the interested reader can find the details in Appendix E.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 671, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "This is similar in spirit to Chen & Duvenaud (2019) but instead of constructing a neural network", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "that specifically allows cheap removal of partial derivatives, we make use of the fact that multihead", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 694, + 404, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 404, + 704 + ], + "score": 1.0, + "content": "attention already allows cheap removal of (cross-event) partial derivatives.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "An ablation experiment is shown in Figure 3 for training on the PINWHEEL data set, where the lower", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 720, + 500, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 500, + 732 + ], + "score": 1.0, + "content": "variance estimates (and gradients) ultimate led to faster convergence and better converged models.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 114, + 83, + 496, + 173 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 83, + 496, + 173 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 83, + 496, + 173 + ], + "spans": [ + { + "bbox": [ + 114, + 83, + 496, + 173 + ], + "score": 0.97, + "type": "image", + "image_path": "63a02fc2549e8b7f69997b935dc6ed8a77774c252453939a052db56ee3e48cab.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 83, + 496, + 113.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 113.0, + 496, + 143.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 143.0, + 496, + 173.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 182, + 505, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "Figure 2: Visualization of the sampling paths of Neural STPP models for a 1-D spatio-temporal data set where", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 192, + 136, + 204 + ], + "score": 0.92, + "content": "\\{ t _ { i } \\} _ { i = 1 } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "are event times. 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Similar to eq. (14), an Attention CNF can now", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 493, + 415 + ], + "score": 1.0, + "content": "solve for the trajectories of all events in parallel while simultaneously depending non-trivially on", + "type": "text" + }, + { + "bbox": [ + 493, + 403, + 503, + 412 + ], + "score": 0.78, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 402, + 506, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 102, + 377, + 506, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 171, + 433 + ], + "score": 1.0, + "content": "To parameterize", + "type": "text" + }, + { + "bbox": [ + 171, + 420, + 190, + 431 + ], + "score": 0.9, + "content": "f _ { \\mathrm { A t t } \\mathrm { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 419, + 506, + 433 + ], + "score": 1.0, + "content": ", we use an embedding layer followed by two multihead attention (MHA) blocks", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "and an output layer to map back into the input space. We use the Lipschitz-continuous multihead atten-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "tion from Kim et al. (2020) as they recently showed that the dot product multihead attention (Vaswani", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 478, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 478, + 464 + ], + "score": 1.0, + "content": "et al., 2017) is not Lipschitz-continuous and thus may be ill-suited for parameterizing ODEs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 419, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 325, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 326, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 326, + 487 + ], + "score": 1.0, + "content": "Low-variance Log-likelihood Estimation The vari-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 486, + 326, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 326, + 498 + ], + "score": 1.0, + "content": "ance of the Hutchinson stochastic trace estimator", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 497, + 325, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 325, + 510 + ], + "score": 1.0, + "content": "in eq. (6) grows with the squared Frobenius norm of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 506, + 327, + 527 + ], + "spans": [ + { + "bbox": [ + 103, + 506, + 160, + 527 + ], + "score": 1.0, + "content": "the Jacobian,", + "type": "text" + }, + { + "bbox": [ + 160, + 508, + 215, + 524 + ], + "score": 0.94, + "content": "\\sum _ { i j } \\left[ \\partial f / \\partial x \\right] _ { i j } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 506, + 327, + 527 + ], + "score": 1.0, + "content": "(Hutchinson, 1990). For at-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 325, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 325, + 534 + ], + "score": 1.0, + "content": "tentive CNFs, we can remove some of the non-diagonal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 326, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 326, + 545 + ], + "score": 1.0, + "content": "elements of the Jacobian and achieve a lower variance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 326, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 326, + 556 + ], + "score": 1.0, + "content": "estimator. The attention mechanism creates a block-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 326, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 326, + 567 + ], + "score": 1.0, + "content": "triangular Jacobian, where each block corresponds", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 565, + 326, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 326, + 577 + ], + "score": 1.0, + "content": "to one event, but the elements outside of the block-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 325, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 325, + 589 + ], + "score": 1.0, + "content": "diagonal are solely due to the multihead attention. 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This effectively allows us to apply", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 456, + 633 + ], + "score": 1.0, + "content": "Hutchinson’s estimator on a matrix that has the same diagonal elements as the Jacobian", + "type": "text" + }, + { + "bbox": [ + 456, + 621, + 479, + 633 + ], + "score": 0.88, + "content": "\\partial f / \\partial x", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "—and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "thus has the same expected value—but has zeros outside of the block-diagonal, leading to a lower", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "variance trace estimator. The procedure consists of selectively removing partial derivatives and is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "straightforward but notationally cumbersome; the interested reader can find the details in Appendix E.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 609, + 506, + 666 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 671, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "This is similar in spirit to Chen & Duvenaud (2019) but instead of constructing a neural network", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "that specifically allows cheap removal of partial derivatives, we make use of the fact that multihead", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 694, + 404, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 404, + 704 + ], + "score": 1.0, + "content": "attention already allows cheap removal of (cross-event) partial derivatives.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 671, + 505, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "An ablation experiment is shown in Figure 3 for training on the PINWHEEL data set, where the lower", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 720, + 500, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 500, + 732 + ], + "score": 1.0, + "content": "variance estimates (and gradients) ultimate led to faster convergence and better converged models.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5, + "bbox_fs": [ + 106, + 710, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 210, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 213, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 213, + 95 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "Neural Temporal Point Processes Modeling real-world data using restricted models such as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "Exponential Hawkes Processes (Ozaki, 1979) may lead to poor results due to model mis-specification.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "While this has led to many works on improving the Hawkes process (e.g. Linderman & Adams 2014;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "Li & Zha 2014; Zhao et al. 2015; Farajtabar et al. 2017; Li & Ke 2020; Nickel & Le 2020), recent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "works have begun to explore neural network parameterizations of TPPs. A common approach is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "to use recurrent neural networks to accumulate the event history in a latent state from which the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "intensity value can then be derived. Models of this form include, for instance, Recurrent Marked", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "score": 1.0, + "content": "Temporal Point Processes (RMTPPs; Du et al. 2016) and Neural Hawkes Processes (NHPs; Mei &", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "Eisner 2017). In contrast to our approach, these methods can not compute the exact likelihood of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "model and have to resort to Monte-Carlo sampling for its approximation. However, this approach", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "is especially problematic for commonly occurring clustered and bursty event sequences as it either", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "requires a very high sampling rate or ignores important temporal dependencies (Nickel & Le, 2020).", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "To overcome this issue, Jia & Benson (2019) proposed Neural Jump SDEs which extend the Neural", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "ODE framework and allow to compute the exact likelihood for neural TPPs, up to numerical errors.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "This method is closely related to our approach and we build on its ideas to compute the ground", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "intensity of the STPP. However, current Neural Jump SDEs —as well as NHPs and RMTPPs—are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "not well-suited for modeling complex continuous mark distributions as they are restricted to methods", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "such as Gaussian mixture models in the spatial domain. Finally, Shchur et al. (2019) and Mehrasa", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "et al. (2019) considered combining TPPs with flexible likelihood-based models, however for different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 315, + 460, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 460, + 327 + ], + "score": 1.0, + "content": "purposes as in our case, i.e., for intensity-free learning of only temporal point processes.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 504, + 349 + ], + "score": 1.0, + "content": "Continuous Normalizing Flows The ability to describe an infinite number of distributions with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "score": 1.0, + "content": "a Continuous Normalizing Flow has been used by a few recent works. Some works in computer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "score": 1.0, + "content": "graphics have used the interpolation effect of CNFs to model transformations of point clouds (Yang", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "et al., 2019; Rempe et al., 2020; Li et al., 2020). CNFs have also been used in sequential latent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 381, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 393 + ], + "score": 1.0, + "content": "variable models (Deng et al., 2020; Rempe et al., 2020). However, such works do not align the “time”", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "axis of the CNF with the temporal axis of observations, and do not train on observations at more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "than one value of “time” in the CNF. In contrast, we align the time axis of the CNF with the time", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "of the observations, directly using its ability to model distributions on a real-valued axis. A closely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "related application of CNFs to spatio-temporal data was done by Tong et al. (2020), who modeled the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "distribution of cells in a developing human embryo system at five fixed time values. In contrast to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "score": 1.0, + "content": "this, we extend to applications where observations are made at arbitrary time values, jointly modeling", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "space and time within the spatio-temporal point process framework. Furthermore, Mathieu & Nickel", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 481 + ], + "score": 1.0, + "content": "(2020); Lou et al. (2020) recently proposed extensions of CNFs to Riemannian manifolds. For our", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "proposed approach, this is especially interesting in the context of earth and climate science, as it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 491, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 502 + ], + "score": 1.0, + "content": "allows us to model STPPs on the sphere simply by replacing the CNF with its Riemannian equivalent.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 200, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 201, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 201, + 532 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "Data Sets Many collected data can be represented within the framework of spatio-temporal events.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 507, + 567 + ], + "score": 1.0, + "content": "We pre-process data from open sources and make them suitable for spatio-temporal event modeling.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "Each sequence in these data sets can contain up to thousands of variables, all the while having a large", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "variance in sequence lengths. Varying across a wide range of domains, the data sets we consider are:", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "earthquakes, pandemic spread, consumer demand for a bike sharing app, and high-amplitude brain", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 596, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 506, + 612 + ], + "score": 1.0, + "content": "signals from fMRI scans. We briefly describe these data sets here; further details, pre-processing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "steps, and data set diagnostics can be found in Appendix C. Code for preprocessing and training are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 619, + 457, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 457, + 633 + ], + "score": 1.0, + "content": "open sourced at https://github.com/facebookresearch/neural_stpp.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 108, + 640, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "PINWHEEL This is a synthetic data set with multimodal and non-Gaussian spatial distributions", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "designed to test the ability to capture drastic changes due to event history (see fig. 5). The data set", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "consists of 10 clusters which form a pinwheel structure. Events are sampled from a multivariate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "score": 1.0, + "content": "Hawkes process such that events from one cluster will increase the probability of observing events in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 684, + 503, + 697 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 503, + 697 + ], + "score": 1.0, + "content": "the next cluster in a clock-wise rotation. Number of events per sequences ranges between 4 to 108.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 720, + 409, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 409, + 734 + ], + "score": 1.0, + "content": "Survey (2020). Number of events per sequences ranges between 18 to 543.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 210, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 213, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 213, + 95 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "Neural Temporal Point Processes Modeling real-world data using restricted models such as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "Exponential Hawkes Processes (Ozaki, 1979) may lead to poor results due to model mis-specification.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "While this has led to many works on improving the Hawkes process (e.g. Linderman & Adams 2014;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "Li & Zha 2014; Zhao et al. 2015; Farajtabar et al. 2017; Li & Ke 2020; Nickel & Le 2020), recent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "works have begun to explore neural network parameterizations of TPPs. A common approach is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "to use recurrent neural networks to accumulate the event history in a latent state from which the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "intensity value can then be derived. Models of this form include, for instance, Recurrent Marked", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "score": 1.0, + "content": "Temporal Point Processes (RMTPPs; Du et al. 2016) and Neural Hawkes Processes (NHPs; Mei &", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "Eisner 2017). In contrast to our approach, these methods can not compute the exact likelihood of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "model and have to resort to Monte-Carlo sampling for its approximation. However, this approach", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "is especially problematic for commonly occurring clustered and bursty event sequences as it either", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "requires a very high sampling rate or ignores important temporal dependencies (Nickel & Le, 2020).", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "To overcome this issue, Jia & Benson (2019) proposed Neural Jump SDEs which extend the Neural", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "ODE framework and allow to compute the exact likelihood for neural TPPs, up to numerical errors.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "This method is closely related to our approach and we build on its ideas to compute the ground", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "intensity of the STPP. However, current Neural Jump SDEs —as well as NHPs and RMTPPs—are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "not well-suited for modeling complex continuous mark distributions as they are restricted to methods", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "such as Gaussian mixture models in the spatial domain. Finally, Shchur et al. (2019) and Mehrasa", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "et al. (2019) considered combining TPPs with flexible likelihood-based models, however for different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 315, + 460, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 460, + 327 + ], + "score": 1.0, + "content": "purposes as in our case, i.e., for intensity-free learning of only temporal point processes.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 106, + 506, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 504, + 349 + ], + "score": 1.0, + "content": "Continuous Normalizing Flows The ability to describe an infinite number of distributions with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "score": 1.0, + "content": "a Continuous Normalizing Flow has been used by a few recent works. Some works in computer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "score": 1.0, + "content": "graphics have used the interpolation effect of CNFs to model transformations of point clouds (Yang", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "et al., 2019; Rempe et al., 2020; Li et al., 2020). CNFs have also been used in sequential latent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 381, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 393 + ], + "score": 1.0, + "content": "variable models (Deng et al., 2020; Rempe et al., 2020). However, such works do not align the “time”", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "axis of the CNF with the temporal axis of observations, and do not train on observations at more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "than one value of “time” in the CNF. In contrast, we align the time axis of the CNF with the time", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "of the observations, directly using its ability to model distributions on a real-valued axis. A closely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "related application of CNFs to spatio-temporal data was done by Tong et al. (2020), who modeled the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "distribution of cells in a developing human embryo system at five fixed time values. In contrast to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "score": 1.0, + "content": "this, we extend to applications where observations are made at arbitrary time values, jointly modeling", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "space and time within the spatio-temporal point process framework. Furthermore, Mathieu & Nickel", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 481 + ], + "score": 1.0, + "content": "(2020); Lou et al. (2020) recently proposed extensions of CNFs to Riemannian manifolds. For our", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "proposed approach, this is especially interesting in the context of earth and climate science, as it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 491, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 502 + ], + "score": 1.0, + "content": "allows us to model STPPs on the sphere simply by replacing the CNF with its Riemannian equivalent.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 338, + 506, + 502 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 200, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 201, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 201, + 532 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "Data Sets Many collected data can be represented within the framework of spatio-temporal events.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 507, + 567 + ], + "score": 1.0, + "content": "We pre-process data from open sources and make them suitable for spatio-temporal event modeling.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "Each sequence in these data sets can contain up to thousands of variables, all the while having a large", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "variance in sequence lengths. Varying across a wide range of domains, the data sets we consider are:", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "earthquakes, pandemic spread, consumer demand for a bike sharing app, and high-amplitude brain", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 596, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 506, + 612 + ], + "score": 1.0, + "content": "signals from fMRI scans. We briefly describe these data sets here; further details, pre-processing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "steps, and data set diagnostics can be found in Appendix C. Code for preprocessing and training are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 619, + 457, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 457, + 633 + ], + "score": 1.0, + "content": "open sourced at https://github.com/facebookresearch/neural_stpp.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 542, + 507, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 640, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "PINWHEEL This is a synthetic data set with multimodal and non-Gaussian spatial distributions", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "designed to test the ability to capture drastic changes due to event history (see fig. 5). The data set", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "consists of 10 clusters which form a pinwheel structure. Events are sampled from a multivariate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 686 + ], + "score": 1.0, + "content": "Hawkes process such that events from one cluster will increase the probability of observing events in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 684, + 503, + 697 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 503, + 697 + ], + "score": 1.0, + "content": "the next cluster in a clock-wise rotation. Number of events per sequences ranges between 4 to 108.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 640, + 505, + 697 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 720, + 409, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 409, + 734 + ], + "score": 1.0, + "content": "Survey (2020). 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(a) Before", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 191, + 226 + ], + "score": 1.0, + "content": "observing any events at", + "type": "text" + }, + { + "bbox": [ + 191, + 214, + 205, + 223 + ], + "score": 0.79, + "content": "\\scriptstyle t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 213, + 505, + 226 + ], + "score": 1.0, + "content": ", the distribution is even across all clusters. (b-f) Each event increases the probability", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "of observing a future event from the subsequent cluster in clock-wise ordering. (g-h) After a period of no new", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 234, + 375, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 375, + 245 + ], + "score": 1.0, + "content": "events, the distribution smoothly returns back to the initial distribution (a).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 135, + 256, + 473, + 399 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 135, + 256, + 473, + 399 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 135, + 256, + 473, + 399 + ], + "spans": [ + { + "bbox": [ + 135, + 256, + 473, + 399 + ], + "score": 0.977, + "type": "image", + "image_path": "c2f9aeea16989017a062ac3824365a59e03ce754df2dafff536fa59e95729512.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 135, + 256, + 473, + 303.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 135, + 303.6666666666667, + 473, + 351.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 135, + 351.33333333333337, + 473, + 399.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 409, + 505, + 450 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "Figure 7: Snapshots of conditional spatial distributions modeled by the Jump CNF (top) and a conditional kernel", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 357, + 430 + ], + "score": 1.0, + "content": "density estimator (KDE; bottom). (a) Distribution before any events at", + "type": "text" + }, + { + "bbox": [ + 358, + 419, + 372, + 428 + ], + "score": 0.79, + "content": "\\scriptstyle t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 419, + 505, + 430 + ], + "score": 1.0, + "content": ". (b-d) The Jump CNF’s distributions", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "concentrate around tectonic plate boundaries where earthquakes and aftershocks gather, whereas the KDE must", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 438, + 427, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 427, + 452 + ], + "score": 1.0, + "content": "use a large variance in order to capture propagation of aftershocks in multiple directions.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + } + ], + "index": 9.75 + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "cases in New Jersey state. The data is aggregated at the county level, which we dequantize uniformly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 496, + 415, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 415, + 509 + ], + "score": 1.0, + "content": "across the county. Number of events per sequences ranges between 3 to 323.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "BOLD5000 This consists of fMRI scans as participants are given visual stimuli (Chang et al., 2019).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "We convert brain responses into spatio-temporal events following the z-score thresholding approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 536, + 488, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 488, + 550 + ], + "score": 1.0, + "content": "in Tagliazucchi et al. (2012; 2016). Number of events per sequences ranges between 6 to 1741.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 502, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "In addition to these datasets, we also report in Appendix A results for CITIBIKE, a data set consisting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 571, + 356, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 356, + 585 + ], + "score": 1.0, + "content": "of rental events from a bike sharing service in New York City.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "Baselines To evaluate the capability of our proposed models, we compare against commonly-used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "score": 1.0, + "content": "baselines and state-of-the-art models. In some settings, ground intensity and conditional mark density", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "are independent of each other and we can freely combine different baselines for the temporal and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "spatial domains. As temporal baselines, we use a homogeneous Poisson process, a self-correction", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "process, a Hawkes process, and the Neural Hawkes Process, which were trained using their officially", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "released code. As spatial baselines, we use a conditional kernel density estimator (KDE) with learned", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 180, + 678 + ], + "score": 1.0, + "content": "parameters, where", + "type": "text" + }, + { + "bbox": [ + 181, + 666, + 207, + 678 + ], + "score": 0.94, + "content": "p ( { \\pmb x } | t )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "is essentially modeled as a history-dependent Gaussian mixture model (see", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "Appendix B), as well as the Time-varying CNF. In addition, we also compare to our implementation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "of Neural Jump SDEs (Jia & Benson, 2019) where the spatial distribution is a Gaussian mixture model.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "We use the same architecture as our GRU-based continuous-time hidden states for fair comparison,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "as we found the simpler parameterization in Jia & Benson (2019) to be numerically unstable for large", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 426, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 426, + 734 + ], + "score": 1.0, + "content": "number of events. 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(a) Before", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 191, + 226 + ], + "score": 1.0, + "content": "observing any events at", + "type": "text" + }, + { + "bbox": [ + 191, + 214, + 205, + 223 + ], + "score": 0.79, + "content": "\\scriptstyle t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 213, + 505, + 226 + ], + "score": 1.0, + "content": ", the distribution is even across all clusters. (b-f) Each event increases the probability", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "of observing a future event from the subsequent cluster in clock-wise ordering. (g-h) After a period of no new", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 234, + 375, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 375, + 245 + ], + "score": 1.0, + "content": "events, the distribution smoothly returns back to the initial distribution (a).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 135, + 256, + 473, + 399 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 135, + 256, + 473, + 399 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 135, + 256, + 473, + 399 + ], + "spans": [ + { + "bbox": [ + 135, + 256, + 473, + 399 + ], + "score": 0.977, + "type": "image", + "image_path": "c2f9aeea16989017a062ac3824365a59e03ce754df2dafff536fa59e95729512.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 135, + 256, + 473, + 303.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 135, + 303.6666666666667, + 473, + 351.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 135, + 351.33333333333337, + 473, + 399.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 409, + 505, + 450 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "Figure 7: Snapshots of conditional spatial distributions modeled by the Jump CNF (top) and a conditional kernel", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 357, + 430 + ], + "score": 1.0, + "content": "density estimator (KDE; bottom). (a) Distribution before any events at", + "type": "text" + }, + { + "bbox": [ + 358, + 419, + 372, + 428 + ], + "score": 0.79, + "content": "\\scriptstyle t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 419, + 505, + 430 + ], + "score": 1.0, + "content": ". (b-d) The Jump CNF’s distributions", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "concentrate around tectonic plate boundaries where earthquakes and aftershocks gather, whereas the KDE must", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 438, + 427, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 427, + 452 + ], + "score": 1.0, + "content": "use a large variance in order to capture propagation of aftershocks in multiple directions.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + } + ], + "index": 9.75 + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "cases in New Jersey state. The data is aggregated at the county level, which we dequantize uniformly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 496, + 415, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 415, + 509 + ], + "score": 1.0, + "content": "across the county. Number of events per sequences ranges between 3 to 323.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 473, + 505, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "BOLD5000 This consists of fMRI scans as participants are given visual stimuli (Chang et al., 2019).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "We convert brain responses into spatio-temporal events following the z-score thresholding approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 536, + 488, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 488, + 550 + ], + "score": 1.0, + "content": "in Tagliazucchi et al. (2012; 2016). Number of events per sequences ranges between 6 to 1741.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 514, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 502, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "In addition to these datasets, we also report in Appendix A results for CITIBIKE, a data set consisting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 571, + 356, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 356, + 585 + ], + "score": 1.0, + "content": "of rental events from a bike sharing service in New York City.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 560, + 505, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "Baselines To evaluate the capability of our proposed models, we compare against commonly-used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "score": 1.0, + "content": "baselines and state-of-the-art models. In some settings, ground intensity and conditional mark density", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "are independent of each other and we can freely combine different baselines for the temporal and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "spatial domains. As temporal baselines, we use a homogeneous Poisson process, a self-correction", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "process, a Hawkes process, and the Neural Hawkes Process, which were trained using their officially", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "released code. As spatial baselines, we use a conditional kernel density estimator (KDE) with learned", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 180, + 678 + ], + "score": 1.0, + "content": "parameters, where", + "type": "text" + }, + { + "bbox": [ + 181, + 666, + 207, + 678 + ], + "score": 0.94, + "content": "p ( { \\pmb x } | t )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "is essentially modeled as a history-dependent Gaussian mixture model (see", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "Appendix B), as well as the Time-varying CNF. In addition, we also compare to our implementation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "of Neural Jump SDEs (Jia & Benson, 2019) where the spatial distribution is a Gaussian mixture model.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "We use the same architecture as our GRU-based continuous-time hidden states for fair comparison,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "as we found the simpler parameterization in Jia & Benson (2019) to be numerically unstable for large", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 426, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 426, + 734 + ], + "score": 1.0, + "content": "number of events. Range of hyperparameter values are outlined in Appendix D.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 599, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 80, + 504, + 210 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 80, + 504, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 504, + 210 + ], + "score": 0.982, + "html": "
PinwheelEarthquakes JPCOVID-19 NJBOLD5000
ModelTemporalSpatialTemporalSpatialTemporalSpatialTemporalSpatial
Poisson Process-0.784±0.0011-0.111±0.00110.878±0.01610.862±0.0181
Self-correcting Process-2.117±0.222-7.051±0.780-10.053±1.150-6.470±0.827
Hawkes Process-0.276±0.0330.114±0.0052.092±0.0232.860±0.050
Neural Hawkes Process-0.023±0.0010.198±0.0012.229±0.0133.080±0.019
Conditional KDE-2.958±0.000-2.259±0.001-2.583±0.000-3.467±0.000
Time-varying CNF-2.185±0.003-1.459±0.016-2.002±0.0021-1.846±0.019
Neural Jump SDE (GRU)-0.006±0.042-2.077±0.0260.186±0.005 -1.652±0.0122.251±0.004 -2.214±0.0055.675±0.0030.743±0.089
Jump CNF0.027±0.002-1.562±0.0150.166±0.001-1.007±0.0502.242±0.002 -1.904±0.0045.536±0.0161.246±0.185
Attentive CNF0.034±0.001 -1.572±0.0020.204±0.001-1.237±0.0752.258±0.002 -1.864±0.0015.842±0.0051.252±0.026
", + "type": "table", + "image_path": "2111b30cc24d7fe8fbeb2177c2219f0516b14ccee56250f827cbf5622cbbb6bf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 80, + 504, + 123.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 123.33333333333334, + 504, + 166.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 166.66666666666669, + 504, + 210.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 105, + 218, + 504, + 229 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "Table 1: Log-likelihood per event on held-out test data (higher is better). Standard devs. estimated over 3 runs.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 503, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "Results & Analyses The results of our evaluation are shown in table 1. We highlight all results", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 268, + 439, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 439, + 282 + ], + "score": 1.0, + "content": "where the intervals containing one standard deviation away from the mean overlap.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "Across all data sets, the Time-varying CNF outperforms the conditional KDE baseline despite not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "being conditional on history. This suggests that the overall spatial distribution is rather complex. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "also see from Figure 7 that Gaussian clusters tend to compensate for far-reaching events by learning a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 318, + 473, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 473, + 331 + ], + "score": 1.0, + "content": "larger band-width whereas a flexible CNF can easily model multi-modal event propagation.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 108, + 335, + 504, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "The Jump and Attentive CNF models achieve better log-likelihoods than the Time-varying CNF,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 346, + 474, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 474, + 360 + ], + "score": 1.0, + "content": "suggesting prediction in these data sets benefit from modeling dependence on event history.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "For COVID-19, the self-exciting Hawkes process is a strong baseline which aligns with similar results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "for other infectious diseases (Park et al., 2019), but Neural STPPs can achieve substantially better", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "spatial likelihoods. Overall, NHP is competitive with the Neural Jump SDE; however, it tends to fall", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 396, + 427, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 427, + 409 + ], + "score": 1.0, + "content": "short of the Attentive CNF which jointly models spatial and temporal variables.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "In a closer comparison to the temporal likelihood of Neural Jump SDEs (Jia & Benson, 2019), we find", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "that overly-restricted spatial models can negatively affect the temporal model since both domains are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "tightly coupled. Since our realization of Neural Jump SDEs and our STPPs use the same underlying", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 507, + 459 + ], + "score": 1.0, + "content": "architecture to model the temporal domain, the temporal likelihood values are often close. However,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "there is still a statistically significant difference between our Neural STPP models and Neural Jump", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 343, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 343, + 480 + ], + "score": 1.0, + "content": "SDEs even for the temporal log-likelihood on all data sets.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "Finally, we note that the results of the Jump and Attentive CNFs are typically close. The attentive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "model generally achieves better temporal log-likelihoods while maintaining competitive spatial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "log-likelihoods. This difference is likely due to the Attentive CNF’s ability to attend to all previous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "events, while the Jump CNF has to compress all history information inside the hidden state at the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 529, + 500, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 500, + 541 + ], + "score": 1.0, + "content": "time of event. The Attentive CNF also enjoys substantially faster computations (see Appendix A).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 195, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 197, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 197, + 580 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "To learn high-fidelity models of stochastic events occurring in continuous space and time, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "have proposed a new class of parameterizations for spatio-temporal point processes. Our approach", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "combines ideas of Neural Jump SDEs with Continuous Normalizing Flows and allows to retain the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "flexibility of neural temporal point processes while enabling highly expressive models of continuous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "marks. We leverage Neural ODEs as a computational method that allows computing, up to negligible", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 507, + 661 + ], + "score": 1.0, + "content": "numerical error, the likelihood of the joint model, and we show that our approach achieves state-of-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 660, + 463, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 463, + 673 + ], + "score": 1.0, + "content": "the-art performance on spatio-temporal datasets collected from a wide range of domains.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "A promising area for future work are applications of our method in earth and climate science which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "often are concerned with modeling highly complex spatio-temporal data. In this context, the use of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "score": 1.0, + "content": "Riemannian CNFs (Mathieu & Nickel, 2020; Lou et al., 2020; Falorsi & Forre´, 2020) is especially", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "interesting as it allows us to model Neural STPPs on manifolds (e.g. the earth’s surface) by simply", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 368, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 368, + 733 + ], + "score": 1.0, + "content": "replacing the CNF in our models with a Riemannian counterpart.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 80, + 504, + 210 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 80, + 504, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 504, + 210 + ], + "score": 0.982, + "html": "
PinwheelEarthquakes JPCOVID-19 NJBOLD5000
ModelTemporalSpatialTemporalSpatialTemporalSpatialTemporalSpatial
Poisson Process-0.784±0.0011-0.111±0.00110.878±0.01610.862±0.0181
Self-correcting Process-2.117±0.222-7.051±0.780-10.053±1.150-6.470±0.827
Hawkes Process-0.276±0.0330.114±0.0052.092±0.0232.860±0.050
Neural Hawkes Process-0.023±0.0010.198±0.0012.229±0.0133.080±0.019
Conditional KDE-2.958±0.000-2.259±0.001-2.583±0.000-3.467±0.000
Time-varying CNF-2.185±0.003-1.459±0.016-2.002±0.0021-1.846±0.019
Neural Jump SDE (GRU)-0.006±0.042-2.077±0.0260.186±0.005 -1.652±0.0122.251±0.004 -2.214±0.0055.675±0.0030.743±0.089
Jump CNF0.027±0.002-1.562±0.0150.166±0.001-1.007±0.0502.242±0.002 -1.904±0.0045.536±0.0161.246±0.185
Attentive CNF0.034±0.001 -1.572±0.0020.204±0.001-1.237±0.0752.258±0.002 -1.864±0.0015.842±0.0051.252±0.026
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Standard devs. estimated over 3 runs.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 503, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "Results & Analyses The results of our evaluation are shown in table 1. We highlight all results", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 268, + 439, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 439, + 282 + ], + "score": 1.0, + "content": "where the intervals containing one standard deviation away from the mean overlap.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 257, + 505, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "Across all data sets, the Time-varying CNF outperforms the conditional KDE baseline despite not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "being conditional on history. This suggests that the overall spatial distribution is rather complex. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "also see from Figure 7 that Gaussian clusters tend to compensate for far-reaching events by learning a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 318, + 473, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 473, + 331 + ], + "score": 1.0, + "content": "larger band-width whereas a flexible CNF can easily model multi-modal event propagation.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 286, + 506, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 335, + 504, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "The Jump and Attentive CNF models achieve better log-likelihoods than the Time-varying CNF,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 346, + 474, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 474, + 360 + ], + "score": 1.0, + "content": "suggesting prediction in these data sets benefit from modeling dependence on event history.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 335, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "For COVID-19, the self-exciting Hawkes process is a strong baseline which aligns with similar results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "for other infectious diseases (Park et al., 2019), but Neural STPPs can achieve substantially better", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "spatial likelihoods. Overall, NHP is competitive with the Neural Jump SDE; however, it tends to fall", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 396, + 427, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 427, + 409 + ], + "score": 1.0, + "content": "short of the Attentive CNF which jointly models spatial and temporal variables.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 362, + 506, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "In a closer comparison to the temporal likelihood of Neural Jump SDEs (Jia & Benson, 2019), we find", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "that overly-restricted spatial models can negatively affect the temporal model since both domains are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "tightly coupled. Since our realization of Neural Jump SDEs and our STPPs use the same underlying", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 507, + 459 + ], + "score": 1.0, + "content": "architecture to model the temporal domain, the temporal likelihood values are often close. However,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "there is still a statistically significant difference between our Neural STPP models and Neural Jump", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 343, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 343, + 480 + ], + "score": 1.0, + "content": "SDEs even for the temporal log-likelihood on all data sets.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 414, + 507, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "Finally, we note that the results of the Jump and Attentive CNFs are typically close. The attentive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "model generally achieves better temporal log-likelihoods while maintaining competitive spatial", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "log-likelihoods. This difference is likely due to the Attentive CNF’s ability to attend to all previous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "events, while the Jump CNF has to compress all history information inside the hidden state at the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 529, + 500, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 500, + 541 + ], + "score": 1.0, + "content": "time of event. The Attentive CNF also enjoys substantially faster computations (see Appendix A).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 484, + 505, + 541 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 195, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 197, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 197, + 580 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "To learn high-fidelity models of stochastic events occurring in continuous space and time, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "have proposed a new class of parameterizations for spatio-temporal point processes. Our approach", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "combines ideas of Neural Jump SDEs with Continuous Normalizing Flows and allows to retain the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "flexibility of neural temporal point processes while enabling highly expressive models of continuous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "marks. We leverage Neural ODEs as a computational method that allows computing, up to negligible", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 507, + 661 + ], + "score": 1.0, + "content": "numerical error, the likelihood of the joint model, and we show that our approach achieves state-of-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 660, + 463, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 463, + 673 + ], + "score": 1.0, + "content": "the-art performance on spatio-temporal datasets collected from a wide range of domains.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 594, + 507, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "A promising area for future work are applications of our method in earth and climate science which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "often are concerned with modeling highly complex spatio-temporal data. In this context, the use of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "score": 1.0, + "content": "Riemannian CNFs (Mathieu & Nickel, 2020; Lou et al., 2020; Falorsi & Forre´, 2020) is especially", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "interesting as it allows us to model Neural STPPs on manifolds (e.g. the earth’s surface) by simply", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 368, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 368, + 733 + ], + "score": 1.0, + "content": "replacing the CNF in our models with a Riemannian counterpart.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 84, + 200, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 84, + 200, + 94 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 103, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 506, + 116 + ], + "score": 1.0, + "content": "We acknowledge the Python community (Van Rossum & Drake Jr, 1995; Oliphant, 2007) for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "developing the core set of tools that enabled this work, including PyTorch (Paszke et al., 2019),", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 507, + 138 + ], + "score": 1.0, + "content": "torchdiffeq (Chen, 2018), fairseq (Ott et al., 2019), Jupyter (Kluyver et al., 2016), Matplotlib (Hunter,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "score": 1.0, + "content": "2007), seaborn (Waskom et al., 2018), Cython (Behnel et al., 2011), numpy (Oliphant, 2006; Van", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 430, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 430, + 160 + ], + "score": 1.0, + "content": "Der Walt et al., 2011), pandas (McKinney, 2012), and SciPy (Jones et al., 2014).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 177, + 175, + 189 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 176, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 176, + 190 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 209 + ], + "score": 1.0, + "content": "Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. 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Citibike NY
ModelTemporalSpatial
Poisson Process0.609±0.012
Self-correcting Process-5.649±1.433
Hawkes Process1.062±0.000
Neural Hawkes Process 1.030±0.015
Conditional KDE-2.856±0.000
Time-varying CNF-2.132±0.012
Neural Jump SDE1.092±0.002-2.731±0.001
Jump CNF1.105±0.002-2.155±0.015
Attentive CNF1.112±0.002-2.095±0.006
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The spatial distribution conditions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 222, + 328, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 328, + 234 + ], + "score": 1.0, + "content": "all past events as well as the current time of occurance.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 105, + 239, + 504, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "score": 1.0, + "content": "Our baseline model assumes a simple Gaussian conditional model, that new events are likely to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 222, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 222, + 261 + ], + "score": 1.0, + "content": "appear near previous events.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 268, + 496, + 304 + ], + "lines": [ + { + "bbox": [ + 117, + 268, + 496, + 304 + ], + "spans": [ + { + "bbox": [ + 117, + 268, + 496, + 304 + ], + "score": 0.94, + "content": "\\log p ( x _ { i } | t _ { i } , t _ { 1 } , \\dots , t _ { i - 1 } , x _ { 1 } , \\dots , x _ { i - 1 } ) = \\sum _ { j = 1 } ^ { i - 1 } \\alpha _ { j } \\mathcal { N } ( x _ { j } | \\sigma ^ { 2 } ) , \\quad \\alpha _ { j } = \\frac { \\exp \\{ ( t _ { j } - t _ { i } ) / \\tau \\} } { \\sum _ { j ^ { \\prime } = 1 } ^ { i - 1 } \\exp \\{ ( t _ { j ^ { \\prime } } - t _ { i } ) / \\tau \\} }", + "type": "interline_equation", + "image_path": "feabdb2a183d8d4a59042c16056b2eb557b46cbd77f6b90b5a949efa9d7563e5.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 117, + 268, + 496, + 280.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 117, + 280.0, + 496, + 292.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 117, + 292.0, + 496, + 304.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 312, + 504, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 327, + 326 + ], + "score": 1.0, + "content": "This parameteric model has two learnable parameters:", + "type": "text" + }, + { + "bbox": [ + 327, + 312, + 339, + 323 + ], + "score": 0.87, + "content": "\\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 311, + 357, + 326 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 315, + 363, + 323 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 311, + 506, + 326 + ], + "score": 1.0, + "content": ", which control the rate of decay in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 323, + 295, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 295, + 337 + ], + "score": 1.0, + "content": "the spatial and temporal domains, respectively.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "However, this Gaussian spatial model assumes events are propagated in all directions equally and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "can only model local self-excitation behavior. These assumptions are often used for simplifications", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "but are generally incorrect for many spatio-temporal data. To name a few, earthquakes occur more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "frequently along boundaries of tectonic plates, epidemics propagate along traffic routes, taxi demands", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 384, + 327, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 327, + 396 + ], + "score": 1.0, + "content": "saturate locally and change as customers move around.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 106, + 415, + 357, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 359, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 359, + 430 + ], + "score": 1.0, + "content": "C PRE-PROCESSING STEPS FOR EACH DATA SET", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "PINWHEEL We sample from a multivariate Hawkes process with 10 dimensions. We turn this into", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "continuous spatial variables by assigning each dimension to a cluster from a “pinwheel” distribution,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "and sample from the corresponding cluster for each event. Number of events per sequences ranges", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 473, + 180, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 180, + 484 + ], + "score": 1.0, + "content": "between 4 to 108.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Survey (2020). Starting from January 01, 1990, we created sequences with a gap of 7 days. Each", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "sequence was of length 30 days. We ensured there was no contamination between train/val/test sets by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "removing intermediate sequences. We removed earthquakes from 2010 November to 2011 December,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "as these sequences were too long and only served as outliers in the data. This resulted in 950 training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "sequences, 50 validation sequences, and 50 test sequences. Number of events per sequence ranges", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 567, + 185, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 185, + 578 + ], + "score": 1.0, + "content": "between 18 to 543.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "cases in the New Jersey state, from March to July of 2020. The data is aggregated at the county level,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "which we dequantize uniformly across the county. We also dequantize the temporal axis by assigning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "new cases uniformly within the day. Starting at March 15, and every 3 days, we took a 7 day length", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "sequence. For each sequence, we sampled each event with a probability of 0.01. This was done 50", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 636, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 506, + 652 + ], + "score": 1.0, + "content": "times per sequence. We ensured there was no contamination between train/val/test sets by removing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "intermediate sequences. This resulted in 1450 training sequences, 100 validation sequences, and 100", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 398, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 398, + 672 + ], + "score": 1.0, + "content": "test sequences. Number of events per sequence ranges between 3 to 323.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "CITIBIKE Citibike is a bike sharing service in New York City. We treat the start of each trip as an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "event, and use the data from April to August of 2019. We split into sequences of length 1 day starting", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 116, + 712 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 116, + 699, + 148, + 709 + ], + "score": 0.45, + "content": "5 { : } 0 0 { \\mathrm { a m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "of each day. For each sequence, we subsampled with a probability of 0.005 per event, 20", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "times. This resulted in 2440 training sequences, 300 validation sequences, and 320 test sequences.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 336, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 336, + 734 + ], + "score": 1.0, + "content": "Number of events per sequence ranges between 9 to 231.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 181, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 182, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 182, + 96 + ], + "score": 1.0, + "content": "B BASELINE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 107, + 506, + 141 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 506, + 120 + ], + "score": 1.0, + "content": "Our self-excitation baseline uses a Hawkes process to model the temporal variable, then uses a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "Gaussian mixture model to describe the spatial distribution conditioned on history of events. 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This dependence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 398, + 213 + ], + "score": 1.0, + "content": "structure allows the usage of simple temporal point processes to model", + "type": "text" + }, + { + "bbox": [ + 398, + 201, + 407, + 211 + ], + "score": 0.83, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 200, + 506, + 213 + ], + "score": 1.0, + "content": ", e.g. a Hawkes process,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "since temporal variables do not depend on the spatial information. The spatial distribution conditions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 222, + 328, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 328, + 234 + ], + "score": 1.0, + "content": "all past events as well as the current time of occurance.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 189, + 506, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 239, + 504, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "score": 1.0, + "content": "Our baseline model assumes a simple Gaussian conditional model, that new events are likely to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 222, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 222, + 261 + ], + "score": 1.0, + "content": "appear near previous events.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 237, + 505, + 261 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 268, + 496, + 304 + ], + "lines": [ + { + "bbox": [ + 117, + 268, + 496, + 304 + ], + "spans": [ + { + "bbox": [ + 117, + 268, + 496, + 304 + ], + "score": 0.94, + "content": "\\log p ( x _ { i } | t _ { i } , t _ { 1 } , \\dots , t _ { i - 1 } , x _ { 1 } , \\dots , x _ { i - 1 } ) = \\sum _ { j = 1 } ^ { i - 1 } \\alpha _ { j } \\mathcal { N } ( x _ { j } | \\sigma ^ { 2 } ) , \\quad \\alpha _ { j } = \\frac { \\exp \\{ ( t _ { j } - t _ { i } ) / \\tau \\} } { \\sum _ { j ^ { \\prime } = 1 } ^ { i - 1 } \\exp \\{ ( t _ { j ^ { \\prime } } - t _ { i } ) / \\tau \\} }", + "type": "interline_equation", + "image_path": "feabdb2a183d8d4a59042c16056b2eb557b46cbd77f6b90b5a949efa9d7563e5.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 117, + 268, + 496, + 280.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 117, + 280.0, + 496, + 292.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 117, + 292.0, + 496, + 304.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 312, + 504, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 327, + 326 + ], + "score": 1.0, + "content": "This parameteric model has two learnable parameters:", + "type": "text" + }, + { + "bbox": [ + 327, + 312, + 339, + 323 + ], + "score": 0.87, + "content": "\\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 311, + 357, + 326 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 315, + 363, + 323 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 311, + 506, + 326 + ], + "score": 1.0, + "content": ", which control the rate of decay in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 323, + 295, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 295, + 337 + ], + "score": 1.0, + "content": "the spatial and temporal domains, respectively.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 311, + 506, + 337 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "However, this Gaussian spatial model assumes events are propagated in all directions equally and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "can only model local self-excitation behavior. These assumptions are often used for simplifications", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "but are generally incorrect for many spatio-temporal data. To name a few, earthquakes occur more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "frequently along boundaries of tectonic plates, epidemics propagate along traffic routes, taxi demands", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 384, + 327, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 327, + 396 + ], + "score": 1.0, + "content": "saturate locally and change as customers move around.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 340, + 506, + 396 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 415, + 357, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 359, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 359, + 430 + ], + "score": 1.0, + "content": "C PRE-PROCESSING STEPS FOR EACH DATA SET", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "PINWHEEL We sample from a multivariate Hawkes process with 10 dimensions. We turn this into", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "continuous spatial variables by assigning each dimension to a cluster from a “pinwheel” distribution,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "and sample from the corresponding cluster for each event. Number of events per sequences ranges", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 473, + 180, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 180, + 484 + ], + "score": 1.0, + "content": "between 4 to 108.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 441, + 506, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "EARTHQUAKES For modeling earthquakes and aftershocks, we gathered location and time of all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Survey (2020). Starting from January 01, 1990, we created sequences with a gap of 7 days. Each", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "sequence was of length 30 days. We ensured there was no contamination between train/val/test sets by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "removing intermediate sequences. We removed earthquakes from 2010 November to 2011 December,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "as these sequences were too long and only served as outliers in the data. This resulted in 950 training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "sequences, 50 validation sequences, and 50 test sequences. Number of events per sequence ranges", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 567, + 185, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 185, + 578 + ], + "score": 1.0, + "content": "between 18 to 543.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 490, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "COVID-19 CASES We use data released publicly by The New York Times (2020) on daily COVID-19", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "cases in the New Jersey state, from March to July of 2020. The data is aggregated at the county level,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "which we dequantize uniformly across the county. We also dequantize the temporal axis by assigning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "new cases uniformly within the day. Starting at March 15, and every 3 days, we took a 7 day length", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "sequence. For each sequence, we sampled each event with a probability of 0.01. This was done 50", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 636, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 506, + 652 + ], + "score": 1.0, + "content": "times per sequence. We ensured there was no contamination between train/val/test sets by removing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "intermediate sequences. This resulted in 1450 training sequences, 100 validation sequences, and 100", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 398, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 398, + 672 + ], + "score": 1.0, + "content": "test sequences. Number of events per sequence ranges between 3 to 323.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 582, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "CITIBIKE Citibike is a bike sharing service in New York City. We treat the start of each trip as an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "event, and use the data from April to August of 2019. We split into sequences of length 1 day starting", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 116, + 712 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 116, + 699, + 148, + 709 + ], + "score": 0.45, + "content": "5 { : } 0 0 { \\mathrm { a m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "of each day. For each sequence, we subsampled with a probability of 0.005 per event, 20", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "times. This resulted in 2440 training sequences, 300 validation sequences, and 320 test sequences.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 336, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 336, + 734 + ], + "score": 1.0, + "content": "Number of events per sequence ranges between 9 to 231.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 676, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 129, + 83, + 482, + 309 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 129, + 83, + 482, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 83, + 482, + 309 + ], + "spans": [ + { + "bbox": [ + 129, + 83, + 482, + 309 + ], + "score": 0.977, + "type": "image", + "image_path": "749703a2fc1655f14b5816bcd8926bc3db67af8d8051f198cc235bae973ff268.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 129, + 83, + 482, + 158.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 129, + 158.33333333333331, + 482, + 233.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 129, + 233.66666666666663, + 482, + 308.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 146, + 318, + 463, + 330 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 317, + 464, + 331 + ], + "spans": [ + { + "bbox": [ + 146, + 317, + 464, + 331 + ], + "score": 1.0, + "content": "Figure 11: Histograms of the number of events per sequence in each processed data set.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 506, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 362 + ], + "score": 1.0, + "content": "BOLD5000 This consists of fMRI scans of four participants as they are given visual stimuli (Chang", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "et al., 2019). We use the sessions of a single patient and for each run, we split into 3 sequences,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 474, + 384 + ], + "score": 1.0, + "content": "treated individually. We converted brain responses into spatio-temporal events following the", + "type": "text" + }, + { + "bbox": [ + 474, + 372, + 480, + 381 + ], + "score": 0.26, + "content": "\\mathbf { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "-score", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 452, + 394 + ], + "score": 1.0, + "content": "thresholding approach in Tagliazucchi et al. 2016, Equation (2). 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We used a threshold of", + "type": "text" + }, + { + "bbox": [ + 452, + 382, + 485, + 393 + ], + "score": 0.91, + "content": "\\gamma = 6 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 381, + 505, + 394 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "split the data into 1050 training sequences, 150 val sequences, 220 test sequences. 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6 4 - 6 4 - 6 4 - d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 509, + 366, + 520 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 366, + 509, + 372, + 519 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 509, + 507, + 520 + ], + "score": 1.0, + "content": "is the number of spatial variables.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "We swept over activation functions between using softplus or a time-dependent Swish (Ramachandran", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 529, + 159, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 159, + 543 + ], + "score": 1.0, + "content": "et al., 2017).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 497, + 507, + 543 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 541, + 395, + 554 + ], + "lines": [ + { + "bbox": [ + 216, + 541, + 395, + 554 + ], + "spans": [ + { + "bbox": [ + 216, + 541, + 395, + 554 + ], + "score": 0.79, + "content": "\\mathrm { T i m e D e p e n d e n t S w i s h } ( t , z ) = h \\sigma ( \\beta ( t ) \\odot z )", + "type": "interline_equation", + "image_path": "25a40ede27a0e706f5451b6c4a8c470341d5d0406ac302803cf29405c1acdd85.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 216, + 541, + 395, + 554 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 503, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 132, + 569 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 558, + 139, + 565 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 553, + 263, + 569 + ], + "score": 1.0, + "content": "is the logistic sigmoid function,", + "type": "text" + }, + { + "bbox": [ + 264, + 556, + 273, + 565 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 553, + 450, + 569 + ], + "score": 1.0, + "content": "is the Hadamard (element-wise) product, and", + "type": "text" + }, + { + "bbox": [ + 450, + 556, + 503, + 567 + ], + "score": 0.89, + "content": "\\beta : \\mathbb { R } \\mathbb { R } _ { d _ { z } }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 198, + 579 + ], + "score": 1.0, + "content": "is a MLP with widths", + "type": "text" + }, + { + "bbox": [ + 199, + 566, + 254, + 578 + ], + "score": 0.93, + "content": "[ \\bar { 1 } - 6 4 - d _ { z } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 566, + 283, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 284, + 567, + 295, + 577 + ], + "score": 0.89, + "content": "d _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 566, + 377, + 579 + ], + "score": 1.0, + "content": "is the dimension of", + "type": "text" + }, + { + "bbox": [ + 378, + 568, + 384, + 576 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 566, + 505, + 579 + ], + "score": 1.0, + "content": ", using the softplus activation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 577, + 460, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 460, + 590 + ], + "score": 1.0, + "content": "function. We ultimately decided on using the time-dependent Swish for all experiments.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 553, + 505, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 252, + 606 + ], + "score": 1.0, + "content": "We swept over the MLP for defining", + "type": "text" + }, + { + "bbox": [ + 252, + 595, + 263, + 605 + ], + "score": 0.88, + "content": "f _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "for the continuous-time hidden state in eq. (11) using hidden", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 146, + 617 + ], + "score": 1.0, + "content": "widths of", + "type": "text" + }, + { + "bbox": [ + 146, + 605, + 178, + 617 + ], + "score": 0.69, + "content": "[ 8 - 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 604, + 182, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 183, + 605, + 220, + 617 + ], + "score": 0.38, + "content": "[ 3 2 - 3 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 604, + 225, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 225, + 605, + 262, + 617 + ], + "score": 0.57, + "content": "[ 6 4 - 6 4 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 604, + 267, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 267, + 605, + 326, + 617 + ], + "score": 0.7, + "content": "\\left[ 3 2 - 3 2 - 3 2 \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 604, + 347, + 617 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 347, + 605, + 406, + 617 + ], + "score": 0.82, + "content": "[ 6 4 - 6 4 - 6 4 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 604, + 505, + 617 + ], + "score": 1.0, + "content": ". The majority of models", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 127, + 630 + ], + "score": 1.0, + "content": "used", + "type": "text" + }, + { + "bbox": [ + 128, + 616, + 161, + 627 + ], + "score": 0.8, + "content": "3 2 - 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "as it provided enough flexibility while remaining easy to solve. We used the softplus", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "activation function. We tried MLP for parameterizing the instantaneous change in eq. (12); however,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "it was too unstable for long sequences. We therefore switched to the GRU parameterization, which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 649, + 489, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 489, + 661 + ], + "score": 1.0, + "content": "takes an input (new event), the hidden state at the time of event, and outputs a new hidden state.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 594, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 184, + 677 + ], + "score": 1.0, + "content": "We regularized the", + "type": "text" + }, + { + "bbox": [ + 185, + 666, + 197, + 677 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 666, + 468, + 677 + ], + "score": 1.0, + "content": "norm of the hidden state drift with a strength of 1e-4, chosen from", + "type": "text" + }, + { + "bbox": [ + 469, + 666, + 480, + 677 + ], + "score": 0.74, + "content": "\\{ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 666, + 506, + 677 + ], + "score": 1.0, + "content": ", 1e-4,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 141, + 689 + ], + "score": 1.0, + "content": "1e-3, 1e-", + "type": "text" + }, + { + "bbox": [ + 142, + 677, + 152, + 689 + ], + "score": 0.7, + "content": "\\langle 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 676, + 506, + 689 + ], + "score": 1.0, + "content": ". We optionally used optimal transport-inspired regularization from Finlay et al. (2020),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 467, + 700 + ], + "score": 1.0, + "content": "which adds a Frobenius norm regularization to the gradient of the drift in addition to the", + "type": "text" + }, + { + "bbox": [ + 467, + 688, + 480, + 699 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "norm", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 321, + 711 + ], + "score": 1.0, + "content": "regularization, to the CNF models with a strength of", + "type": "text" + }, + { + "bbox": [ + 322, + 699, + 333, + 711 + ], + "score": 0.54, + "content": "\\{ 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 699, + 396, + 711 + ], + "score": 1.0, + "content": ", 1e-4, 1e-3, 1e-", + "type": "text" + }, + { + "bbox": [ + 396, + 699, + 407, + 711 + ], + "score": 0.3, + "content": "\\cdot 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". The Time-varying and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Attentive CNF models did not require regularization and were mostly kept at 0, but the Jump CNF", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 444, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 444, + 733 + ], + "score": 1.0, + "content": "models benefited from some amount of regularization to avoid numerical instability.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "To model a non-trivial spatial distribution for the entire data interval, we shift the data interval to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 136, + 106 + ], + "score": 1.0, + "content": "start at", + "type": "text" + }, + { + "bbox": [ + 136, + 94, + 160, + 104 + ], + "score": 0.9, + "content": "t = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 93, + 462, + 106 + ], + "score": 1.0, + "content": "for all CNF models. Thus the interval used for parameterizing the CNF is", + "type": "text" + }, + { + "bbox": [ + 463, + 93, + 503, + 106 + ], + "score": 0.89, + "content": "[ 2 , T + 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "Generally, the “time” variable is a dummy one; we can place the base distribution at any time, and we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "can choose any interval on the real line to be the data interval; this does not limit the model in any", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 128, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 128, + 141 + ], + "score": 1.0, + "content": "way.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "score": 1.0, + "content": "For the Jump CNF, we used a composition of 4 radial flows (Rezende & Mohamed, 2015) to parame-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "terize the instantaneous updates in eq. (17). All parameters of the radial flows were parameterized to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "be the output of a MLP that takes as input the hidden state at the time of the event (before the hidden", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "state is updated based on the current event). The radial flows were initialized in such a way that the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 222, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 222, + 199 + ], + "score": 1.0, + "content": "log determinant is near zero.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 315, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 315, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 315, + 216 + ], + "score": 1.0, + "content": "For the Attentive CNF, the drift function consists of", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 109, + 223, + 504, + 235 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 173, + 237 + ], + "score": 1.0, + "content": "Time-dependent", + "type": "text" + }, + { + "bbox": [ + 173, + 223, + 275, + 235 + ], + "score": 0.72, + "content": "\\mathrm { M L P } ( d - 6 4 - 6 4 ) 2 \\times ]", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 221, + 351, + 237 + ], + "score": 1.0, + "content": "MultiheadAttention", + "type": "text" + }, + { + "bbox": [ + 351, + 225, + 364, + 233 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 221, + 506, + 237 + ], + "score": 1.0, + "content": "Time-dependent MLP(64 − 64 − d)", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 254 + ], + "score": 1.0, + "content": "where the Time-dependent MLPs make use of the TimeDependentSwish. As was done in Vaswani et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 265 + ], + "score": 1.0, + "content": "(2017), the multihead attention is used within a residual branch, except we swapped LayerNorm (Ba", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "et al., 2016) with ActNorm (Kingma & Dhariwal, 2018) as LayerNorm has an unbounded Lipschitz", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "and can be ill-suited for use in ODEs. We tested both standard multihead attention (Vaswani et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "2017) and the Lipschitz multihead attention (Kim et al., 2020). The Lipschitz multihead attention", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "typically produced similar validation NLL as the standard multihead attention but were more stable", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "on multiple occassions. We therefore kept the Lipschitz multihead attention for all experiments. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "additionally, use an auxiliary (non-attentive, simply with the two multihead attention layers removed)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 330, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 469, + 342 + ], + "score": 1.0, + "content": "CNF to map from 0 (i.e. the time of base distribution) to the beginning of the data interval.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 503, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "We initialized all Neural ODEs (for the hidden state and CNFs) with zero drift by initializing the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 284, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 284, + 371 + ], + "score": 1.0, + "content": "weights and biases of the final layer to zero.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 503, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "The log-likelihood values reported are after the spatial variables have been standardized using the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 386, + 349, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 349, + 398 + ], + "score": 1.0, + "content": "empirical mean and standard deviation from the training set.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 504, + 416 + ], + "score": 1.0, + "content": "We train and test on log-likelihood (in nats) per event, which normalizes eq. (8) of each sequence by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 414, + 195, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 195, + 426 + ], + "score": 1.0, + "content": "the number of events.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 504, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "All integrals were solved using Chen (2018) to within a relative and absolute tolerance of 1E-4 or", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 395, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 395, + 455 + ], + "score": 1.0, + "content": "1E-6, chosen based on preliminary testing for convergence and stability.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "Our implementation of the Neural Jump SDE shares the same continuous-time hidden state parame-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 470, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 506, + 481 + ], + "score": 1.0, + "content": "terization but uses a mixture of Gaussians as the spatial model. We used 5 mixtures, and a MLP that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "maps from the hidden state to the parameters of this mixture of Gaussians (the means, log standard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 256, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 256, + 503 + ], + "score": 1.0, + "content": "deviations, and mixture coefficients).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 376, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 377, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 377, + 533 + ], + "score": 1.0, + "content": "E REMOVING CROSS-EVENT PARTIAL DERIVATIVES", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "This results in a lower-variance gradient estimator for training, and allows parallel computation of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 554, + 271, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 271, + 567 + ], + "score": 1.0, + "content": "conditional log probabilities at test time.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 570, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 104, + 568, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 387, + 584 + ], + "score": 1.0, + "content": "We first summarily describe the attention mechanism. 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The output is then added to", + "type": "text" + }, + { + "bbox": [ + 398, + 636, + 408, + 645 + ], + "score": 0.86, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "as a residual connection.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 244, + 658 + ], + "score": 1.0, + "content": "The multihead attention computes", + "type": "text" + }, + { + "bbox": [ + 245, + 647, + 254, + 657 + ], + "score": 0.84, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 646, + 329, + 658 + ], + "score": 1.0, + "content": "in a way such that", + "type": "text" + }, + { + "bbox": [ + 330, + 647, + 344, + 659 + ], + "score": 0.9, + "content": "P _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 646, + 393, + 658 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 393, + 647, + 406, + 658 + ], + "score": 0.89, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 646, + 424, + 658 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 424, + 647, + 437, + 659 + ], + "score": 0.89, + "content": "X _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 646, + 458, + 658 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 458, + 647, + 469, + 658 + ], + "score": 0.86, + "content": "V _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "depends", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 658, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 119, + 669 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 658, + 132, + 668 + ], + "score": 0.87, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 658, + 506, + 669 + ], + "score": 1.0, + "content": ". This is true for both the vanilla MHA (Vaswani et al., 2017) and the L2 MHA (Kim et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 667, + 488, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 205, + 682 + ], + "score": 1.0, + "content": "2020). 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Thus the interval used for parameterizing the CNF is", + "type": "text" + }, + { + "bbox": [ + 463, + 93, + 503, + 106 + ], + "score": 0.89, + "content": "[ 2 , T + 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "Generally, the “time” variable is a dummy one; we can place the base distribution at any time, and we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "can choose any interval on the real line to be the data interval; this does not limit the model in any", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 128, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 128, + 141 + ], + "score": 1.0, + "content": "way.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 82, + 506, + 141 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "score": 1.0, + "content": "For the Jump CNF, we used a composition of 4 radial flows (Rezende & Mohamed, 2015) to parame-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "terize the instantaneous updates in eq. (17). All parameters of the radial flows were parameterized to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "be the output of a MLP that takes as input the hidden state at the time of the event (before the hidden", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "state is updated based on the current event). 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As was done in Vaswani et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 265 + ], + "score": 1.0, + "content": "(2017), the multihead attention is used within a residual branch, except we swapped LayerNorm (Ba", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "et al., 2016) with ActNorm (Kingma & Dhariwal, 2018) as LayerNorm has an unbounded Lipschitz", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "and can be ill-suited for use in ODEs. We tested both standard multihead attention (Vaswani et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "2017) and the Lipschitz multihead attention (Kim et al., 2020). The Lipschitz multihead attention", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "typically produced similar validation NLL as the standard multihead attention but were more stable", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "on multiple occassions. We therefore kept the Lipschitz multihead attention for all experiments. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "additionally, use an auxiliary (non-attentive, simply with the two multihead attention layers removed)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 330, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 469, + 342 + ], + "score": 1.0, + "content": "CNF to map from 0 (i.e. the time of base distribution) to the beginning of the data interval.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 243, + 506, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 503, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "We initialized all Neural ODEs (for the hidden state and CNFs) with zero drift by initializing the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 284, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 284, + 371 + ], + "score": 1.0, + "content": "weights and biases of the final layer to zero.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 345, + 505, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 503, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "The log-likelihood values reported are after the spatial variables have been standardized using the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 386, + 349, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 349, + 398 + ], + "score": 1.0, + "content": "empirical mean and standard deviation from the training set.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 375, + 505, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 504, + 416 + ], + "score": 1.0, + "content": "We train and test on log-likelihood (in nats) per event, which normalizes eq. (8) of each sequence by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 414, + 195, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 195, + 426 + ], + "score": 1.0, + "content": "the number of events.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 401, + 504, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 504, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "All integrals were solved using Chen (2018) to within a relative and absolute tolerance of 1E-4 or", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 395, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 395, + 455 + ], + "score": 1.0, + "content": "1E-6, chosen based on preliminary testing for convergence and stability.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 430, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "Our implementation of the Neural Jump SDE shares the same continuous-time hidden state parame-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 470, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 506, + 481 + ], + "score": 1.0, + "content": "terization but uses a mixture of Gaussians as the spatial model. We used 5 mixtures, and a MLP that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "maps from the hidden state to the parameters of this mixture of Gaussians (the means, log standard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 256, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 256, + 503 + ], + "score": 1.0, + "content": "deviations, and mixture coefficients).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 458, + 507, + 503 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 519, + 376, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 377, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 377, + 533 + ], + "score": 1.0, + "content": "E REMOVING CROSS-EVENT PARTIAL DERIVATIVES", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "This results in a lower-variance gradient estimator for training, and allows parallel computation of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 554, + 271, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 271, + 567 + ], + "score": 1.0, + "content": "conditional log probabilities at test time.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 542, + 505, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 570, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 104, + 568, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 387, + 584 + ], + "score": 1.0, + "content": "We first summarily describe the attention mechanism. 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This is true for both the vanilla MHA (Vaswani et al., 2017) and the L2 MHA (Kim et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 667, + 488, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 205, + 682 + ], + "score": 1.0, + "content": "2020). 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PinwheelEarthquakes JPCOVID-19 NJBOLD5000
ModelTemporalSpatialTemporalSpatialTemporalSpatialTemporalSpatial
Poisson Process-0.784±0.0011-0.111±0.00110.878±0.01610.862±0.0181
Self-correcting Process-2.117±0.222-7.051±0.780-10.053±1.150-6.470±0.827
Hawkes Process-0.276±0.0330.114±0.0052.092±0.0232.860±0.050
Neural Hawkes Process-0.023±0.0010.198±0.0012.229±0.0133.080±0.019
Conditional KDE-2.958±0.000-2.259±0.001-2.583±0.000-3.467±0.000
Time-varying CNF-2.185±0.003-1.459±0.016-2.002±0.0021-1.846±0.019
Neural Jump SDE (GRU)-0.006±0.042-2.077±0.0260.186±0.005 -1.652±0.0122.251±0.004 -2.214±0.0055.675±0.0030.743±0.089
Jump CNF0.027±0.002-1.562±0.0150.166±0.001-1.007±0.0502.242±0.002 -1.904±0.0045.536±0.0161.246±0.185
Attentive CNF0.034±0.001 -1.572±0.0020.204±0.001-1.237±0.0752.258±0.002 -1.864±0.0015.842±0.0051.252±0.026
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Citibike NY
ModelTemporalSpatial
Poisson Process0.609±0.012
Self-correcting Process-5.649±1.433
Hawkes Process1.062±0.000
Neural Hawkes Process 1.030±0.015
Conditional KDE-2.856±0.000
Time-varying CNF-2.132±0.012
Neural Jump SDE1.092±0.002-2.731±0.001
Jump CNF1.105±0.002-2.155±0.015
Attentive CNF1.112±0.002-2.095±0.006
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"text": "" + } + ], + "page_info": { + "page_no": 18, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/qmI0P1ZExUl/qmI0P1ZExUl.md b/parse/train/qmI0P1ZExUl/qmI0P1ZExUl.md new file mode 100644 index 0000000000000000000000000000000000000000..09e5bf82b77504c6465956d77d3c3cb82308bd95 --- /dev/null +++ b/parse/train/qmI0P1ZExUl/qmI0P1ZExUl.md @@ -0,0 +1,321 @@ +# ENCODING IN STYLE: A STYLEGAN ENCODER FOR IMAGE-TO-IMAGE TRANSLATION + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +We present a generic image-to-image translation framework, Pixel2Style2Pixel $( p S p )$ . Our pSp framework is based on a novel encoder network that directly generates a series of style vectors which are fed into a pretrained StyleGAN generator, forming the extended $\mathcal { W } +$ latent space. We first show that our encoder can directly embed real images into $\mathcal { W } +$ , with no additional optimization. We further introduce a dedicated identity loss which is shown to achieve improved performance in the reconstruction of an input image. We demonstrate pSp to be a simple architecture that, by leveraging a well-trained, fixed generator network, can be easily applied on a wide-range of image-to-image translation tasks. Solving these tasks through the style representation results in a global approach that does not rely on a local pixel-to-pixel correspondence and further supports multi-modal synthesis via the resampling of styles. Notably, we demonstrate that pSp can be trained to align a face image to a frontal pose with no labeled data and generate multi-modal results for ambiguous tasks such as conditional face generation from sketches and segmentation maps. + +# 1 INTRODUCTION + +In recent years, Generative Adversarial Networks (GANs) have significantly advanced image synthesis, particularly on face images. State-of-the-art image generation methods have achieved high visual quality and fidelity, and can now generate images with phenomenal realism. Most notably, StyleGAN (Karras et al., 2019; 2020) proposes a novel style-based generator architecture and attains state-of-the-art visual quality on high-resolution images. Moreover, it has been demonstrated that it has a disentangled latent space, $\mathcal { W }$ (Yang et al., 2019; Collins et al., 2020; Shen et al., 2020), which may offer control and editing capabilities. + +Recently, numerous methods have shown competence in controlling StyleGAN’s latent space and performing meaningful manipulations in $\mathcal { W }$ (Jahanian et al., 2019; Shen et al., 2020; Tewari et al., 2020; Hark ¨ onen et al., 2020). To perform such edits on real images, one needs to invert the image ¨ into StyleGAN’s latent space, i.e., retrieve the latent code that reconstructs the image. However, it has been shown that inverting a real image into a 512-dimensional vector $\mathbf { w } \in \mathcal { W }$ does not lead to an accurate reconstruction. Motivated by this, it has become common practice (Abdal et al., 2019; 2020; Baylies, 2019; Zhu et al., 2020a; Adbal et al., 2020) to encode real images into an extended latent space, $\mathcal { W } +$ , defined by the concatenation of 18 different 512-dimensional w vectors, one for each input layer of StyleGAN. Nevertheless, many methods resort to using per-image optimization over $\mathcal { W } +$ , requiring several minutes for a single image. To accelerate this optimization process, some methods (Baylies, 2019; Zhu et al., 2020a) have trained an encoder to infer an approximate vector in $\mathcal { W } +$ which serves as a good initial point from which additional optimization is required. However, a fast, direct, and accurate learned inversion of real images into $\mathcal { W } +$ remains a challenge. + +In this paper, we focus on the broader task of latent space embedding, which aims to retrieve the latent vector that generates a desired, not necessarily known, image. We do so by introducing a novel encoder architecture tasked with encoding an arbitrary image directly into $\mathcal { W } +$ . The encoder is based on a Feature Pyramid Network (Lin et al., 2017), where style feature vectors are extracted from different pyramid scales and inserted directly into a fixed, pretrained StyleGAN generator in correspondence to their spatial scales. Our encoder into $\mathcal { W } +$ , together with the StyleGAN decoder, form a generic encoder-decoder network that benefits many image-to-image translation tasks. Focusing on face images, we first demonstrate our method’s ability to successfully reconstruct a given image while preserving identity and other attributes. We then present numerous image-to-image translation applications. In a sense, our method performs Pixel2Style2Pixel translation, as every image is first encoded into style vectors and then into an image, and is therefore dubbed $p S p$ . + +While many previous approaches to solving image-to-image translations tasks involve dedicated architectures specific for solving a single problem, we follow the spirit of pix2pix (Isola et al., 2017) and define a generic framework able to solve a wide range of tasks, all using the same architecture. Besides the simplification of the training process, as no adversary discriminator needs to be trained, using a pretrained StyleGAN generator offers several intriguing advantages over previous works. Many image-to-image architectures explicitly feed the generator with residual feature maps from the encoder (Isola et al., 2017; Wang et al., 2018), creating a strong locality bias (Richardson & Weiss, 2020). In contrast, our generator is governed only by the styles with no direct spatial input. The advantage of such a global approach is most evident in the task of Face Frontalization, where our encoder can be trained to align a given face image to a frontal pose with no labeled data. Another notable advantage of the intermediate style representation is the inherent support for multi-modal synthesis for ambiguous tasks such as face generation from sketches, segmentation maps, or lowresolution images. In such tasks, the generated styles can be resampled to create variations of the output image with no change to the architecture or training process. + +The main contributions of this paper are: (i) a novel StyleGAN encoder able to directly encode real face images into the $\mathcal { W } +$ target latent domain; and (ii) a generic end-to-end framework for solving image-to-image translation tasks. + +# 2 RELATED WORK + +Latent Space Embedding With the rapid evolution of GANs, many works have tried to understand and control their latent space. A specific task that has received substantial attention is $G A N$ Inversion — where the latent vector from which a pretrained GAN most accurately reconstructs a given, known image, is sought. Motivated by its state-of-the-art image quality and latent space semantic richness, many recent works have used StyleGAN (Karras et al., 2019; 2020) for this task. Generally, inversion methods either directly optimize the latent vector to minimize the error for the given image (Lipton & Tripathi, 2017; Creswell & Bharath, 2018; Abdal et al., 2019; 2020), train an encoder to map the given image to the latent space (Perarnau et al., 2016; Creswell & Bharath, 2018; Pidhorskyi et al., 2020; Guan et al., 2020; Nitzan et al., 2020), or use a hybrid approach combining both (Baylies, 2019; Zhu et al., 2020a). Typically, methods performing optimization are superior in reconstruction quality to a learned encoder mapping, but require a substantially longer time. Unlike the above methods, our encoder can accurately and efficiently embed a given face image into the extended latent space $\mathcal { W } +$ of a fixed, pretrained StyleGAN generator, with no further optimization. + +Image-to-Image Image-to-Image translation techniques aim at learning a conditional image generation function that maps an input image of a source domain to a corresponding image of a target domain. Isola et al. (2017) first introduced the use of conditional GANs to solve various imageto-image translation tasks. Since then, their work has been extended for many scenarios: highresolution synthesis (Wang et al., 2018), unsupervised learning (Liu et al., 2017; Zhu et al., 2017a; Katzir et al., 2019; Lira et al., 2020), multi-modal image synthesis (Zhu et al., 2017b; Huang et al., 2018; Choi et al., 2020), and conditional image synthesis (Park et al., 2019; Li et al., 2019; Liu et al., 2019b; Zhu et al., 2020b; Chen et al., 2020). The aforementioned works have constructed dedicated architectures, which require training the generator network. + +Latent-Space Manipulation Recently, numerous papers have presented diverse methods to learn semantic edits of the latent code. A popular approach is finding linear directions that correspond to changes in a given binary labeled attribute, such as young $ \mathrm { o l d }$ , or no-smile smile (Shen et al., 2020; Goetschalckx et al., 2019; Denton et al., 2019; Adbal et al., 2020). Tewari et al. (2020) utilize a pretrained 3DMM to learn semantic face edits in the latent space. Jahanian et al. (2019) find latent space paths that correspond to a specific transformation, such as zoom or rotation, in a selfsupervised manner. Hark ¨ onen et al. (2020) find useful paths in an unsupervised manner by using the ¨ principal component axes (PCA) of an intermediate activation space. Finally, Collins et al. (2020) perform local semantic editing by manipulating corresponding components of the latent code. + +![](images/aaa87c70d8d04df81fa258ed78659642cc51ccb7d39c7c3bd9e96fd5502d0ea6.jpg) +Figure 1: Our pSp architecture. Feature maps are first extracted using a standard feature pyramid over a ResNet backbone. For each of the 18 target styles, a small mapping network is trained to extract the learned styles from the corresponding feature map, where styles (0-2) are generated from the small feature map, (3-6) from the medium feature map, and (7-18) from the largest feature map. The mapping network, map2style, is a small fully convolutional network, which gradually reduces spatial size using a set of 2-strided convolutions followed by LeakyReLU activations. Each generated 512 vector, is fed into StyleGAN, starting from its matching affine transformation, $A$ . + +# 3 THE PSP FRAMEWORK + +Our pSp framework builds upon the representative power of a pretrained StyleGAN generator and the $\mathcal { W } +$ latent space. To utilize this representation one needs a strong encoder that is able to match each input image to an accurate encoding in the latent domain. A simple technique to embed into this domain is directly encoding a given input image into $\mathcal { W } +$ using a single 512-dimensional vector obtained from the last layer of the encoder network, thereby learning all 18 style vectors together. However, such an architecture presents a strong bottleneck making it difficult to fully represent the finer details of the original image and therefore limiting the reconstruction quality. + +In StyleGAN, the authors have shown that the different style inputs correspond to different levels of detail, which are roughly divided into three groups — coarse, medium, and fine. Following this observation, in pSp we extend an encoder backbone with a feature pyramid (Lin et al., 2017), generating three levels of feature maps from which styles are extracted using a simple intermediate network — map2style — shown in Figure 1. The styles, aligned with the hierarchical representation, are then fed into the generator in correspondence to their scale to generate the output image, thus completing the translation from input pixels to output pixels, through the intermediate style representation. Therefore, our architecture, pSp, is an end-to-end image-to-image translation framework. The complete architecture is illustrated in Figure 1. + +As in StyleGAN, we further define $\overline { { \mathbf { W } } }$ to be the average style vector of the pretrained generator. Given an input image, $\mathbf { X }$ , the output of our model is then defined as $p S p ( \mathbf { x } ) : = \mathbf { \bar { \boldsymbol { G } } } ( E ( \mathbf { \boldsymbol { x } } ) + \mathbf { \bar { \boldsymbol { w } } } )$ where $E ( \cdot )$ and $G ( \cdot )$ denote the encoder and StyleGAN generator, respectively. In this formulation, our encoder aims to learn the latent code with respect to the average style vector. We find that this results in better initialization. + +# 3.1 LOSS FUNCTIONS + +While the style-based translation is the core part of our framework, the choice of losses is also crucial. Our encoder is trained using a weighted combination of several objectives. First, we utilize the pixel-wise $\mathcal { L } _ { 2 }$ loss, + +$$ +\mathcal { L } _ { 2 } \left( \mathbf { x } \right) = | | \mathbf { x } - p S p ( \mathbf { x } ) | | _ { 2 } . +$$ + +In addition, to learn perceptual similarities, we utilize the LPIPS (Zhang et al., 2018) loss, which has been shown to better preserve image quality (Guan et al., 2020) compared to the more standard perceptual loss (Johnson et al., 2016): + +$$ +\mathcal { L } _ { \mathrm { L P I P S } } \left( \mathbf { x } \right) = | | F ( \mathbf { x } ) - F ( p S p ( \mathbf { x } ) ) | | _ { 2 } , +$$ + +where $F ( \cdot )$ denotes the perceptual feature extractor. + +To encourage the encoder to output latent style vectors closer to the average latent vector, we additionally define the following regularization loss: + +$$ +\mathcal { L } _ { \mathrm { r e g } } \left( \mathbf { x } \right) = | | E ( \mathbf { x } ) - \overline { { \mathbf { w } } } | | _ { 2 } . +$$ + +Similar to the truncation trick introduced in StyleGAN, we find that adding this regularization in the training of our encoder improves image quality without harming the fidelity of our outputs, especially in some of the more ambiguous tasks explored below. + +The Identity Loss One of the main challenges of face generation tasks is the ability to preserve identity between the input and output images. Since identity preservation is a crucial part of face reconstruction tasks, it is important to integrate this objective into the overall loss function. Therefore, we incorporate a dedicated recognition loss measuring the cosine similarity between the output image and its source, + +$$ +\mathcal { L } _ { \mathrm { I D } } \left( \mathbf { x } \right) = 1 - \left. R ( \mathbf { x } ) , R ( p S p ( \mathbf { x } ) ) \right. , +$$ + +where $R$ is a pretrained ArcFace (Deng et al., 2019) network for face recognition. The input, $\mathbf { X }$ , and output, $p S p ( \mathbf { x } )$ , are cropped around the face and resized to $1 1 2 \times 1 1 2$ before being fed into $R$ . + +In summary, the total loss function is defined as + +$$ +\begin{array} { r } { \mathcal { L } ( \mathbf { x } ) = \lambda _ { 1 } \mathcal { L } _ { 2 } ( \mathbf { x } ) + \lambda _ { 2 } \mathcal { L } _ { \mathrm { L P I P S } } ( \mathbf { x } ) + \lambda _ { 3 } \mathcal { L } _ { \mathrm { I D } } ( \mathbf { x } ) + \lambda _ { 4 } \mathcal { L } _ { \mathrm { r e g } } ( \mathbf { x } ) , } \end{array} +$$ + +where $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \lambda _ { 4 }$ are constants defining the loss weights. Constants and other implementation details can be found in Appendix A.1. + +# 3.2 THE BENEFITS OF THE STYLEGAN DOMAIN + +The translation between images through the style domain differentiates pSp from many standard image-to-image translation frameworks, as it makes our model operate globally instead of locally, without requiring pixel-to-pixel correspondence. This is a desired property as it has been shown that the locality bias limits current methods when handling non-local transformations (Richardson & Weiss, 2020). Moreover, previous works (Karras et al., 2019; Collins et al., 2020) have demonstrated that the disentanglement of semantic objects learned by StyleGAN is due to its layer-wise representation. This ability to independently manipulate semantic attributes leads to another desired property: the support for multi-modal synthesis. As some translation tasks are ambiguous, where a single input image may correspond to several outputs, it is desirable to be able to sample these possible outputs. While this requires specialized changes in standard image-to-image architectures (Zhu et al., 2017b; Huang et al., 2018), our framework inherently supports this by simply sampling style vectors. In practice, this is done by randomly sampling a vector $\mathbf { w } \in \mathbb { R } ^ { 5 1 2 }$ and generating a corresponding latent code in $\mathcal { W } +$ by replicating w. Style mixing is then performed by replacing select layers of the computed latent with those of the randomly generated latent, possibly with an $\alpha$ parameter for blending between the two styles. This is illustrated in Figure 7a in Appendix A. There, layers 1-7 are selected from the input latent while layers 8-18 are taken from the sampled vector allowing one to obtain outputs with similar coarse and medium features, but varying fine features. + +# 4 APPLICATIONS AND EXPERIMENTS + +To explore the effectiveness of our approach we evaluate our pSp framework on numerous imageto-image translation tasks. + +# 4.1 STYLEGAN INVERSION + +We start by evaluating the usage of the pSp framework for StyleGAN Inversion, that is, finding the latent code of real images in the latent domain. We compare our method to the ALAE encoder (Pidhorskyi et al., 2020) and to the encoder from IDInvert (In-Domain Invert) (Zhu et al., 2020a). The ALAE method proposes a StyleGAN-based autoencoder, where the encoder is trained alongside the generator to generate latent codes. In IDInvert, real images are embedded into the latent domain of a pretrained StyleGAN by first encoding the image into $\mathcal { W } +$ and then directly optimizing over the generated image to tune the latent. For a fair comparison with our method, we compare with IDInvert where no further optimization is performed after computing the encoding of a given image. + +![](images/bb69cc85325bb39515b25424afb41cb5f70bd3c6eb6a0999b0c6640a439ba37d.jpg) +Figure 2: Results of pSp for StyleGAN inversion compared to other approaches on CelebA-HQ. + +![](images/fd8e8a9c98b438418175d16bb3f7e30702a7cca4f16c79e2de2b313cc8b5bd50.jpg) +Figure 3: (a) Ablation of the pSp encoder over CelebA-HQ. (b) The importance of the identity loss. + +Results Figure 2 shows a qualitative comparison between the methods. One can see that the ALAE method, operating in the $\mathcal { W }$ domain, cannot accurately reconstruct the input images. While IDInvert (Zhu et al., 2020a) better preserves the image attributes, it still fails to accurately preserve identity and the finer details of the input image. In contrast, our method is able to preserve identity while also reconstructing fine details such as lighting, hairstyle, and glasses. + +Next, we conduct an ablation study to analyze the effectiveness of the pSp architecture. We compare our architecture to two simpler variations. First, we define an encoder generating a 512-dimensional style vector in the $\mathcal { W }$ latent domain, extracted from the last layer of the encoder network. We then expand this and define an encoder with an additional layer to transform the 512-dimensional feature vector to a full $1 8 \times 5 1 2 ~ { \textmu } { \bmod { } }$ vector. Figure 3a shows that while this simple extension into $\mathcal { W } +$ significantly improves the results, it still cannot preserve the finer details generated by our architecture. In Figure 3b we show the importance of the identity loss in the reconstruction task. + +Finally, Table 4a presents a quantitative evaluation measuring the different encoders examined above. Our pSp model is able to better preserve the original images in terms of both perceptual similarity and identity. To make sure the similarity score is independent of our loss function, we utilize the Curricularface (Huang et al., 2020) method for evaluation. + +Figure 4: (a) Quantitative results for image reconstruction on CelebA-HQ. (b) Results for Face Frontalization on the FEI Face Database split by rotation angle of the face in the input image. + +
Method↑ Similarity↓LPIPS↓MSERuntime
ALAE IDInvert0.06 0.180.32 0.220.15 0.060.207 0.032
W Encoder Naive W+0.350.23 0.190.06 0.040.064 0.064
0.49 0.560.170.030.105
pSp
(a)
+ +
Method 90°个 Similarity ↓Runtime 70° 50° 30°
R&R 0.34 0.56 0.660.7 1.5
pSp 0.32 0.52 0.600.63 0.1
(b)
+ +![](images/6cb149008410b1e68a49694b950d34ccbea9c3396117e7f7edb3172de4e05421.jpg) +Figure 5: Comparison of face frontalization methods. + +# 4.2 FACE FRONTALIZATION + +Face frontalization is a challenging task for image-to-image translation frameworks due to the required non-local transformations and the lack of paired training data. RotateAndRender (R&R) (Zhou et al., 2020) overcome this challenge by incorporating a geometric 3D alignment process before the translation process. Alternatively, we show that our style-based translation mechanism is able overcome these challenges, even when trained with no labeled data. + +Methodology and details For this task, training is the same as the encoder formulation with two important changes. First, we randomly flip the target image, thus creating inconsistencies in terms of pose compared to the input image. This guides the model towards generating a frontalized face, as the true target pose is unknown. While this may seem minor, without this augmentation the model would simply learn to encode the input image, matching its pose as well as identity. Next, in frontalization, as we are less interested in the background region compared to the face region and its identity, we also change the weights of the loss objective. In particular, we decrease the weights of the LPIPS and $L _ { 2 }$ loss functions, and give more weight to the losses computed on the inner part of the face, focusing the model on the inner region while reducing the importance of background preservation. As shown below, these changes to the training objective are enough for the model to generate realistic frontal faces, while also preserving identity. + +Results Results are illustrated in Figure 5. When trained with the same data, pix2pixHD is unable to converge to satisfying results as it is much more dependent on the correspondence between the input and output pairs. Conversely, our method is able to handle the task successfully, generating realistic frontal faces, which are comparable to the more involved RotateAndRender approach. This shows the benefit of using a pretrained StyleGAN for image translation, as it allows us to achieve visually-pleasing results even with weak supervision. Table 4b provides a quantitative evaluation on the FEI Faces Database (Thomaz & Giraldi, 2010). While R&R outperforms pSp, our simple approach provides an elegant alternative, without requiring specialized alignment steps. + +![](images/2fae6c2fd49241b71e411ab8fe52917c7054060d2baca609e732f5db62918b51.jpg) +Figure 6: (a) Comparison of sketches presented in DeepFaceDrawing. (b) Comparisons to other label-to-image methods on CelebAMask-HQ. (c) Multi-modal outputs using pSp with style-mixing. (d) Human evaluation results on CelebA-HQ for Conditional Image Synthesis tasks. Each cell denotes the percentage of users who favored pSp over the listed method. + +# 4.3 CONDITIONAL IMAGE SYNTHESIS + +Conditional image synthesis aims at generating photo-realistic images conditioned on certain input types. In this section, our pSp architecture is tested on two conditional image generation tasks: generating high-quality face images from sketches and semantic label maps. We demonstrate that, with only minimal changes, our encoder successfully utilizes the expressiveness of StyleGAN to generate high-quality and diverse outputs. Additionally, an ideal mapping framework should be able to generate multiple diverse outputs for a given input. To achieve this, we utilize the multi-modal synthesis approach described in Section 3.2. + +Methodology and details The training of the two conditional generation tasks is identical to that of the encoder for StyleGAN inversion except for the omission of the identity loss and the addition of the regularization loss. To generate multiple images at inference time, we perform style-mixing, taking layers $( 1 - 7 )$ from the latent code of the input image and layers $( 8 - 1 8 )$ from a randomly drawn w vector. + +# 4.3.1 FACE FROM SKETCH + +Common approaches for sketch-to-image synthesis incorporate hard constraints that require pixelwise correspondence between the input sketch and generated image, making them ill-suited when given incomplete sketches. DeepFaceDrawing (Chen et al., 2020) address this using a set of dedicated mapping networks. We show that pSp provides a simple alternative to past approaches. + +Dataset Construction As there are currently no publicly available datasets representative of handdrawn face sketches, we elect to construct our own dataset, which we describe in Appendix A.2. + +Results Figure 6a compares the results of our method to those of pix2pixHD and DeepFaceDrawing. As no code release is available for DeepFaceDrawing, we compare directly with sketches and results published in their paper. Due to the hard constraints of pix2pixHD, they are unable to handle the abstract sketches and obtain poor visual results. While DeepFaceDrawing obtain more visually pleasing results compared to pix2pixHD, they are still limited in their diversity. Conversely, although our model is trained on a different dataset, we are still able to generalize well to their sketches. Notably, we observe our ability to obtain more diverse outputs that better retain finer details (e.g. facial hair). Another limitation of DeepFaceDrawing is its focus on frontal images. We therefore illustrate our model’s ability to generate high-fidelity outputs from non-frontal sketches in Figure 13. As we are unable to directly evaluate DeepFaceDrawing on our constructed dataset, we compare our results only to those of pix2pixHD, trained and evaluated with the same data. + +# 4.3.2 FACE FROM SEGMENTATION MAP + +Here, we evaluate using pSp for synthesizing face images from segmentation maps. In addition to pix2pixHD, we compare our approach to two additional state-of-the-art label-to-image methods: SPADE (Park et al., 2019), and CC FPSE (Liu et al., 2019b), both of which are based on pix2pixHD. + +Results In Figure 6b we provide a visual comparison of the competing approaches on the CelebAMask-HQ dataset containing 19 semantic categories. As the competing methods are based on pix2pixHD, the results of all three are visually similar. Conversely, our approach is able to generate high-quality outputs across a wide range of inputs of various poses and expressions. Additionally, using our multi-modal technique, pSp can easily generate various possible outputs with the same pose and attributes but varying fine styles for a single input semantic map or sketch image. We provide examples in Figure 6c with additional examples in Appendix C. + +# 4.3.3 HUMAN PERCEPTUAL STUDY + +We additionally perform a human evaluation to compare the visual quality of each method presented above. Here, each worker is given two images synthesized by different methods on the same input and is given an unlimited time to select which output looks more realistic. Each of our three workers reviews approximately 2, 800 pairs for each task, resulting in over 8, 400 human judgements for each method. Table 6d shows that pSp significantly outperforms the other respective methods in both synthesis tasks. + +# 5 DISCUSSION AND CONCLUSIONS + +Although our suggested framework for image-to-image translation achieves compelling results in various applications, it has some inherent assumptions that should be considered. First, the highquality images that are generated by utilizing the pretrained StyleGAN come with a cost — the method is limited to images that can be generated by StyleGAN. Thus, generating faces which are not close to frontal, or have certain expressions may be challenging if such examples were not available when training the StyleGAN model. Also, the global approach of $\mathsf { p } \mathsf { S p }$ , although advantageous for many tasks, does introduce a challenge in preserving finer details of the input image, such as earrings or background details. This is especially significant in tasks such as inpainting or superresolution where standard image-to-image architectures can simply propagate local information. Figure 7b in Appendix A presents some examples of such reconstruction failures. + +In this work, we proposed a novel encoder architecture that can be used to directly map a face image into the $\mathcal { W } +$ latent space with no optimization required. The encoder architecture, motivated by StyleGAN, consists of a hierarchy of three levels that correspond to the coarse, medium, and fine groupings of the 18 style vectors defining the input in the $\mathcal { W } +$ latent space. Styles are then extracted from the encoder in a hierarchical fashion and fed into the corresponding inputs of a fixed StyleGAN generator. Notably, our network is trained with an ID similarity loss, which encourages better preservation of identity compared to previous direct approaches. Combining our encoder with a StyleGAN decoder, we present a general framework for solving various image-to-image translation tasks. In contrast to previous methods, which tackle such tasks using a local ”pixel-topixel” approach, our framework takes a global approach, which we show can be used to solve a wide variety of image-to-image translation problems. + +# REFERENCES + +Rameen Abdal, Yipeng Qin, and Peter Wonka. Image2stylegan: How to embed images into the stylegan latent space? In Proceedings of the IEEE international conference on computer vision, pp. 4432–4441, 2019. + +Rameen Abdal, Yipeng Qin, and Peter Wonka. Image2stylegan $^ { + + }$ : How to edit the embedded images? In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8296–8305, 2020. + +Rameen Adbal, Pie Zhu, Niloy J. Mitra, and Peter Wonka. 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(b) Challenging cases for StyleGAN Inversion. + +A ADDITIONAL DETAILS + +# A.1 IMPLEMENTATION DETAILS + +Training Details For our backbone network we use the ResNet-IR architecture from (Deng et al., 2019) pretrained on face recognition, which accelerated convergence. We use a fixed StyleGAN2 generator trained on the FFHQ (Karras et al., 2019) dataset. That is, only the pSp encoder network is trained on the given image-to-image translation task. For all applications, the input image resolution is $2 5 6 \times 2 5 6$ , where the generated $1 0 2 4 \times 1 0 2 4$ output is resized before being fed into the loss functions. For training, we use the Ranger optimizer, a combination of Rectified Adam (Liu et al., 2019a) with the Lookahead technique (Zhang et al., 2019), with a constant learning rate of 0.001. Only horizontal flips are used as augmentations during training. All experiments are performed using a single NVIDIA Tesla P40 GPU. + +For the StyleGAN inversion task, the $\lambda$ values are set as $\lambda _ { 1 } = 1$ , $\lambda _ { 2 } = 0 . 8$ , $\lambda _ { 3 } = 0 . 1$ . For face frontalization, we increase the weight of the identity loss, setting $\lambda _ { 3 } = 1$ , and decrease the LPIPS and $L _ { 2 }$ loss functions, setting $\lambda _ { 1 } = 0 . 0 1$ , $\lambda _ { 2 } = 0 . 8$ over the inner part of the face and $\lambda _ { 1 } = 0 . 0 0 1$ , $\lambda _ { 2 } = 0 . 0 8$ elsewhere. Additionally, the constants used in the conditional image synthesis tasks are identical to those used in the inversion task except for the omission of the identity loss (i.e. we set $\lambda _ { 3 } = 0$ ). Finally, $\lambda _ { 4 }$ is set to 0.005 in all applications except for the StyleGAN inversion task, which does not utilize the regularization loss. + +# A.2 DATASETS + +We conduct our experiments on the CelebA-HQ dataset (Karras et al., 2018), which contains 30,000 high quality images. We use a standard train-test split of the dataset, resulting in approximately 24,000 training images. The FFHQ dataset from (Karras et al., 2019), which contains 70,000 face images, is used for the StyleGAN inversion and face frontalization tasks. + +For the generation of face images from sketches, we construct a dataset representative of handdrawn sketches using the CelebA-HQ dataset (Karras et al., 2018). Given an input image, we first apply a “pencil sketch” filter which retains most facial details of the original image while removing the remaining noise. We then apply the sketch-simplification method by Simo-Serra et al. (2016), resulting in images resembling hand-drawn sketches. + +# B ADDITIONAL APPLICATIONS + +# B.1 SUPER RESOLUTION + +Here we show that our framework can be used to construct high-resolution (HR) facial images from corresponding low-resolution (LR) input images. PULSE (Menon et al., 2020) approaches this task in an unsupervised manner by traversing the HR image manifold in search of an image that downsamples to the input LR image. In this work we focus on applying pSp in a supervised manner as obtaining paired data is immediate. We show that our method achieves comparable results to PULSE and other previous works. + +Methodology and details We train our model in a supervised fashion, where for each input we perform random bi-cubic down-sampling of $\times 1$ (i.e. no down-sampling), $\times 2 , \times 4 , \times 8$ , $\times 1 6$ , $\times 3 2$ and set the original, full resolution image as the target. + +Results Figure 9 demonstrates the visual quality of the resulting images from our method along with those of the previous approaches. Although PULSE is able to achieve very high-quality results due to their usage of StyleGAN to generate images, they are unable to accurately retain identity even when performing down-sampling of $\times 8$ to a resolution of $3 2 \times 3 2$ . By learning a pixel-wise correspondence between the LR and HR images, pix2pixHD is able to obtain satisfying results even when down-sampled to a resolution of $1 6 \times 1 6$ (i.e. $\times 1 6$ down-sampling). However, visually, their results appear less photo-realistic. Contrary to these previous works, we are able to obtain highquality results even when down-sampling to resolutions of $1 6 \times 1 6$ and $8 \times 8$ . Finally, we generate multiple outputs for a given LR image using our multi-modal technique by perform style-mixing on layers (4-7) with an $\alpha$ value of 0.5 with a randomly sampled w vector, which alters medium-level styles that mainly control facial features. Figure 10 illustrates the results. + +# B.2 EVEN MORE APPLICATIONS + +To better show the flexibility of our pSp framework, We present three additional applications, which are summarized in Figure 8. + +Local Editing Our framework allows for a simple approach to local image editing where altering specific attributes of an input sketch (e.g. eyes, smile) or segmentation map (e.g. hair) results in local edits of the generated images. + +Face Interpolation Given two real images one can obtain their respective latent codes $w _ { 1 } , w _ { 2 } \in$ $\mathcal { W } +$ by feeding the images through our encoder. We can then naturally interpolate between the two images by computing their intermediate latent code $w ^ { \prime } = \lambda w _ { 1 } + ( 1 \bar { - \lambda } ) \bar { w } _ { 2 }$ for $0 \leq \lambda \leq 1$ and generate the corresponding image using the new code $w ^ { \prime }$ . + +Inpainting Finally, we show the ability of our framework to reconstruct missing parts of an image using a simple, symmetric triangular mask. Our approach is able to accurately reconstruct the occluded areas while preserving the identity with respect to the original image. + +![](images/86f5a49cffbd32d7a6733400d7854ef3fa70e9da809eb23cd98d657af77aae3e.jpg) +Figure 8: Additional applications for the pSp framework. + +![](images/306f3b0994bcc83d2eb8bcc0bfefa18fb443c65fcaf1adadee5c332ec672f0c4.jpg) +Figure 9: Comparison of super-resolution approaches with (a) $\times 8$ down-sampling, (b) $\times 1 6$ downsampling, and (c) $\times 3 2$ down-sampling. + +![](images/cc851dc564bcf59d7ad3cc72569ed9fb3ead6857cedb2026f61dfd6d209bbd20.jpg) +Figure 10: Multi-modal synthesis for super-resolution using pSp with style-mixing. + +# C ADDITIONAL RESULTS + +![](images/0b25c29782244b03b64745b80191702264eb58f7874ed7bb4b9668287da6e962.jpg) +Figure 11: Additional StyleGAN inversion results using pSp on the CelebA-HQ (Karras et al., 2018) test set. + +![](images/0bc1a6317ea8cc91663d5ac69208bcc68da61cbde0a677bb5ea6a84e8a4b80f5.jpg) +Figure 12: Additional face frontalization results using pSp on the CelebA-HQ (Karras et al., 2018) test set. + +![](images/d2f8dd3a4bf409c2f63b861f3263b95631a3910158bfd6c3ff50947224bac76f.jpg) +Figure 13: Even for challenging, non-frontal face sketches, pSp is able to obtain high-quality, diverse outputs. + +![](images/e0db11481e9689ab3efaf35c551a8e00ff24be6e85b5d47c8c790141a433397a.jpg) +Figure 14: Additional results using pSp for the generation of face images from sketches constructed from the CelebA-HQ (Karras et al., 2018) test dataset. + +![](images/285edcda767a71026820f65323a255deec2bce6771dc0a1f83b629dcf5a456c8.jpg) +Figure 15: Additional results on the Helen Faces (Le et al., 2012) dataset using our proposed labelto-image method. + +![](images/a9f3f2b94b8e1934a25bd2d1d39736774770d5133ef0ff5c9e79e28416c3dbed.jpg) +Figure 16: Additional results on the CelebAMask-HQ (Karras et al., 2018) test set using our proposed label-to-image method. + +![](images/fe2db569bda27147de22c163e828d57efa05c8729d3e9bc45cda5f45a1c704d7.jpg) +Figure 17: Conditional image synthesis results from sketches and segmentation maps displaying the multi-modal property of our approach. \ No newline at end of file diff --git a/parse/train/qmI0P1ZExUl/qmI0P1ZExUl_content_list.json b/parse/train/qmI0P1ZExUl/qmI0P1ZExUl_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..33306f3d43201754f2547ed9eed168032b464ac0 --- /dev/null +++ b/parse/train/qmI0P1ZExUl/qmI0P1ZExUl_content_list.json @@ -0,0 +1,1734 @@ +[ + { + "type": "text", + "text": "ENCODING IN STYLE: A STYLEGAN ENCODER FOR IMAGE-TO-IMAGE TRANSLATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present a generic image-to-image translation framework, Pixel2Style2Pixel $( p S p )$ . Our pSp framework is based on a novel encoder network that directly generates a series of style vectors which are fed into a pretrained StyleGAN generator, forming the extended $\\mathcal { W } +$ latent space. We first show that our encoder can directly embed real images into $\\mathcal { W } +$ , with no additional optimization. We further introduce a dedicated identity loss which is shown to achieve improved performance in the reconstruction of an input image. We demonstrate pSp to be a simple architecture that, by leveraging a well-trained, fixed generator network, can be easily applied on a wide-range of image-to-image translation tasks. Solving these tasks through the style representation results in a global approach that does not rely on a local pixel-to-pixel correspondence and further supports multi-modal synthesis via the resampling of styles. Notably, we demonstrate that pSp can be trained to align a face image to a frontal pose with no labeled data and generate multi-modal results for ambiguous tasks such as conditional face generation from sketches and segmentation maps. ", + "bbox": [ + 233, + 267, + 764, + 474 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 502, + 336, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, Generative Adversarial Networks (GANs) have significantly advanced image synthesis, particularly on face images. State-of-the-art image generation methods have achieved high visual quality and fidelity, and can now generate images with phenomenal realism. Most notably, StyleGAN (Karras et al., 2019; 2020) proposes a novel style-based generator architecture and attains state-of-the-art visual quality on high-resolution images. Moreover, it has been demonstrated that it has a disentangled latent space, $\\mathcal { W }$ (Yang et al., 2019; Collins et al., 2020; Shen et al., 2020), which may offer control and editing capabilities. ", + "bbox": [ + 174, + 534, + 825, + 631 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, numerous methods have shown competence in controlling StyleGAN’s latent space and performing meaningful manipulations in $\\mathcal { W }$ (Jahanian et al., 2019; Shen et al., 2020; Tewari et al., 2020; Hark ¨ onen et al., 2020). To perform such edits on real images, one needs to invert the image ¨ into StyleGAN’s latent space, i.e., retrieve the latent code that reconstructs the image. However, it has been shown that inverting a real image into a 512-dimensional vector $\\mathbf { w } \\in \\mathcal { W }$ does not lead to an accurate reconstruction. Motivated by this, it has become common practice (Abdal et al., 2019; 2020; Baylies, 2019; Zhu et al., 2020a; Adbal et al., 2020) to encode real images into an extended latent space, $\\mathcal { W } +$ , defined by the concatenation of 18 different 512-dimensional w vectors, one for each input layer of StyleGAN. Nevertheless, many methods resort to using per-image optimization over $\\mathcal { W } +$ , requiring several minutes for a single image. To accelerate this optimization process, some methods (Baylies, 2019; Zhu et al., 2020a) have trained an encoder to infer an approximate vector in $\\mathcal { W } +$ which serves as a good initial point from which additional optimization is required. However, a fast, direct, and accurate learned inversion of real images into $\\mathcal { W } +$ remains a challenge. ", + "bbox": [ + 173, + 638, + 825, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we focus on the broader task of latent space embedding, which aims to retrieve the latent vector that generates a desired, not necessarily known, image. We do so by introducing a novel encoder architecture tasked with encoding an arbitrary image directly into $\\mathcal { W } +$ . The encoder is based on a Feature Pyramid Network (Lin et al., 2017), where style feature vectors are extracted from different pyramid scales and inserted directly into a fixed, pretrained StyleGAN generator in correspondence to their spatial scales. Our encoder into $\\mathcal { W } +$ , together with the StyleGAN decoder, form a generic encoder-decoder network that benefits many image-to-image translation tasks. Focusing on face images, we first demonstrate our method’s ability to successfully reconstruct a given image while preserving identity and other attributes. We then present numerous image-to-image translation applications. In a sense, our method performs Pixel2Style2Pixel translation, as every image is first encoded into style vectors and then into an image, and is therefore dubbed $p S p$ . ", + "bbox": [ + 174, + 825, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While many previous approaches to solving image-to-image translations tasks involve dedicated architectures specific for solving a single problem, we follow the spirit of pix2pix (Isola et al., 2017) and define a generic framework able to solve a wide range of tasks, all using the same architecture. Besides the simplification of the training process, as no adversary discriminator needs to be trained, using a pretrained StyleGAN generator offers several intriguing advantages over previous works. Many image-to-image architectures explicitly feed the generator with residual feature maps from the encoder (Isola et al., 2017; Wang et al., 2018), creating a strong locality bias (Richardson & Weiss, 2020). In contrast, our generator is governed only by the styles with no direct spatial input. The advantage of such a global approach is most evident in the task of Face Frontalization, where our encoder can be trained to align a given face image to a frontal pose with no labeled data. Another notable advantage of the intermediate style representation is the inherent support for multi-modal synthesis for ambiguous tasks such as face generation from sketches, segmentation maps, or lowresolution images. In such tasks, the generated styles can be resampled to create variations of the output image with no change to the architecture or training process. ", + "bbox": [ + 174, + 166, + 825, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main contributions of this paper are: (i) a novel StyleGAN encoder able to directly encode real face images into the $\\mathcal { W } +$ target latent domain; and (ii) a generic end-to-end framework for solving image-to-image translation tasks. ", + "bbox": [ + 178, + 367, + 821, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 431, + 343, + 446 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Latent Space Embedding With the rapid evolution of GANs, many works have tried to understand and control their latent space. A specific task that has received substantial attention is $G A N$ Inversion — where the latent vector from which a pretrained GAN most accurately reconstructs a given, known image, is sought. Motivated by its state-of-the-art image quality and latent space semantic richness, many recent works have used StyleGAN (Karras et al., 2019; 2020) for this task. Generally, inversion methods either directly optimize the latent vector to minimize the error for the given image (Lipton & Tripathi, 2017; Creswell & Bharath, 2018; Abdal et al., 2019; 2020), train an encoder to map the given image to the latent space (Perarnau et al., 2016; Creswell & Bharath, 2018; Pidhorskyi et al., 2020; Guan et al., 2020; Nitzan et al., 2020), or use a hybrid approach combining both (Baylies, 2019; Zhu et al., 2020a). Typically, methods performing optimization are superior in reconstruction quality to a learned encoder mapping, but require a substantially longer time. Unlike the above methods, our encoder can accurately and efficiently embed a given face image into the extended latent space $\\mathcal { W } +$ of a fixed, pretrained StyleGAN generator, with no further optimization. ", + "bbox": [ + 173, + 463, + 825, + 643 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Image-to-Image Image-to-Image translation techniques aim at learning a conditional image generation function that maps an input image of a source domain to a corresponding image of a target domain. Isola et al. (2017) first introduced the use of conditional GANs to solve various imageto-image translation tasks. Since then, their work has been extended for many scenarios: highresolution synthesis (Wang et al., 2018), unsupervised learning (Liu et al., 2017; Zhu et al., 2017a; Katzir et al., 2019; Lira et al., 2020), multi-modal image synthesis (Zhu et al., 2017b; Huang et al., 2018; Choi et al., 2020), and conditional image synthesis (Park et al., 2019; Li et al., 2019; Liu et al., 2019b; Zhu et al., 2020b; Chen et al., 2020). The aforementioned works have constructed dedicated architectures, which require training the generator network. ", + "bbox": [ + 174, + 661, + 825, + 786 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Latent-Space Manipulation Recently, numerous papers have presented diverse methods to learn semantic edits of the latent code. A popular approach is finding linear directions that correspond to changes in a given binary labeled attribute, such as young $ \\mathrm { o l d }$ , or no-smile smile (Shen et al., 2020; Goetschalckx et al., 2019; Denton et al., 2019; Adbal et al., 2020). Tewari et al. (2020) utilize a pretrained 3DMM to learn semantic face edits in the latent space. Jahanian et al. (2019) find latent space paths that correspond to a specific transformation, such as zoom or rotation, in a selfsupervised manner. Hark ¨ onen et al. (2020) find useful paths in an unsupervised manner by using the ¨ principal component axes (PCA) of an intermediate activation space. Finally, Collins et al. (2020) perform local semantic editing by manipulating corresponding components of the latent code. ", + "bbox": [ + 174, + 803, + 825, + 928 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/aaa87c70d8d04df81fa258ed78659642cc51ccb7d39c7c3bd9e96fd5502d0ea6.jpg", + "image_caption": [ + "Figure 1: Our pSp architecture. Feature maps are first extracted using a standard feature pyramid over a ResNet backbone. For each of the 18 target styles, a small mapping network is trained to extract the learned styles from the corresponding feature map, where styles (0-2) are generated from the small feature map, (3-6) from the medium feature map, and (7-18) from the largest feature map. The mapping network, map2style, is a small fully convolutional network, which gradually reduces spatial size using a set of 2-strided convolutions followed by LeakyReLU activations. Each generated 512 vector, is fed into StyleGAN, starting from its matching affine transformation, $A$ . " + ], + "image_footnote": [], + "bbox": [ + 178, + 111, + 815, + 268 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 THE PSP FRAMEWORK ", + "text_level": 1, + "bbox": [ + 176, + 397, + 395, + 414 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our pSp framework builds upon the representative power of a pretrained StyleGAN generator and the $\\mathcal { W } +$ latent space. To utilize this representation one needs a strong encoder that is able to match each input image to an accurate encoding in the latent domain. A simple technique to embed into this domain is directly encoding a given input image into $\\mathcal { W } +$ using a single 512-dimensional vector obtained from the last layer of the encoder network, thereby learning all 18 style vectors together. However, such an architecture presents a strong bottleneck making it difficult to fully represent the finer details of the original image and therefore limiting the reconstruction quality. ", + "bbox": [ + 173, + 429, + 825, + 527 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In StyleGAN, the authors have shown that the different style inputs correspond to different levels of detail, which are roughly divided into three groups — coarse, medium, and fine. Following this observation, in pSp we extend an encoder backbone with a feature pyramid (Lin et al., 2017), generating three levels of feature maps from which styles are extracted using a simple intermediate network — map2style — shown in Figure 1. The styles, aligned with the hierarchical representation, are then fed into the generator in correspondence to their scale to generate the output image, thus completing the translation from input pixels to output pixels, through the intermediate style representation. Therefore, our architecture, pSp, is an end-to-end image-to-image translation framework. The complete architecture is illustrated in Figure 1. ", + "bbox": [ + 174, + 534, + 825, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As in StyleGAN, we further define $\\overline { { \\mathbf { W } } }$ to be the average style vector of the pretrained generator. Given an input image, $\\mathbf { X }$ , the output of our model is then defined as $p S p ( \\mathbf { x } ) : = \\mathbf { \\bar { \\boldsymbol { G } } } ( E ( \\mathbf { \\boldsymbol { x } } ) + \\mathbf { \\bar { \\boldsymbol { w } } } )$ where $E ( \\cdot )$ and $G ( \\cdot )$ denote the encoder and StyleGAN generator, respectively. In this formulation, our encoder aims to learn the latent code with respect to the average style vector. We find that this results in better initialization. ", + "bbox": [ + 174, + 666, + 825, + 736 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 LOSS FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 750, + 339, + 763 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While the style-based translation is the core part of our framework, the choice of losses is also crucial. Our encoder is trained using a weighted combination of several objectives. First, we utilize the pixel-wise $\\mathcal { L } _ { 2 }$ loss, ", + "bbox": [ + 174, + 772, + 825, + 813 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ed6a47020474faf5981a01d77bc38265eb39485419124eb91c6b0d8fb980efe9.jpg", + "text": "$$\n\\mathcal { L } _ { 2 } \\left( \\mathbf { x } \\right) = | | \\mathbf { x } - p S p ( \\mathbf { x } ) | | _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 811, + 584, + 830 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In addition, to learn perceptual similarities, we utilize the LPIPS (Zhang et al., 2018) loss, which has been shown to better preserve image quality (Guan et al., 2020) compared to the more standard perceptual loss (Johnson et al., 2016): ", + "bbox": [ + 174, + 835, + 825, + 878 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/54894a7637408695b964206da7835068194769aee59c79b2946c7696f97338d3.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { L P I P S } } \\left( \\mathbf { x } \\right) = | | F ( \\mathbf { x } ) - F ( p S p ( \\mathbf { x } ) ) | | _ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 885, + 620, + 902 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $F ( \\cdot )$ denotes the perceptual feature extractor. ", + "bbox": [ + 174, + 909, + 514, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To encourage the encoder to output latent style vectors closer to the average latent vector, we additionally define the following regularization loss: ", + "bbox": [ + 171, + 103, + 821, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f1c5dc391b34830af1c3286b0d244e181427b91eee692ecff3bf232f9553d035.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { r e g } } \\left( \\mathbf { x } \\right) = | | E ( \\mathbf { x } ) - \\overline { { \\mathbf { w } } } | | _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 138, + 584, + 156 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Similar to the truncation trick introduced in StyleGAN, we find that adding this regularization in the training of our encoder improves image quality without harming the fidelity of our outputs, especially in some of the more ambiguous tasks explored below. ", + "bbox": [ + 174, + 162, + 825, + 205 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Identity Loss One of the main challenges of face generation tasks is the ability to preserve identity between the input and output images. Since identity preservation is a crucial part of face reconstruction tasks, it is important to integrate this objective into the overall loss function. Therefore, we incorporate a dedicated recognition loss measuring the cosine similarity between the output image and its source, ", + "bbox": [ + 174, + 219, + 825, + 289 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/15aba6cec2e849f8f0ef121a73a940961c311e13c00ecf7cfd4787c7d0770118.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { I D } } \\left( \\mathbf { x } \\right) = 1 - \\left. R ( \\mathbf { x } ) , R ( p S p ( \\mathbf { x } ) ) \\right. ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 287, + 619, + 305 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $R$ is a pretrained ArcFace (Deng et al., 2019) network for face recognition. The input, $\\mathbf { X }$ , and output, $p S p ( \\mathbf { x } )$ , are cropped around the face and resized to $1 1 2 \\times 1 1 2$ before being fed into $R$ . ", + "bbox": [ + 173, + 313, + 830, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In summary, the total loss function is defined as ", + "bbox": [ + 173, + 348, + 486, + 362 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c0aef7429c8f9b3c2bd7340d2a5f037e36848b4744e168d022774c4cb1bec3fc.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\mathbf { x } ) = \\lambda _ { 1 } \\mathcal { L } _ { 2 } ( \\mathbf { x } ) + \\lambda _ { 2 } \\mathcal { L } _ { \\mathrm { L P I P S } } ( \\mathbf { x } ) + \\lambda _ { 3 } \\mathcal { L } _ { \\mathrm { I D } } ( \\mathbf { x } ) + \\lambda _ { 4 } \\mathcal { L } _ { \\mathrm { r e g } } ( \\mathbf { x } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 367, + 687, + 386 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\lambda _ { 1 } , \\lambda _ { 2 } , \\lambda _ { 3 } , \\lambda _ { 4 }$ are constants defining the loss weights. Constants and other implementation details can be found in Appendix A.1. ", + "bbox": [ + 174, + 391, + 821, + 421 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 THE BENEFITS OF THE STYLEGAN DOMAIN ", + "text_level": 1, + "bbox": [ + 174, + 436, + 526, + 452 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The translation between images through the style domain differentiates pSp from many standard image-to-image translation frameworks, as it makes our model operate globally instead of locally, without requiring pixel-to-pixel correspondence. This is a desired property as it has been shown that the locality bias limits current methods when handling non-local transformations (Richardson & Weiss, 2020). Moreover, previous works (Karras et al., 2019; Collins et al., 2020) have demonstrated that the disentanglement of semantic objects learned by StyleGAN is due to its layer-wise representation. This ability to independently manipulate semantic attributes leads to another desired property: the support for multi-modal synthesis. As some translation tasks are ambiguous, where a single input image may correspond to several outputs, it is desirable to be able to sample these possible outputs. While this requires specialized changes in standard image-to-image architectures (Zhu et al., 2017b; Huang et al., 2018), our framework inherently supports this by simply sampling style vectors. In practice, this is done by randomly sampling a vector $\\mathbf { w } \\in \\mathbb { R } ^ { 5 1 2 }$ and generating a corresponding latent code in $\\mathcal { W } +$ by replicating w. Style mixing is then performed by replacing select layers of the computed latent with those of the randomly generated latent, possibly with an $\\alpha$ parameter for blending between the two styles. This is illustrated in Figure 7a in Appendix A. There, layers 1-7 are selected from the input latent while layers 8-18 are taken from the sampled vector allowing one to obtain outputs with similar coarse and medium features, but varying fine features. ", + "bbox": [ + 173, + 462, + 825, + 699 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 APPLICATIONS AND EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 715, + 496, + 731 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To explore the effectiveness of our approach we evaluate our pSp framework on numerous imageto-image translation tasks. ", + "bbox": [ + 174, + 742, + 821, + 770 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 STYLEGAN INVERSION ", + "text_level": 1, + "bbox": [ + 174, + 786, + 382, + 800 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We start by evaluating the usage of the pSp framework for StyleGAN Inversion, that is, finding the latent code of real images in the latent domain. We compare our method to the ALAE encoder (Pidhorskyi et al., 2020) and to the encoder from IDInvert (In-Domain Invert) (Zhu et al., 2020a). The ALAE method proposes a StyleGAN-based autoencoder, where the encoder is trained alongside the generator to generate latent codes. In IDInvert, real images are embedded into the latent domain of a pretrained StyleGAN by first encoding the image into $\\mathcal { W } +$ and then directly optimizing over the generated image to tune the latent. For a fair comparison with our method, we compare with IDInvert where no further optimization is performed after computing the encoding of a given image. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/bb69cc85325bb39515b25424afb41cb5f70bd3c6eb6a0999b0c6640a439ba37d.jpg", + "image_caption": [ + "Figure 2: Results of pSp for StyleGAN inversion compared to other approaches on CelebA-HQ. " + ], + "image_footnote": [], + "bbox": [ + 189, + 101, + 805, + 335 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/fd8e8a9c98b438418175d16bb3f7e30702a7cca4f16c79e2de2b313cc8b5bd50.jpg", + "image_caption": [ + "Figure 3: (a) Ablation of the pSp encoder over CelebA-HQ. (b) The importance of the identity loss. " + ], + "image_footnote": [], + "bbox": [ + 191, + 386, + 813, + 625 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Results Figure 2 shows a qualitative comparison between the methods. One can see that the ALAE method, operating in the $\\mathcal { W }$ domain, cannot accurately reconstruct the input images. While IDInvert (Zhu et al., 2020a) better preserves the image attributes, it still fails to accurately preserve identity and the finer details of the input image. In contrast, our method is able to preserve identity while also reconstructing fine details such as lighting, hairstyle, and glasses. ", + "bbox": [ + 174, + 690, + 825, + 761 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Next, we conduct an ablation study to analyze the effectiveness of the pSp architecture. We compare our architecture to two simpler variations. First, we define an encoder generating a 512-dimensional style vector in the $\\mathcal { W }$ latent domain, extracted from the last layer of the encoder network. We then expand this and define an encoder with an additional layer to transform the 512-dimensional feature vector to a full $1 8 \\times 5 1 2 ~ { \\textmu } { \\bmod { } }$ vector. Figure 3a shows that while this simple extension into $\\mathcal { W } +$ significantly improves the results, it still cannot preserve the finer details generated by our architecture. In Figure 3b we show the importance of the identity loss in the reconstruction task. ", + "bbox": [ + 174, + 767, + 825, + 866 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, Table 4a presents a quantitative evaluation measuring the different encoders examined above. Our pSp model is able to better preserve the original images in terms of both perceptual similarity and identity. To make sure the similarity score is independent of our loss function, we utilize the Curricularface (Huang et al., 2020) method for evaluation. ", + "bbox": [ + 174, + 872, + 823, + 928 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/a280d0dec7a77df194ea62fa3a3fedb5fab186a7f9466359c3c6fbcd58dc0f68.jpg", + "table_caption": [ + "Figure 4: (a) Quantitative results for image reconstruction on CelebA-HQ. (b) Results for Face Frontalization on the FEI Face Database split by rotation angle of the face in the input image. " + ], + "table_footnote": [], + "table_body": "
Method↑ Similarity↓LPIPS↓MSERuntime
ALAE IDInvert0.06 0.180.32 0.220.15 0.060.207 0.032
W Encoder Naive W+0.350.23 0.190.06 0.040.064 0.064
0.49 0.560.170.030.105
pSp
(a)
", + "bbox": [ + 174, + 101, + 522, + 224 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/c0a46396f328ea673ec429666795e599adea90d562ae9756b92e6cbd9f4aa437.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Method 90°个 Similarity ↓Runtime 70° 50° 30°
R&R 0.34 0.56 0.660.7 1.5
pSp 0.32 0.52 0.600.63 0.1
(b)
", + "bbox": [ + 539, + 114, + 820, + 212 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/6cb149008410b1e68a49694b950d34ccbea9c3396117e7f7edb3172de4e05421.jpg", + "image_caption": [ + "Figure 5: Comparison of face frontalization methods. " + ], + "image_footnote": [], + "bbox": [ + 223, + 263, + 764, + 492 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 FACE FRONTALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 526, + 379, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Face frontalization is a challenging task for image-to-image translation frameworks due to the required non-local transformations and the lack of paired training data. RotateAndRender (R&R) (Zhou et al., 2020) overcome this challenge by incorporating a geometric 3D alignment process before the translation process. Alternatively, we show that our style-based translation mechanism is able overcome these challenges, even when trained with no labeled data. ", + "bbox": [ + 173, + 554, + 825, + 623 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Methodology and details For this task, training is the same as the encoder formulation with two important changes. First, we randomly flip the target image, thus creating inconsistencies in terms of pose compared to the input image. This guides the model towards generating a frontalized face, as the true target pose is unknown. While this may seem minor, without this augmentation the model would simply learn to encode the input image, matching its pose as well as identity. Next, in frontalization, as we are less interested in the background region compared to the face region and its identity, we also change the weights of the loss objective. In particular, we decrease the weights of the LPIPS and $L _ { 2 }$ loss functions, and give more weight to the losses computed on the inner part of the face, focusing the model on the inner region while reducing the importance of background preservation. As shown below, these changes to the training objective are enough for the model to generate realistic frontal faces, while also preserving identity. ", + "bbox": [ + 174, + 641, + 825, + 795 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results Results are illustrated in Figure 5. When trained with the same data, pix2pixHD is unable to converge to satisfying results as it is much more dependent on the correspondence between the input and output pairs. Conversely, our method is able to handle the task successfully, generating realistic frontal faces, which are comparable to the more involved RotateAndRender approach. This shows the benefit of using a pretrained StyleGAN for image translation, as it allows us to achieve visually-pleasing results even with weak supervision. Table 4b provides a quantitative evaluation on the FEI Faces Database (Thomaz & Giraldi, 2010). While R&R outperforms pSp, our simple approach provides an elegant alternative, without requiring specialized alignment steps. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/2fae6c2fd49241b71e411ab8fe52917c7054060d2baca609e732f5db62918b51.jpg", + "image_caption": [ + "Figure 6: (a) Comparison of sketches presented in DeepFaceDrawing. (b) Comparisons to other label-to-image methods on CelebAMask-HQ. (c) Multi-modal outputs using pSp with style-mixing. (d) Human evaluation results on CelebA-HQ for Conditional Image Synthesis tasks. Each cell denotes the percentage of users who favored pSp over the listed method. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 828, + 462 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 CONDITIONAL IMAGE SYNTHESIS ", + "text_level": 1, + "bbox": [ + 176, + 561, + 449, + 575 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Conditional image synthesis aims at generating photo-realistic images conditioned on certain input types. In this section, our pSp architecture is tested on two conditional image generation tasks: generating high-quality face images from sketches and semantic label maps. We demonstrate that, with only minimal changes, our encoder successfully utilizes the expressiveness of StyleGAN to generate high-quality and diverse outputs. Additionally, an ideal mapping framework should be able to generate multiple diverse outputs for a given input. To achieve this, we utilize the multi-modal synthesis approach described in Section 3.2. ", + "bbox": [ + 174, + 589, + 825, + 686 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Methodology and details The training of the two conditional generation tasks is identical to that of the encoder for StyleGAN inversion except for the omission of the identity loss and the addition of the regularization loss. To generate multiple images at inference time, we perform style-mixing, taking layers $( 1 - 7 )$ from the latent code of the input image and layers $( 8 - 1 8 )$ from a randomly drawn w vector. ", + "bbox": [ + 174, + 705, + 823, + 775 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3.1 FACE FROM SKETCH ", + "text_level": 1, + "bbox": [ + 176, + 794, + 372, + 809 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Common approaches for sketch-to-image synthesis incorporate hard constraints that require pixelwise correspondence between the input sketch and generated image, making them ill-suited when given incomplete sketches. DeepFaceDrawing (Chen et al., 2020) address this using a set of dedicated mapping networks. We show that pSp provides a simple alternative to past approaches. ", + "bbox": [ + 174, + 820, + 823, + 876 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Dataset Construction As there are currently no publicly available datasets representative of handdrawn face sketches, we elect to construct our own dataset, which we describe in Appendix A.2. ", + "bbox": [ + 174, + 895, + 821, + 922 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results Figure 6a compares the results of our method to those of pix2pixHD and DeepFaceDrawing. As no code release is available for DeepFaceDrawing, we compare directly with sketches and results published in their paper. Due to the hard constraints of pix2pixHD, they are unable to handle the abstract sketches and obtain poor visual results. While DeepFaceDrawing obtain more visually pleasing results compared to pix2pixHD, they are still limited in their diversity. Conversely, although our model is trained on a different dataset, we are still able to generalize well to their sketches. Notably, we observe our ability to obtain more diverse outputs that better retain finer details (e.g. facial hair). Another limitation of DeepFaceDrawing is its focus on frontal images. We therefore illustrate our model’s ability to generate high-fidelity outputs from non-frontal sketches in Figure 13. As we are unable to directly evaluate DeepFaceDrawing on our constructed dataset, we compare our results only to those of pix2pixHD, trained and evaluated with the same data. ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3.2 FACE FROM SEGMENTATION MAP ", + "text_level": 1, + "bbox": [ + 176, + 272, + 462, + 286 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Here, we evaluate using pSp for synthesizing face images from segmentation maps. In addition to pix2pixHD, we compare our approach to two additional state-of-the-art label-to-image methods: SPADE (Park et al., 2019), and CC FPSE (Liu et al., 2019b), both of which are based on pix2pixHD. ", + "bbox": [ + 176, + 296, + 823, + 338 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results In Figure 6b we provide a visual comparison of the competing approaches on the CelebAMask-HQ dataset containing 19 semantic categories. As the competing methods are based on pix2pixHD, the results of all three are visually similar. Conversely, our approach is able to generate high-quality outputs across a wide range of inputs of various poses and expressions. Additionally, using our multi-modal technique, pSp can easily generate various possible outputs with the same pose and attributes but varying fine styles for a single input semantic map or sketch image. We provide examples in Figure 6c with additional examples in Appendix C. ", + "bbox": [ + 174, + 353, + 825, + 450 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3.3 HUMAN PERCEPTUAL STUDY ", + "text_level": 1, + "bbox": [ + 176, + 465, + 431, + 479 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We additionally perform a human evaluation to compare the visual quality of each method presented above. Here, each worker is given two images synthesized by different methods on the same input and is given an unlimited time to select which output looks more realistic. Each of our three workers reviews approximately 2, 800 pairs for each task, resulting in over 8, 400 human judgements for each method. Table 6d shows that pSp significantly outperforms the other respective methods in both synthesis tasks. ", + "bbox": [ + 174, + 489, + 825, + 573 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 DISCUSSION AND CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 593, + 478, + 609 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Although our suggested framework for image-to-image translation achieves compelling results in various applications, it has some inherent assumptions that should be considered. First, the highquality images that are generated by utilizing the pretrained StyleGAN come with a cost — the method is limited to images that can be generated by StyleGAN. Thus, generating faces which are not close to frontal, or have certain expressions may be challenging if such examples were not available when training the StyleGAN model. Also, the global approach of $\\mathsf { p } \\mathsf { S p }$ , although advantageous for many tasks, does introduce a challenge in preserving finer details of the input image, such as earrings or background details. This is especially significant in tasks such as inpainting or superresolution where standard image-to-image architectures can simply propagate local information. Figure 7b in Appendix A presents some examples of such reconstruction failures. ", + "bbox": [ + 174, + 625, + 823, + 763 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this work, we proposed a novel encoder architecture that can be used to directly map a face image into the $\\mathcal { W } +$ latent space with no optimization required. The encoder architecture, motivated by StyleGAN, consists of a hierarchy of three levels that correspond to the coarse, medium, and fine groupings of the 18 style vectors defining the input in the $\\mathcal { W } +$ latent space. Styles are then extracted from the encoder in a hierarchical fashion and fed into the corresponding inputs of a fixed StyleGAN generator. Notably, our network is trained with an ID similarity loss, which encourages better preservation of identity compared to previous direct approaches. Combining our encoder with a StyleGAN decoder, we present a general framework for solving various image-to-image translation tasks. In contrast to previous methods, which tackle such tasks using a local ”pixel-topixel” approach, our framework takes a global approach, which we show can be used to solve a wide variety of image-to-image translation problems. 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In Advances in neural information processing systems, pp. 465–476, 2017b. ", + "bbox": [ + 173, + 694, + 825, + 738 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Peihao Zhu, Rameen Abdal, Yipeng Qin, and Peter Wonka. Sean: Image synthesis with semantic region-adaptive normalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5104–5113, 2020b. ", + "bbox": [ + 174, + 746, + 825, + 790 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/6c4cc31dab2277525f919812eaa15843177fde4d034b0ab73aa011ebcda5b8b3.jpg", + "image_caption": [ + "Figure 7: (a) To generate multiple outputs for a single input image, style-mixing is performed over pSp. (b) Challenging cases for StyleGAN Inversion. " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 812, + 247 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A ADDITIONAL DETAILS ", + "bbox": [ + 176, + 299, + 398, + 315 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 330, + 410, + 345 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Training Details For our backbone network we use the ResNet-IR architecture from (Deng et al., 2019) pretrained on face recognition, which accelerated convergence. We use a fixed StyleGAN2 generator trained on the FFHQ (Karras et al., 2019) dataset. That is, only the pSp encoder network is trained on the given image-to-image translation task. For all applications, the input image resolution is $2 5 6 \\times 2 5 6$ , where the generated $1 0 2 4 \\times 1 0 2 4$ output is resized before being fed into the loss functions. For training, we use the Ranger optimizer, a combination of Rectified Adam (Liu et al., 2019a) with the Lookahead technique (Zhang et al., 2019), with a constant learning rate of 0.001. Only horizontal flips are used as augmentations during training. All experiments are performed using a single NVIDIA Tesla P40 GPU. ", + "bbox": [ + 174, + 357, + 825, + 482 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the StyleGAN inversion task, the $\\lambda$ values are set as $\\lambda _ { 1 } = 1$ , $\\lambda _ { 2 } = 0 . 8$ , $\\lambda _ { 3 } = 0 . 1$ . For face frontalization, we increase the weight of the identity loss, setting $\\lambda _ { 3 } = 1$ , and decrease the LPIPS and $L _ { 2 }$ loss functions, setting $\\lambda _ { 1 } = 0 . 0 1$ , $\\lambda _ { 2 } = 0 . 8$ over the inner part of the face and $\\lambda _ { 1 } = 0 . 0 0 1$ , $\\lambda _ { 2 } = 0 . 0 8$ elsewhere. Additionally, the constants used in the conditional image synthesis tasks are identical to those used in the inversion task except for the omission of the identity loss (i.e. we set $\\lambda _ { 3 } = 0$ ). Finally, $\\lambda _ { 4 }$ is set to 0.005 in all applications except for the StyleGAN inversion task, which does not utilize the regularization loss. ", + "bbox": [ + 174, + 489, + 825, + 587 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 603, + 290, + 617 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We conduct our experiments on the CelebA-HQ dataset (Karras et al., 2018), which contains 30,000 high quality images. We use a standard train-test split of the dataset, resulting in approximately 24,000 training images. The FFHQ dataset from (Karras et al., 2019), which contains 70,000 face images, is used for the StyleGAN inversion and face frontalization tasks. ", + "bbox": [ + 174, + 628, + 825, + 685 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the generation of face images from sketches, we construct a dataset representative of handdrawn sketches using the CelebA-HQ dataset (Karras et al., 2018). Given an input image, we first apply a “pencil sketch” filter which retains most facial details of the original image while removing the remaining noise. We then apply the sketch-simplification method by Simo-Serra et al. (2016), resulting in images resembling hand-drawn sketches. ", + "bbox": [ + 174, + 693, + 825, + 762 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B ADDITIONAL APPLICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 784, + 446, + 799 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.1 SUPER RESOLUTION ", + "text_level": 1, + "bbox": [ + 176, + 814, + 359, + 829 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here we show that our framework can be used to construct high-resolution (HR) facial images from corresponding low-resolution (LR) input images. PULSE (Menon et al., 2020) approaches this task in an unsupervised manner by traversing the HR image manifold in search of an image that downsamples to the input LR image. In this work we focus on applying pSp in a supervised manner as obtaining paired data is immediate. We show that our method achieves comparable results to PULSE and other previous works. ", + "bbox": [ + 174, + 840, + 825, + 922 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Methodology and details We train our model in a supervised fashion, where for each input we perform random bi-cubic down-sampling of $\\times 1$ (i.e. no down-sampling), $\\times 2 , \\times 4 , \\times 8$ , $\\times 1 6$ , $\\times 3 2$ and set the original, full resolution image as the target. ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Results Figure 9 demonstrates the visual quality of the resulting images from our method along with those of the previous approaches. Although PULSE is able to achieve very high-quality results due to their usage of StyleGAN to generate images, they are unable to accurately retain identity even when performing down-sampling of $\\times 8$ to a resolution of $3 2 \\times 3 2$ . By learning a pixel-wise correspondence between the LR and HR images, pix2pixHD is able to obtain satisfying results even when down-sampled to a resolution of $1 6 \\times 1 6$ (i.e. $\\times 1 6$ down-sampling). However, visually, their results appear less photo-realistic. Contrary to these previous works, we are able to obtain highquality results even when down-sampling to resolutions of $1 6 \\times 1 6$ and $8 \\times 8$ . Finally, we generate multiple outputs for a given LR image using our multi-modal technique by perform style-mixing on layers (4-7) with an $\\alpha$ value of 0.5 with a randomly sampled w vector, which alters medium-level styles that mainly control facial features. Figure 10 illustrates the results. ", + "bbox": [ + 173, + 161, + 825, + 315 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.2 EVEN MORE APPLICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 333, + 406, + 348 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To better show the flexibility of our pSp framework, We present three additional applications, which are summarized in Figure 8. ", + "bbox": [ + 174, + 359, + 823, + 388 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Local Editing Our framework allows for a simple approach to local image editing where altering specific attributes of an input sketch (e.g. eyes, smile) or segmentation map (e.g. hair) results in local edits of the generated images. ", + "bbox": [ + 174, + 405, + 825, + 446 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Face Interpolation Given two real images one can obtain their respective latent codes $w _ { 1 } , w _ { 2 } \\in$ $\\mathcal { W } +$ by feeding the images through our encoder. We can then naturally interpolate between the two images by computing their intermediate latent code $w ^ { \\prime } = \\lambda w _ { 1 } + ( 1 \\bar { - \\lambda } ) \\bar { w } _ { 2 }$ for $0 \\leq \\lambda \\leq 1$ and generate the corresponding image using the new code $w ^ { \\prime }$ . ", + "bbox": [ + 174, + 464, + 825, + 520 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Inpainting Finally, we show the ability of our framework to reconstruct missing parts of an image using a simple, symmetric triangular mask. Our approach is able to accurately reconstruct the occluded areas while preserving the identity with respect to the original image. ", + "bbox": [ + 176, + 536, + 823, + 579 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/86f5a49cffbd32d7a6733400d7854ef3fa70e9da809eb23cd98d657af77aae3e.jpg", + "image_caption": [ + "Figure 8: Additional applications for the pSp framework. " + ], + "image_footnote": [], + "bbox": [ + 199, + 592, + 795, + 895 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/306f3b0994bcc83d2eb8bcc0bfefa18fb443c65fcaf1adadee5c332ec672f0c4.jpg", + "image_caption": [ + "Figure 9: Comparison of super-resolution approaches with (a) $\\times 8$ down-sampling, (b) $\\times 1 6$ downsampling, and (c) $\\times 3 2$ down-sampling. " + ], + "image_footnote": [], + "bbox": [ + 274, + 108, + 720, + 867 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/cc851dc564bcf59d7ad3cc72569ed9fb3ead6857cedb2026f61dfd6d209bbd20.jpg", + "image_caption": [ + "Figure 10: Multi-modal synthesis for super-resolution using pSp with style-mixing. " + ], + "image_footnote": [], + "bbox": [ + 336, + 102, + 661, + 295 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 349, + 400, + 366 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/0b25c29782244b03b64745b80191702264eb58f7874ed7bb4b9668287da6e962.jpg", + "image_caption": [ + "Figure 11: Additional StyleGAN inversion results using pSp on the CelebA-HQ (Karras et al., 2018) test set. " + ], + "image_footnote": [], + "bbox": [ + 176, + 386, + 838, + 813 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/0bc1a6317ea8cc91663d5ac69208bcc68da61cbde0a677bb5ea6a84e8a4b80f5.jpg", + "image_caption": [ + "Figure 12: Additional face frontalization results using pSp on the CelebA-HQ (Karras et al., 2018) test set. " + ], + "image_footnote": [], + "bbox": [ + 176, + 159, + 836, + 824 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/d2f8dd3a4bf409c2f63b861f3263b95631a3910158bfd6c3ff50947224bac76f.jpg", + "image_caption": [ + "Figure 13: Even for challenging, non-frontal face sketches, pSp is able to obtain high-quality, diverse outputs. " + ], + "image_footnote": [], + "bbox": [ + 173, + 108, + 834, + 363 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e0db11481e9689ab3efaf35c551a8e00ff24be6e85b5d47c8c790141a433397a.jpg", + "image_caption": [ + "Figure 14: Additional results using pSp for the generation of face images from sketches constructed from the CelebA-HQ (Karras et al., 2018) test dataset. " + ], + "image_footnote": [], + "bbox": [ + 174, + 434, + 838, + 866 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/285edcda767a71026820f65323a255deec2bce6771dc0a1f83b629dcf5a456c8.jpg", + "image_caption": [ + "Figure 15: Additional results on the Helen Faces (Le et al., 2012) dataset using our proposed labelto-image method. " + ], + "image_footnote": [], + "bbox": [ + 176, + 281, + 825, + 698 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/a9f3f2b94b8e1934a25bd2d1d39736774770d5133ef0ff5c9e79e28416c3dbed.jpg", + "image_caption": [ + "Figure 16: Additional results on the CelebAMask-HQ (Karras et al., 2018) test set using our proposed label-to-image method. " + ], + "image_footnote": [], + "bbox": [ + 176, + 275, + 838, + 704 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/fe2db569bda27147de22c163e828d57efa05c8729d3e9bc45cda5f45a1c704d7.jpg", + "image_caption": [ + "Figure 17: Conditional image synthesis results from sketches and segmentation maps displaying the multi-modal property of our approach. " + ], + "image_footnote": [], + "bbox": [ + 173, + 147, + 826, + 837 + ], + "page_idx": 19 + } +] \ No newline at end of file diff --git a/parse/train/qmI0P1ZExUl/qmI0P1ZExUl_middle.json b/parse/train/qmI0P1ZExUl/qmI0P1ZExUl_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..dde366f461fd9e202571d12eb9309150120ec3ff --- /dev/null +++ b/parse/train/qmI0P1ZExUl/qmI0P1ZExUl_middle.json @@ -0,0 +1,36356 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 99 + ], + "score": 1.0, + "content": "ENCODING IN STYLE: A STYLEGAN ENCODER FOR", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 353, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 353, + 118 + ], + "score": 1.0, + "content": "IMAGE-TO-IMAGE TRANSLATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 376 + ], + "lines": [ + { + "bbox": [ + 142, + 212, + 469, + 224 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 469, + 224 + ], + "score": 1.0, + "content": "We present a generic image-to-image translation framework, Pixel2Style2Pixel", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 222, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 142, + 223, + 165, + 235 + ], + "score": 0.84, + "content": "( p S p )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 222, + 469, + 236 + ], + "score": 1.0, + "content": ". Our pSp framework is based on a novel encoder network that directly gen-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 469, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 469, + 246 + ], + "score": 1.0, + "content": "erates a series of style vectors which are fed into a pretrained StyleGAN generator,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 228, + 257 + ], + "score": 1.0, + "content": "forming the extended", + "type": "text" + }, + { + "bbox": [ + 228, + 245, + 248, + 255 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 244, + 469, + 257 + ], + "score": 1.0, + "content": "latent space. 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Solving these tasks", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 469, + 323 + ], + "score": 1.0, + "content": "through the style representation results in a global approach that does not rely on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 334 + ], + "score": 1.0, + "content": "a local pixel-to-pixel correspondence and further supports multi-modal synthesis", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "via the resampling of styles. Notably, we demonstrate that pSp can be trained to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 469, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 355 + ], + "score": 1.0, + "content": "align a face image to a frontal pose with no labeled data and generate multi-modal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "results for ambiguous tasks such as conditional face generation from sketches and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 224, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 224, + 378 + ], + "score": 1.0, + "content": "segmentation maps.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "In recent years, Generative Adversarial Networks (GANs) have significantly advanced image syn-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "thesis, particularly on face images. State-of-the-art image generation methods have achieved high", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "visual quality and fidelity, and can now generate images with phenomenal realism. Most notably,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "StyleGAN (Karras et al., 2019; 2020) proposes a novel style-based generator architecture and attains", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "state-of-the-art visual quality on high-resolution images. Moreover, it has been demonstrated that it", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 233, + 491 + ], + "score": 1.0, + "content": "has a disentangled latent space,", + "type": "text" + }, + { + "bbox": [ + 234, + 479, + 246, + 489 + ], + "score": 0.64, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "(Yang et al., 2019; Collins et al., 2020; Shen et al., 2020), which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 276, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 276, + 502 + ], + "score": 1.0, + "content": "may offer control and editing capabilities.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "Recently, numerous methods have shown competence in controlling StyleGAN’s latent space and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 271, + 530 + ], + "score": 1.0, + "content": "performing meaningful manipulations in", + "type": "text" + }, + { + "bbox": [ + 271, + 518, + 284, + 528 + ], + "score": 0.75, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "(Jahanian et al., 2019; Shen et al., 2020; Tewari et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "score": 1.0, + "content": "2020; Hark ¨ onen et al., 2020). To perform such edits on real images, one needs to invert the image ¨", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "into StyleGAN’s latent space, i.e., retrieve the latent code that reconstructs the image. 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Motivated by this, it has become common practice (Abdal et al., 2019;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "2020; Baylies, 2019; Zhu et al., 2020a; Adbal et al., 2020) to encode real images into an extended", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 158, + 595 + ], + "score": 1.0, + "content": "latent space,", + "type": "text" + }, + { + "bbox": [ + 159, + 583, + 178, + 594 + ], + "score": 0.87, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 583, + 505, + 595 + ], + "score": 1.0, + "content": ", defined by the concatenation of 18 different 512-dimensional w vectors, one for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "each input layer of StyleGAN. 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To accelerate this optimization process,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "score": 1.0, + "content": "some methods (Baylies, 2019; Zhu et al., 2020a) have trained an encoder to infer an approximate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 144, + 640 + ], + "score": 1.0, + "content": "vector in", + "type": "text" + }, + { + "bbox": [ + 145, + 627, + 165, + 637 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "which serves as a good initial point from which additional optimization is required.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 400, + 651 + ], + "score": 1.0, + "content": "However, a fast, direct, and accurate learned inversion of real images into", + "type": "text" + }, + { + "bbox": [ + 400, + 639, + 420, + 649 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 637, + 504, + 651 + ], + "score": 1.0, + "content": "remains a challenge.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "In this paper, we focus on the broader task of latent space embedding, which aims to retrieve the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "latent vector that generates a desired, not necessarily known, image. 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The encoder", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is based on a Feature Pyramid Network (Lin et al., 2017), where style feature vectors are extracted", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "from different pyramid scales and inserted directly into a fixed, pretrained StyleGAN generator in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 331, + 722 + ], + "score": 1.0, + "content": "correspondence to their spatial scales. Our encoder into", + "type": "text" + }, + { + "bbox": [ + 331, + 710, + 350, + 720 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", together with the StyleGAN decoder,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "form a generic encoder-decoder network that benefits many image-to-image translation tasks. 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Our pSp framework is based on a novel encoder network that directly gen-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 469, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 469, + 246 + ], + "score": 1.0, + "content": "erates a series of style vectors which are fed into a pretrained StyleGAN generator,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 228, + 257 + ], + "score": 1.0, + "content": "forming the extended", + "type": "text" + }, + { + "bbox": [ + 228, + 245, + 248, + 255 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 244, + 469, + 257 + ], + "score": 1.0, + "content": "latent space. We first show that our encoder can directly", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 256, + 469, + 267 + ], + "spans": [ + { + "bbox": [ + 142, + 256, + 239, + 267 + ], + "score": 1.0, + "content": "embed real images into", + "type": "text" + }, + { + "bbox": [ + 240, + 256, + 259, + 266 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 256, + 469, + 267 + ], + "score": 1.0, + "content": ", with no additional optimization. We further intro-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "duce a dedicated identity loss which is shown to achieve improved performance", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 470, + 291 + ], + "score": 1.0, + "content": "in the reconstruction of an input image. We demonstrate pSp to be a simple archi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "tecture that, by leveraging a well-trained, fixed generator network, can be easily", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 312 + ], + "score": 1.0, + "content": "applied on a wide-range of image-to-image translation tasks. Solving these tasks", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 469, + 323 + ], + "score": 1.0, + "content": "through the style representation results in a global approach that does not rely on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 334 + ], + "score": 1.0, + "content": "a local pixel-to-pixel correspondence and further supports multi-modal synthesis", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "via the resampling of styles. Notably, we demonstrate that pSp can be trained to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 469, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 355 + ], + "score": 1.0, + "content": "align a face image to a frontal pose with no labeled data and generate multi-modal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "results for ambiguous tasks such as conditional face generation from sketches and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 224, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 224, + 378 + ], + "score": 1.0, + "content": "segmentation maps.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 212, + 470, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "In recent years, Generative Adversarial Networks (GANs) have significantly advanced image syn-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "thesis, particularly on face images. State-of-the-art image generation methods have achieved high", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "visual quality and fidelity, and can now generate images with phenomenal realism. Most notably,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "StyleGAN (Karras et al., 2019; 2020) proposes a novel style-based generator architecture and attains", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "state-of-the-art visual quality on high-resolution images. Moreover, it has been demonstrated that it", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 233, + 491 + ], + "score": 1.0, + "content": "has a disentangled latent space,", + "type": "text" + }, + { + "bbox": [ + 234, + 479, + 246, + 489 + ], + "score": 0.64, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "(Yang et al., 2019; Collins et al., 2020; Shen et al., 2020), which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 276, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 276, + 502 + ], + "score": 1.0, + "content": "may offer control and editing capabilities.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 423, + 506, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "Recently, numerous methods have shown competence in controlling StyleGAN’s latent space and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 271, + 530 + ], + "score": 1.0, + "content": "performing meaningful manipulations in", + "type": "text" + }, + { + "bbox": [ + 271, + 518, + 284, + 528 + ], + "score": 0.75, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "(Jahanian et al., 2019; Shen et al., 2020; Tewari et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "score": 1.0, + "content": "2020; Hark ¨ onen et al., 2020). To perform such edits on real images, one needs to invert the image ¨", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "into StyleGAN’s latent space, i.e., retrieve the latent code that reconstructs the image. However, it", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 404, + 563 + ], + "score": 1.0, + "content": "has been shown that inverting a real image into a 512-dimensional vector", + "type": "text" + }, + { + "bbox": [ + 405, + 550, + 437, + 561 + ], + "score": 0.89, + "content": "\\mathbf { w } \\in \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "does not lead to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "an accurate reconstruction. Motivated by this, it has become common practice (Abdal et al., 2019;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "2020; Baylies, 2019; Zhu et al., 2020a; Adbal et al., 2020) to encode real images into an extended", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 158, + 595 + ], + "score": 1.0, + "content": "latent space,", + "type": "text" + }, + { + "bbox": [ + 159, + 583, + 178, + 594 + ], + "score": 0.87, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 583, + 505, + 595 + ], + "score": 1.0, + "content": ", defined by the concatenation of 18 different 512-dimensional w vectors, one for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "each input layer of StyleGAN. Nevertheless, many methods resort to using per-image optimization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 127, + 619 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 127, + 605, + 147, + 615 + ], + "score": 0.9, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 603, + 506, + 619 + ], + "score": 1.0, + "content": ", requiring several minutes for a single image. To accelerate this optimization process,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 630 + ], + "score": 1.0, + "content": "some methods (Baylies, 2019; Zhu et al., 2020a) have trained an encoder to infer an approximate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 144, + 640 + ], + "score": 1.0, + "content": "vector in", + "type": "text" + }, + { + "bbox": [ + 145, + 627, + 165, + 637 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "which serves as a good initial point from which additional optimization is required.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 400, + 651 + ], + "score": 1.0, + "content": "However, a fast, direct, and accurate learned inversion of real images into", + "type": "text" + }, + { + "bbox": [ + 400, + 639, + 420, + 649 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 637, + 504, + 651 + ], + "score": 1.0, + "content": "remains a challenge.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 507, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "In this paper, we focus on the broader task of latent space embedding, which aims to retrieve the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "latent vector that generates a desired, not necessarily known, image. We do so by introducing a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 428, + 690 + ], + "score": 1.0, + "content": "novel encoder architecture tasked with encoding an arbitrary image directly into", + "type": "text" + }, + { + "bbox": [ + 429, + 677, + 448, + 688 + ], + "score": 0.88, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 677, + 506, + 690 + ], + "score": 1.0, + "content": ". The encoder", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is based on a Feature Pyramid Network (Lin et al., 2017), where style feature vectors are extracted", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "from different pyramid scales and inserted directly into a fixed, pretrained StyleGAN generator in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 331, + 722 + ], + "score": 1.0, + "content": "correspondence to their spatial scales. Our encoder into", + "type": "text" + }, + { + "bbox": [ + 331, + 710, + 350, + 720 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", together with the StyleGAN decoder,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "form a generic encoder-decoder network that benefits many image-to-image translation tasks. Fo-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "cusing on face images, we first demonstrate our method’s ability to successfully reconstruct a given", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "image while preserving identity and other attributes. We then present numerous image-to-image", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "translation applications. In a sense, our method performs Pixel2Style2Pixel translation, as every", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 478, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 458, + 128 + ], + "score": 1.0, + "content": "image is first encoded into style vectors and then into an image, and is therefore dubbed", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 458, + 116, + 475, + 127 + ], + "score": 0.66, + "content": "p S p", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 475, + 115, + 478, + 128 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 654, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "cusing on face images, we first demonstrate our method’s ability to successfully reconstruct a given", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "image while preserving identity and other attributes. We then present numerous image-to-image", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "translation applications. In a sense, our method performs Pixel2Style2Pixel translation, as every", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 478, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 458, + 128 + ], + "score": 1.0, + "content": "image is first encoded into style vectors and then into an image, and is therefore dubbed", + "type": "text" + }, + { + "bbox": [ + 458, + 116, + 475, + 127 + ], + "score": 0.66, + "content": "p S p", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 115, + 478, + 128 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "While many previous approaches to solving image-to-image translations tasks involve dedicated", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "architectures specific for solving a single problem, we follow the spirit of pix2pix (Isola et al., 2017)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "and define a generic framework able to solve a wide range of tasks, all using the same architecture.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 179 + ], + "score": 1.0, + "content": "Besides the simplification of the training process, as no adversary discriminator needs to be trained,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "using a pretrained StyleGAN generator offers several intriguing advantages over previous works.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "Many image-to-image architectures explicitly feed the generator with residual feature maps from", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "the encoder (Isola et al., 2017; Wang et al., 2018), creating a strong locality bias (Richardson &", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "score": 1.0, + "content": "Weiss, 2020). In contrast, our generator is governed only by the styles with no direct spatial input.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "The advantage of such a global approach is most evident in the task of Face Frontalization, where", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "our encoder can be trained to align a given face image to a frontal pose with no labeled data. Another", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "notable advantage of the intermediate style representation is the inherent support for multi-modal", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "synthesis for ambiguous tasks such as face generation from sketches, segmentation maps, or low-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "resolution images. 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A specific task that has received substantial attention is", + "type": "text" + }, + { + "bbox": [ + 483, + 379, + 505, + 389 + ], + "score": 0.27, + "content": "G A N", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "Inversion — where the latent vector from which a pretrained GAN most accurately reconstructs a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "given, known image, is sought. 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The aforementioned works have constructed dedicated", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 612, + 345, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 345, + 624 + ], + "score": 1.0, + "content": "architectures, which require training the generator network.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "Latent-Space Manipulation Recently, numerous papers have presented diverse methods to learn", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "semantic edits of the latent code. 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Tewari et al. (2020)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "utilize a pretrained 3DMM to learn semantic face edits in the latent space. Jahanian et al. (2019) find", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "latent space paths that correspond to a specific transformation, such as zoom or rotation, in a self-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 702, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 505, + 716 + ], + "score": 1.0, + "content": "supervised manner. Hark ¨ onen et al. (2020) find useful paths in an unsupervised manner by using the ¨", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 713, + 505, + 726 + ], + "spans": [ + { + "bbox": [ + 105, + 713, + 505, + 726 + ], + "score": 1.0, + "content": "principal component axes (PCA) of an intermediate activation space. Finally, Collins et al. 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Motivated by its state-of-the-art image quality and latent space se-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "mantic richness, many recent works have used StyleGAN (Karras et al., 2019; 2020) for this task.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "Generally, inversion methods either directly optimize the latent vector to minimize the error for the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "given image (Lipton & Tripathi, 2017; Creswell & Bharath, 2018; Abdal et al., 2019; 2020), train an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "encoder to map the given image to the latent space (Perarnau et al., 2016; Creswell & Bharath, 2018;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 454, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 454, + 506, + 470 + ], + "score": 1.0, + "content": "Pidhorskyi et al., 2020; Guan et al., 2020; Nitzan et al., 2020), or use a hybrid approach combining", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "both (Baylies, 2019; Zhu et al., 2020a). 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Unlike", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "score": 1.0, + "content": "the above methods, our encoder can accurately and efficiently embed a given face image into the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 499, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 194, + 513 + ], + "score": 1.0, + "content": "extended latent space", + "type": "text" + }, + { + "bbox": [ + 194, + 500, + 214, + 510 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 499, + 504, + 513 + ], + "score": 1.0, + "content": "of a fixed, pretrained StyleGAN generator, with no further optimization.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 367, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "Image-to-Image Image-to-Image translation techniques aim at learning a conditional image gen-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "eration function that maps an input image of a source domain to a corresponding image of a target", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "domain. Isola et al. (2017) first introduced the use of conditional GANs to solve various image-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "to-image translation tasks. Since then, their work has been extended for many scenarios: high-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "resolution synthesis (Wang et al., 2018), unsupervised learning (Liu et al., 2017; Zhu et al., 2017a;", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "Katzir et al., 2019; Lira et al., 2020), multi-modal image synthesis (Zhu et al., 2017b; Huang et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "2018; Choi et al., 2020), and conditional image synthesis (Park et al., 2019; Li et al., 2019; Liu et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "2019b; Zhu et al., 2020b; Chen et al., 2020). The aforementioned works have constructed dedicated", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 612, + 345, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 345, + 624 + ], + "score": 1.0, + "content": "architectures, which require training the generator network.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 524, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "Latent-Space Manipulation Recently, numerous papers have presented diverse methods to learn", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "semantic edits of the latent code. A popular approach is finding linear directions that correspond", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 358, + 671 + ], + "score": 1.0, + "content": "to changes in a given binary labeled attribute, such as young", + "type": "text" + }, + { + "bbox": [ + 358, + 659, + 387, + 669 + ], + "score": 0.34, + "content": " \\mathrm { o l d }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 659, + 440, + 671 + ], + "score": 1.0, + "content": ", or no-smile", + "type": "text" + }, + { + "bbox": [ + 440, + 659, + 453, + 669 + ], + "score": 0.83, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "smile (Shen", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "et al., 2020; Goetschalckx et al., 2019; Denton et al., 2019; Adbal et al., 2020). Tewari et al. (2020)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "utilize a pretrained 3DMM to learn semantic face edits in the latent space. Jahanian et al. (2019) find", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "latent space paths that correspond to a specific transformation, such as zoom or rotation, in a self-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 702, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 505, + 716 + ], + "score": 1.0, + "content": "supervised manner. Hark ¨ onen et al. (2020) find useful paths in an unsupervised manner by using the ¨", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 713, + 505, + 726 + ], + "spans": [ + { + "bbox": [ + 105, + 713, + 505, + 726 + ], + "score": 1.0, + "content": "principal component axes (PCA) of an intermediate activation space. Finally, Collins et al. (2020)", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 724, + 482, + 738 + ], + "spans": [ + { + "bbox": [ + 105, + 724, + 482, + 738 + ], + "score": 1.0, + "content": "perform local semantic editing by manipulating corresponding components of the latent code.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 637, + 505, + 738 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 88, + 499, + 213 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 88, + 499, + 213 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 88, + 499, + 213 + ], + "spans": [ + { + "bbox": [ + 109, + 88, + 499, + 213 + ], + "score": 0.97, + "type": "image", + "image_path": "aaa87c70d8d04df81fa258ed78659642cc51ccb7d39c7c3bd9e96fd5502d0ea6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 88, + 499, + 129.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 129.66666666666666, + 499, + 171.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 171.33333333333331, + 499, + 212.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 225, + 506, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Figure 1: Our pSp architecture. Feature maps are first extracted using a standard feature pyramid", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "over a ResNet backbone. For each of the 18 target styles, a small mapping network is trained to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "extract the learned styles from the corresponding feature map, where styles (0-2) are generated from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "the small feature map, (3-6) from the medium feature map, and (7-18) from the largest feature", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "map. The mapping network, map2style, is a small fully convolutional network, which gradually", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "reduces spatial size using a set of 2-strided convolutions followed by LeakyReLU activations. Each", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 291, + 491, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 478, + 304 + ], + "score": 1.0, + "content": "generated 512 vector, is fed into StyleGAN, starting from its matching affine transformation,", + "type": "text" + }, + { + "bbox": [ + 478, + 291, + 487, + 301 + ], + "score": 0.61, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 291, + 491, + 304 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 315, + 242, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 244, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 244, + 330 + ], + "score": 1.0, + "content": "3 THE PSP FRAMEWORK", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "Our pSp framework builds upon the representative power of a pretrained StyleGAN generator and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 121, + 365 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 352, + 141, + 362 + ], + "score": 0.88, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "latent space. To utilize this representation one needs a strong encoder that is able to match", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "each input image to an accurate encoding in the latent domain. A simple technique to embed into", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 332, + 386 + ], + "score": 1.0, + "content": "this domain is directly encoding a given input image into", + "type": "text" + }, + { + "bbox": [ + 332, + 374, + 352, + 384 + ], + "score": 0.9, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "using a single 512-dimensional vector", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 383, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 398 + ], + "score": 1.0, + "content": "obtained from the last layer of the encoder network, thereby learning all 18 style vectors together.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "However, such an architecture presents a strong bottleneck making it difficult to fully represent the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 405, + 437, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 437, + 420 + ], + "score": 1.0, + "content": "finer details of the original image and therefore limiting the reconstruction quality.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 436 + ], + "score": 1.0, + "content": "In StyleGAN, the authors have shown that the different style inputs correspond to different levels", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "of detail, which are roughly divided into three groups — coarse, medium, and fine. Following this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "observation, in pSp we extend an encoder backbone with a feature pyramid (Lin et al., 2017), gen-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "erating three levels of feature maps from which styles are extracted using a simple intermediate net-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "work — map2style — shown in Figure 1. The styles, aligned with the hierarchical representation,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "are then fed into the generator in correspondence to their scale to generate the output image, thus", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "completing the translation from input pixels to output pixels, through the intermediate style repre-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "sentation. Therefore, our architecture, pSp, is an end-to-end image-to-image translation framework.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 511, + 313, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 313, + 524 + ], + "score": 1.0, + "content": "The complete architecture is illustrated in Figure 1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 254, + 541 + ], + "score": 1.0, + "content": "As in StyleGAN, we further define", + "type": "text" + }, + { + "bbox": [ + 254, + 529, + 264, + 538 + ], + "score": 0.56, + "content": "\\overline { { \\mathbf { W } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "to be the average style vector of the pretrained generator.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 196, + 552 + ], + "score": 1.0, + "content": "Given an input image,", + "type": "text" + }, + { + "bbox": [ + 197, + 541, + 204, + 549 + ], + "score": 0.41, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 538, + 375, + 552 + ], + "score": 1.0, + "content": ", the output of our model is then defined as", + "type": "text" + }, + { + "bbox": [ + 375, + 538, + 477, + 551 + ], + "score": 0.92, + "content": "p S p ( \\mathbf { x } ) : = \\mathbf { \\bar { \\boldsymbol { G } } } ( E ( \\mathbf { \\boldsymbol { x } } ) + \\mathbf { \\bar { \\boldsymbol { w } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 126, + 562 + ], + "score": 0.9, + "content": "E ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 550, + 145, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 550, + 165, + 562 + ], + "score": 0.91, + "content": "G ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "denote the encoder and StyleGAN generator, respectively. In this formulation, our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "encoder aims to learn the latent code with respect to the average style vector. 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Our encoder is trained using a weighted combination of several objectives. 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Constants and other implementation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 321, + 262, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 262, + 333 + ], + "score": 1.0, + "content": "details can be found in Appendix A.1.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 322, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 324, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 324, + 359 + ], + "score": 1.0, + "content": "3.2 THE BENEFITS OF THE STYLEGAN DOMAIN", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "The translation between images through the style domain differentiates pSp from many standard", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 377, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 504, + 390 + ], + "score": 1.0, + "content": "image-to-image translation frameworks, as it makes our model operate globally instead of locally,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "without requiring pixel-to-pixel correspondence. This is a desired property as it has been shown", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "that the locality bias limits current methods when handling non-local transformations (Richardson", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "& Weiss, 2020). 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As some translation tasks are ambiguous, where a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "single input image may correspond to several outputs, it is desirable to be able to sample these possi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "ble outputs. While this requires specialized changes in standard image-to-image architectures (Zhu", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "score": 1.0, + "content": "et al., 2017b; Huang et al., 2018), our framework inherently supports this by simply sampling style", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 486, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 366, + 502 + ], + "score": 1.0, + "content": "vectors. In practice, this is done by randomly sampling a vector", + "type": "text" + }, + { + "bbox": [ + 366, + 487, + 407, + 498 + ], + "score": 0.94, + "content": "\\mathbf { w } \\in \\mathbb { R } ^ { 5 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 486, + 505, + 502 + ], + "score": 1.0, + "content": "and generating a corre-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 203, + 512 + ], + "score": 1.0, + "content": "sponding latent code in", + "type": "text" + }, + { + "bbox": [ + 204, + 498, + 224, + 509 + ], + "score": 0.88, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "by replicating w. 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There,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "layers 1-7 are selected from the input latent while layers 8-18 are taken from the sampled vector", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 542, + 499, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 499, + 556 + ], + "score": 1.0, + "content": "allowing one to obtain outputs with similar coarse and medium features, but varying fine features.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 567, + 304, + 579 + ], + "lines": [ + { + "bbox": [ + 104, + 565, + 306, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 306, + 581 + ], + "score": 1.0, + "content": "4 APPLICATIONS AND EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 503, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "score": 1.0, + "content": "To explore the effectiveness of our approach we evaluate our pSp framework on numerous image-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 213, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 213, + 610 + ], + "score": 1.0, + "content": "to-image translation tasks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 234, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 235, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 235, + 636 + ], + "score": 1.0, + "content": "4.1 STYLEGAN INVERSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "We start by evaluating the usage of the pSp framework for StyleGAN Inversion, that is, finding the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "latent code of real images in the latent domain. 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There-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "score": 1.0, + "content": "fore, we incorporate a dedicated recognition loss measuring the cosine similarity between the output", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 194, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 194, + 232 + ], + "score": 1.0, + "content": "image and its source,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 174, + 506, + 232 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 228, + 379, + 242 + ], + "lines": [ + { + "bbox": [ + 231, + 228, + 379, + 242 + ], + "spans": [ + { + "bbox": [ + 231, + 228, + 379, + 242 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { I D } } \\left( \\mathbf { x } \\right) = 1 - \\left. R ( \\mathbf { x } ) , R ( p S p ( \\mathbf { x } ) ) \\right. ,", + "type": "interline_equation", + "image_path": "15aba6cec2e849f8f0ef121a73a940961c311e13c00ecf7cfd4787c7d0770118.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 231, + 228, + 379, + 242 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 508, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 133, + 261 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 249, + 142, + 258 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 248, + 478, + 261 + ], + "score": 1.0, + "content": "is a pretrained ArcFace (Deng et al., 2019) network for face recognition. The input,", + "type": "text" + }, + { + "bbox": [ + 478, + 250, + 485, + 258 + ], + "score": 0.65, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 248, + 505, + 261 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 259, + 485, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 137, + 272 + ], + "score": 1.0, + "content": "output,", + "type": "text" + }, + { + "bbox": [ + 137, + 259, + 167, + 271 + ], + "score": 0.92, + "content": "p S p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 259, + 342, + 272 + ], + "score": 1.0, + "content": ", are cropped around the face and resized to", + "type": "text" + }, + { + "bbox": [ + 343, + 259, + 386, + 270 + ], + "score": 0.9, + "content": "1 1 2 \\times 1 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 259, + 473, + 272 + ], + "score": 1.0, + "content": "before being fed into", + "type": "text" + }, + { + "bbox": [ + 473, + 260, + 481, + 269 + ], + "score": 0.84, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 259, + 485, + 272 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 248, + 505, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 298, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 299, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 299, + 289 + ], + "score": 1.0, + "content": "In summary, the total loss function is defined as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 275, + 299, + 289 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 291, + 421, + 306 + ], + "lines": [ + { + "bbox": [ + 189, + 291, + 421, + 306 + ], + "spans": [ + { + "bbox": [ + 189, + 291, + 421, + 306 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\mathbf { x } ) = \\lambda _ { 1 } \\mathcal { L } _ { 2 } ( \\mathbf { x } ) + \\lambda _ { 2 } \\mathcal { L } _ { \\mathrm { L P I P S } } ( \\mathbf { x } ) + \\lambda _ { 3 } \\mathcal { L } _ { \\mathrm { I D } } ( \\mathbf { x } ) + \\lambda _ { 4 } \\mathcal { L } _ { \\mathrm { r e g } } ( \\mathbf { x } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "c0aef7429c8f9b3c2bd7340d2a5f037e36848b4744e168d022774c4cb1bec3fc.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 189, + 291, + 421, + 306 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 310, + 503, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 133, + 323 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 311, + 193, + 322 + ], + "score": 0.48, + "content": "\\lambda _ { 1 } , \\lambda _ { 2 } , \\lambda _ { 3 } , \\lambda _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "are constants defining the loss weights. Constants and other implementation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 321, + 262, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 262, + 333 + ], + "score": 1.0, + "content": "details can be found in Appendix A.1.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 310, + 505, + 333 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 322, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 324, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 324, + 359 + ], + "score": 1.0, + "content": "3.2 THE BENEFITS OF THE STYLEGAN DOMAIN", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "The translation between images through the style domain differentiates pSp from many standard", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 377, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 504, + 390 + ], + "score": 1.0, + "content": "image-to-image translation frameworks, as it makes our model operate globally instead of locally,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "without requiring pixel-to-pixel correspondence. This is a desired property as it has been shown", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "that the locality bias limits current methods when handling non-local transformations (Richardson", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "& Weiss, 2020). 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As some translation tasks are ambiguous, where a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "single input image may correspond to several outputs, it is desirable to be able to sample these possi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "ble outputs. While this requires specialized changes in standard image-to-image architectures (Zhu", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 491 + ], + "score": 1.0, + "content": "et al., 2017b; Huang et al., 2018), our framework inherently supports this by simply sampling style", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 486, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 366, + 502 + ], + "score": 1.0, + "content": "vectors. In practice, this is done by randomly sampling a vector", + "type": "text" + }, + { + "bbox": [ + 366, + 487, + 407, + 498 + ], + "score": 0.94, + "content": "\\mathbf { w } \\in \\mathbb { R } ^ { 5 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 486, + 505, + 502 + ], + "score": 1.0, + "content": "and generating a corre-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 203, + 512 + ], + "score": 1.0, + "content": "sponding latent code in", + "type": "text" + }, + { + "bbox": [ + 204, + 498, + 224, + 509 + ], + "score": 0.88, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "by replicating w. 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There,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "layers 1-7 are selected from the input latent while layers 8-18 are taken from the sampled vector", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 542, + 499, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 499, + 556 + ], + "score": 1.0, + "content": "allowing one to obtain outputs with similar coarse and medium features, but varying fine features.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 366, + 507, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 567, + 304, + 579 + ], + "lines": [ + { + "bbox": [ + 104, + 565, + 306, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 306, + 581 + ], + "score": 1.0, + "content": "4 APPLICATIONS AND EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 503, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "score": 1.0, + "content": "To explore the effectiveness of our approach we evaluate our pSp framework on numerous image-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 213, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 213, + 610 + ], + "score": 1.0, + "content": "to-image translation tasks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 586, + 505, + 610 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 234, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 235, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 235, + 636 + ], + "score": 1.0, + "content": "4.1 STYLEGAN INVERSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "We start by evaluating the usage of the pSp framework for StyleGAN Inversion, that is, finding the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "latent code of real images in the latent domain. 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While", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "IDInvert (Zhu et al., 2020a) better preserves the image attributes, it still fails to accurately preserve", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "identity and the finer details of the input image. 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Method↑ Similarity↓LPIPS↓MSERuntime
ALAE IDInvert0.06 0.180.32 0.220.15 0.060.207 0.032
W Encoder Naive W+0.350.23 0.190.06 0.040.064 0.064
0.49 0.560.170.030.105
pSp
(a)
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Method 90°个 Similarity ↓Runtime 70° 50° 30°
R&R 0.34 0.56 0.660.7 1.5
pSp 0.32 0.52 0.600.63 0.1
(b)
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RotateAndRender", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "(R&R) (Zhou et al., 2020) overcome this challenge by incorporating a geometric 3D alignment", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "process before the translation process. 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Next, in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "frontalization, as we are less interested in the background region compared to the face region and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "its identity, we also change the weights of the loss objective. 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This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "shows the benefit of using a pretrained StyleGAN for image translation, as it allows us to achieve", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "visually-pleasing results even with weak supervision. Table 4b provides a quantitative evaluation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "on the FEI Faces Database (Thomaz & Giraldi, 2010). 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Method↑ Similarity↓LPIPS↓MSERuntime
ALAE IDInvert0.06 0.180.32 0.220.15 0.060.207 0.032
W Encoder Naive W+0.350.23 0.190.06 0.040.064 0.064
0.49 0.560.170.030.105
pSp
(a)
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Method 90°个 Similarity ↓Runtime 70° 50° 30°
R&R 0.34 0.56 0.660.7 1.5
pSp 0.32 0.52 0.600.63 0.1
(b)
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RotateAndRender", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "(R&R) (Zhou et al., 2020) overcome this challenge by incorporating a geometric 3D alignment", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "process before the translation process. Alternatively, we show that our style-based translation mech-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 483, + 433, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 433, + 495 + ], + "score": 1.0, + "content": "anism is able overcome these challenges, even when trained with no labeled data.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 438, + 506, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 508, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "Methodology and details For this task, training is the same as the encoder formulation with two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "important changes. First, we randomly flip the target image, thus creating inconsistencies in terms", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "score": 1.0, + "content": "of pose compared to the input image. This guides the model towards generating a frontalized face,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "as the true target pose is unknown. While this may seem minor, without this augmentation the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "model would simply learn to encode the input image, matching its pose as well as identity. Next, in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "frontalization, as we are less interested in the background region compared to the face region and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "its identity, we also change the weights of the loss objective. In particular, we decrease the weights", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 178, + 598 + ], + "score": 1.0, + "content": "of the LPIPS and", + "type": "text" + }, + { + "bbox": [ + 178, + 586, + 191, + 596 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "loss functions, and give more weight to the losses computed on the inner part", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "of the face, focusing the model on the inner region while reducing the importance of background", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "preservation. As shown below, these changes to the training objective are enough for the model to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 618, + 353, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 353, + 631 + ], + "score": 1.0, + "content": "generate realistic frontal faces, while also preserving identity.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 509, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "Results Results are illustrated in Figure 5. When trained with the same data, pix2pixHD is unable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "to converge to satisfying results as it is much more dependent on the correspondence between the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "input and output pairs. Conversely, our method is able to handle the task successfully, generating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "realistic frontal faces, which are comparable to the more involved RotateAndRender approach. This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "shows the benefit of using a pretrained StyleGAN for image translation, as it allows us to achieve", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "visually-pleasing results even with weak supervision. Table 4b provides a quantitative evaluation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "on the FEI Faces Database (Thomaz & Giraldi, 2010). While R&R outperforms pSp, our simple", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 457, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 457, + 734 + ], + "score": 1.0, + "content": "approach provides an elegant alternative, without requiring specialized alignment steps.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 643, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 507, + 366 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 507, + 366 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 507, + 366 + ], + "score": 0.974, + "type": "image", + "image_path": "2fae6c2fd49241b71e411ab8fe52917c7054060d2baca609e732f5db62918b51.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 507, + 176.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 176.0, + 507, + 271.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 271.0, + 507, + 366.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 376, + 505, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "Figure 6: (a) Comparison of sketches presented in DeepFaceDrawing. (b) Comparisons to other", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "label-to-image methods on CelebAMask-HQ. (c) Multi-modal outputs using pSp with style-mixing.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "(d) Human evaluation results on CelebA-HQ for Conditional Image Synthesis tasks. Each cell", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 410, + 397, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 397, + 421 + ], + "score": 1.0, + "content": "denotes the percentage of users who favored pSp over the listed method.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 445, + 275, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 277, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 277, + 458 + ], + "score": 1.0, + "content": "4.3 CONDITIONAL IMAGE SYNTHESIS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "Conditional image synthesis aims at generating photo-realistic images conditioned on certain input", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "types. In this section, our pSp architecture is tested on two conditional image generation tasks:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "generating high-quality face images from sketches and semantic label maps. We demonstrate that,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "with only minimal changes, our encoder successfully utilizes the expressiveness of StyleGAN to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "generate high-quality and diverse outputs. Additionally, an ideal mapping framework should be able", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "to generate multiple diverse outputs for a given input. To achieve this, we utilize the multi-modal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 534, + 286, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 286, + 545 + ], + "score": 1.0, + "content": "synthesis approach described in Section 3.2.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "Methodology and details The training of the two conditional generation tasks is identical to that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "of the encoder for StyleGAN inversion except for the omission of the identity loss and the addition", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 579, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 595 + ], + "score": 1.0, + "content": "of the regularization loss. To generate multiple images at inference time, we perform style-mixing,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 592, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 161, + 605 + ], + "score": 1.0, + "content": "taking layers", + "type": "text" + }, + { + "bbox": [ + 162, + 592, + 192, + 604 + ], + "score": 0.86, + "content": "( 1 - 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 592, + 397, + 605 + ], + "score": 1.0, + "content": "from the latent code of the input image and layers", + "type": "text" + }, + { + "bbox": [ + 398, + 592, + 433, + 604 + ], + "score": 0.78, + "content": "( 8 - 1 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 592, + 504, + 605 + ], + "score": 1.0, + "content": "from a randomly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 602, + 173, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 173, + 616 + ], + "score": 1.0, + "content": "drawn w vector.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 228, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 230, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 230, + 642 + ], + "score": 1.0, + "content": "4.3.1 FACE FROM SKETCH", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 504, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "Common approaches for sketch-to-image synthesis incorporate hard constraints that require pixel-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "wise correspondence between the input sketch and generated image, making them ill-suited when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 673, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 685 + ], + "score": 1.0, + "content": "given incomplete sketches. DeepFaceDrawing (Chen et al., 2020) address this using a set of dedi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 683, + 479, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 479, + 696 + ], + "score": 1.0, + "content": "cated mapping networks. We show that pSp provides a simple alternative to past approaches.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Dataset Construction As there are currently no publicly available datasets representative of hand-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 721, + 491, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 491, + 732 + ], + "score": 1.0, + "content": "drawn face sketches, we elect to construct our own dataset, which we describe in Appendix A.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 507, + 366 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 507, + 366 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 507, + 366 + ], + "score": 0.974, + "type": "image", + "image_path": "2fae6c2fd49241b71e411ab8fe52917c7054060d2baca609e732f5db62918b51.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 507, + 176.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 176.0, + 507, + 271.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 271.0, + 507, + 366.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 376, + 505, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "Figure 6: (a) Comparison of sketches presented in DeepFaceDrawing. (b) Comparisons to other", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "label-to-image methods on CelebAMask-HQ. (c) Multi-modal outputs using pSp with style-mixing.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "(d) Human evaluation results on CelebA-HQ for Conditional Image Synthesis tasks. Each cell", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 410, + 397, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 397, + 421 + ], + "score": 1.0, + "content": "denotes the percentage of users who favored pSp over the listed method.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 445, + 275, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 277, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 277, + 458 + ], + "score": 1.0, + "content": "4.3 CONDITIONAL IMAGE SYNTHESIS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "Conditional image synthesis aims at generating photo-realistic images conditioned on certain input", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "types. In this section, our pSp architecture is tested on two conditional image generation tasks:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "generating high-quality face images from sketches and semantic label maps. We demonstrate that,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "with only minimal changes, our encoder successfully utilizes the expressiveness of StyleGAN to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "generate high-quality and diverse outputs. Additionally, an ideal mapping framework should be able", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "to generate multiple diverse outputs for a given input. To achieve this, we utilize the multi-modal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 534, + 286, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 286, + 545 + ], + "score": 1.0, + "content": "synthesis approach described in Section 3.2.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 467, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "Methodology and details The training of the two conditional generation tasks is identical to that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "of the encoder for StyleGAN inversion except for the omission of the identity loss and the addition", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 579, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 595 + ], + "score": 1.0, + "content": "of the regularization loss. To generate multiple images at inference time, we perform style-mixing,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 592, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 161, + 605 + ], + "score": 1.0, + "content": "taking layers", + "type": "text" + }, + { + "bbox": [ + 162, + 592, + 192, + 604 + ], + "score": 0.86, + "content": "( 1 - 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 592, + 397, + 605 + ], + "score": 1.0, + "content": "from the latent code of the input image and layers", + "type": "text" + }, + { + "bbox": [ + 398, + 592, + 433, + 604 + ], + "score": 0.78, + "content": "( 8 - 1 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 592, + 504, + 605 + ], + "score": 1.0, + "content": "from a randomly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 602, + 173, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 173, + 616 + ], + "score": 1.0, + "content": "drawn w vector.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 560, + 506, + 616 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 228, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 230, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 230, + 642 + ], + "score": 1.0, + "content": "4.3.1 FACE FROM SKETCH", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 504, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "Common approaches for sketch-to-image synthesis incorporate hard constraints that require pixel-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "wise correspondence between the input sketch and generated image, making them ill-suited when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 673, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 685 + ], + "score": 1.0, + "content": "given incomplete sketches. DeepFaceDrawing (Chen et al., 2020) address this using a set of dedi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 683, + 479, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 479, + 696 + ], + "score": 1.0, + "content": "cated mapping networks. We show that pSp provides a simple alternative to past approaches.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 650, + 506, + 696 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Dataset Construction As there are currently no publicly available datasets representative of hand-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 721, + 491, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 491, + 732 + ], + "score": 1.0, + "content": "drawn face sketches, we elect to construct our own dataset, which we describe in Appendix A.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Results Figure 6a compares the results of our method to those of pix2pixHD and DeepFaceDraw-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "ing. As no code release is available for DeepFaceDrawing, we compare directly with sketches and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "results published in their paper. Due to the hard constraints of pix2pixHD, they are unable to handle", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 504, + 127 + ], + "score": 1.0, + "content": "the abstract sketches and obtain poor visual results. While DeepFaceDrawing obtain more visu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "ally pleasing results compared to pix2pixHD, they are still limited in their diversity. Conversely,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "although our model is trained on a different dataset, we are still able to generalize well to their", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "sketches. Notably, we observe our ability to obtain more diverse outputs that better retain finer de-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "tails (e.g. facial hair). Another limitation of DeepFaceDrawing is its focus on frontal images. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 169, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 506, + 184 + ], + "score": 1.0, + "content": "therefore illustrate our model’s ability to generate high-fidelity outputs from non-frontal sketches in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "Figure 13. As we are unable to directly evaluate DeepFaceDrawing on our constructed dataset, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 469, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 469, + 204 + ], + "score": 1.0, + "content": "compare our results only to those of pix2pixHD, trained and evaluated with the same data.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 216, + 283, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 284, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 284, + 228 + ], + "score": 1.0, + "content": "4.3.2 FACE FROM SEGMENTATION MAP", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 235, + 504, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 249 + ], + "score": 1.0, + "content": "Here, we evaluate using pSp for synthesizing face images from segmentation maps. In addition", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "to pix2pixHD, we compare our approach to two additional state-of-the-art label-to-image methods:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 504, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 504, + 269 + ], + "score": 1.0, + "content": "SPADE (Park et al., 2019), and CC FPSE (Liu et al., 2019b), both of which are based on pix2pixHD.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Results In Figure 6b we provide a visual comparison of the competing approaches on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "CelebAMask-HQ dataset containing 19 semantic categories. As the competing methods are based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "on pix2pixHD, the results of all three are visually similar. Conversely, our approach is able to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "generate high-quality outputs across a wide range of inputs of various poses and expressions. Ad-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "ditionally, using our multi-modal technique, pSp can easily generate various possible outputs with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "the same pose and attributes but varying fine styles for a single input semantic map or sketch image.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 412, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 412, + 360 + ], + "score": 1.0, + "content": "We provide examples in Figure 6c with additional examples in Appendix C.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 264, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 266, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 266, + 381 + ], + "score": 1.0, + "content": "4.3.3 HUMAN PERCEPTUAL STUDY", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "We additionally perform a human evaluation to compare the visual quality of each method presented", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "above. Here, each worker is given two images synthesized by different methods on the same input", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "and is given an unlimited time to select which output looks more realistic. Each of our three workers", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "reviews approximately 2, 800 pairs for each task, resulting in over 8, 400 human judgements for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "each method. Table 6d shows that pSp significantly outperforms the other respective methods in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 443, + 191, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 191, + 455 + ], + "score": 1.0, + "content": "both synthesis tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 470, + 293, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 295, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 295, + 486 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSIONS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "Although our suggested framework for image-to-image translation achieves compelling results in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "various applications, it has some inherent assumptions that should be considered. First, the high-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "quality images that are generated by utilizing the pretrained StyleGAN come with a cost — the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "method is limited to images that can be generated by StyleGAN. Thus, generating faces which are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "not close to frontal, or have certain expressions may be challenging if such examples were not avail-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 390, + 563 + ], + "score": 1.0, + "content": "able when training the StyleGAN model. Also, the global approach of", + "type": "text" + }, + { + "bbox": [ + 390, + 551, + 407, + 562 + ], + "score": 0.26, + "content": "\\mathsf { p } \\mathsf { S p }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ", although advantageous", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "for many tasks, does introduce a challenge in preserving finer details of the input image, such as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "earrings or background details. This is especially significant in tasks such as inpainting or super-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "resolution where standard image-to-image architectures can simply propagate local information.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 434, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 434, + 607 + ], + "score": 1.0, + "content": "Figure 7b in Appendix A presents some examples of such reconstruction failures.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "In this work, we proposed a novel encoder architecture that can be used to directly map a face", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 165, + 634 + ], + "score": 1.0, + "content": "image into the", + "type": "text" + }, + { + "bbox": [ + 165, + 622, + 185, + 632 + ], + "score": 0.88, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "latent space with no optimization required. The encoder architecture, motivated", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "by StyleGAN, consists of a hierarchy of three levels that correspond to the coarse, medium, and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 364, + 657 + ], + "score": 1.0, + "content": "fine groupings of the 18 style vectors defining the input in the", + "type": "text" + }, + { + "bbox": [ + 365, + 644, + 385, + 654 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "latent space. Styles are then", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "extracted from the encoder in a hierarchical fashion and fed into the corresponding inputs of a fixed", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "StyleGAN generator. Notably, our network is trained with an ID similarity loss, which encourages", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "better preservation of identity compared to previous direct approaches. Combining our encoder", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "with a StyleGAN decoder, we present a general framework for solving various image-to-image", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "translation tasks. In contrast to previous methods, which tackle such tasks using a local ”pixel-to-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "pixel” approach, our framework takes a global approach, which we show can be used to solve a wide", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 299, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 299, + 733 + ], + "score": 1.0, + "content": "variety of image-to-image translation problems.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Results Figure 6a compares the results of our method to those of pix2pixHD and DeepFaceDraw-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "ing. As no code release is available for DeepFaceDrawing, we compare directly with sketches and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "results published in their paper. Due to the hard constraints of pix2pixHD, they are unable to handle", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 504, + 127 + ], + "score": 1.0, + "content": "the abstract sketches and obtain poor visual results. While DeepFaceDrawing obtain more visu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "ally pleasing results compared to pix2pixHD, they are still limited in their diversity. Conversely,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "although our model is trained on a different dataset, we are still able to generalize well to their", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "sketches. Notably, we observe our ability to obtain more diverse outputs that better retain finer de-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "tails (e.g. facial hair). Another limitation of DeepFaceDrawing is its focus on frontal images. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 169, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 506, + 184 + ], + "score": 1.0, + "content": "therefore illustrate our model’s ability to generate high-fidelity outputs from non-frontal sketches in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "Figure 13. As we are unable to directly evaluate DeepFaceDrawing on our constructed dataset, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 469, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 469, + 204 + ], + "score": 1.0, + "content": "compare our results only to those of pix2pixHD, trained and evaluated with the same data.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 82, + 506, + 204 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 216, + 283, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 284, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 284, + 228 + ], + "score": 1.0, + "content": "4.3.2 FACE FROM SEGMENTATION MAP", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 235, + 504, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 249 + ], + "score": 1.0, + "content": "Here, we evaluate using pSp for synthesizing face images from segmentation maps. In addition", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "to pix2pixHD, we compare our approach to two additional state-of-the-art label-to-image methods:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 504, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 504, + 269 + ], + "score": 1.0, + "content": "SPADE (Park et al., 2019), and CC FPSE (Liu et al., 2019b), both of which are based on pix2pixHD.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 234, + 506, + 269 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Results In Figure 6b we provide a visual comparison of the competing approaches on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "CelebAMask-HQ dataset containing 19 semantic categories. As the competing methods are based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "on pix2pixHD, the results of all three are visually similar. Conversely, our approach is able to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "generate high-quality outputs across a wide range of inputs of various poses and expressions. Ad-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "ditionally, using our multi-modal technique, pSp can easily generate various possible outputs with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "the same pose and attributes but varying fine styles for a single input semantic map or sketch image.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 412, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 412, + 360 + ], + "score": 1.0, + "content": "We provide examples in Figure 6c with additional examples in Appendix C.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 280, + 506, + 360 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 369, + 264, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 266, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 266, + 381 + ], + "score": 1.0, + "content": "4.3.3 HUMAN PERCEPTUAL STUDY", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "We additionally perform a human evaluation to compare the visual quality of each method presented", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "above. Here, each worker is given two images synthesized by different methods on the same input", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "and is given an unlimited time to select which output looks more realistic. Each of our three workers", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "reviews approximately 2, 800 pairs for each task, resulting in over 8, 400 human judgements for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "each method. Table 6d shows that pSp significantly outperforms the other respective methods in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 443, + 191, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 191, + 455 + ], + "score": 1.0, + "content": "both synthesis tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 388, + 505, + 455 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 470, + 293, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 295, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 295, + 486 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSIONS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "Although our suggested framework for image-to-image translation achieves compelling results in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "various applications, it has some inherent assumptions that should be considered. First, the high-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "quality images that are generated by utilizing the pretrained StyleGAN come with a cost — the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "method is limited to images that can be generated by StyleGAN. Thus, generating faces which are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "not close to frontal, or have certain expressions may be challenging if such examples were not avail-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 390, + 563 + ], + "score": 1.0, + "content": "able when training the StyleGAN model. Also, the global approach of", + "type": "text" + }, + { + "bbox": [ + 390, + 551, + 407, + 562 + ], + "score": 0.26, + "content": "\\mathsf { p } \\mathsf { S p }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ", although advantageous", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "for many tasks, does introduce a challenge in preserving finer details of the input image, such as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "earrings or background details. This is especially significant in tasks such as inpainting or super-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "resolution where standard image-to-image architectures can simply propagate local information.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 434, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 434, + 607 + ], + "score": 1.0, + "content": "Figure 7b in Appendix A presents some examples of such reconstruction failures.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 495, + 506, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "In this work, we proposed a novel encoder architecture that can be used to directly map a face", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 165, + 634 + ], + "score": 1.0, + "content": "image into the", + "type": "text" + }, + { + "bbox": [ + 165, + 622, + 185, + 632 + ], + "score": 0.88, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "latent space with no optimization required. The encoder architecture, motivated", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "by StyleGAN, consists of a hierarchy of three levels that correspond to the coarse, medium, and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 364, + 657 + ], + "score": 1.0, + "content": "fine groupings of the 18 style vectors defining the input in the", + "type": "text" + }, + { + "bbox": [ + 365, + 644, + 385, + 654 + ], + "score": 0.89, + "content": "\\mathcal { W } +", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "latent space. 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Combining our encoder", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "with a StyleGAN decoder, we present a general framework for solving various image-to-image", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "translation tasks. In contrast to previous methods, which tackle such tasks using a local ”pixel-to-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "pixel” approach, our framework takes a global approach, which we show can be used to solve a wide", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 299, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 299, + 733 + ], + "score": 1.0, + "content": "variety of image-to-image translation problems.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45, + "bbox_fs": [ + 104, + 610, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 505, + 113 + ], + "score": 1.0, + "content": "Rameen Abdal, Yipeng Qin, and Peter Wonka. 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We use a standard train-test split of the dataset, resulting in approximately", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "24,000 training images. 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For all applications, the input image resolution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 117, + 340 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 327, + 162, + 338 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 327, + 253, + 340 + ], + "score": 1.0, + "content": ", where the generated", + "type": "text" + }, + { + "bbox": [ + 253, + 327, + 308, + 338 + ], + "score": 0.9, + "content": "1 0 2 4 \\times 1 0 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "output is resized before being fed into the loss", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "functions. 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Finally,", + "type": "text" + }, + { + "bbox": [ + 176, + 443, + 188, + 454 + ], + "score": 0.88, + "content": "\\lambda _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "is set to 0.005 in all applications except for the StyleGAN inversion task, which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 453, + 263, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 263, + 466 + ], + "score": 1.0, + "content": "does not utilize the regularization loss.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 387, + 505, + 466 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 478, + 178, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 180, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 180, + 491 + ], + "score": 1.0, + "content": "A.2 DATASETS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "We conduct our experiments on the CelebA-HQ dataset (Karras et al., 2018), which contains 30,000", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "high quality images. We use a standard train-test split of the dataset, resulting in approximately", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "24,000 training images. The FFHQ dataset from (Karras et al., 2019), which contains 70,000 face", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 532, + 399, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 399, + 544 + ], + "score": 1.0, + "content": "images, is used for the StyleGAN inversion and face frontalization tasks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 498, + 506, + 544 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 504, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 504, + 561 + ], + "score": 1.0, + "content": "For the generation of face images from sketches, we construct a dataset representative of hand-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "drawn sketches using the CelebA-HQ dataset (Karras et al., 2018). Given an input image, we first", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 585 + ], + "score": 1.0, + "content": "apply a “pencil sketch” filter which retains most facial details of the original image while removing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "the remaining noise. We then apply the sketch-simplification method by Simo-Serra et al. (2016),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 592, + 320, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 320, + 605 + ], + "score": 1.0, + "content": "resulting in images resembling hand-drawn sketches.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 549, + 506, + 605 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 621, + 273, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 275, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 275, + 635 + ], + "score": 1.0, + "content": "B ADDITIONAL APPLICATIONS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 645, + 220, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 222, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 222, + 658 + ], + "score": 1.0, + "content": "B.1 SUPER RESOLUTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Here we show that our framework can be used to construct high-resolution (HR) facial images from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "corresponding low-resolution (LR) input images. PULSE (Menon et al., 2020) approaches this", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "task in an unsupervised manner by traversing the HR image manifold in search of an image that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "downsamples to the input LR image. In this work we focus on applying pSp in a supervised manner", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "as obtaining paired data is immediate. We show that our method achieves comparable results to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 721, + 245, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 245, + 732 + ], + "score": 1.0, + "content": "PULSE and other previous works.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 665, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Methodology and details We train our model in a supervised fashion, where for each input we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 288, + 106 + ], + "score": 1.0, + "content": "perform random bi-cubic down-sampling of", + "type": "text" + }, + { + "bbox": [ + 288, + 94, + 302, + 104 + ], + "score": 0.86, + "content": "\\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 93, + 408, + 106 + ], + "score": 1.0, + "content": "(i.e. no down-sampling),", + "type": "text" + }, + { + "bbox": [ + 409, + 94, + 457, + 105 + ], + "score": 0.83, + "content": "\\times 2 , \\times 4 , \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 93, + 461, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 461, + 94, + 480, + 105 + ], + "score": 0.79, + "content": "\\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 93, + 484, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 484, + 94, + 504, + 105 + ], + "score": 0.74, + "content": "\\times 3 2", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 325, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 325, + 117 + ], + "score": 1.0, + "content": "and set the original, full resolution image as the target.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 142 + ], + "score": 1.0, + "content": "Results Figure 9 demonstrates the visual quality of the resulting images from our method along", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "with those of the previous approaches. Although PULSE is able to achieve very high-quality results", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 164 + ], + "score": 1.0, + "content": "due to their usage of StyleGAN to generate images, they are unable to accurately retain identity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 276, + 175 + ], + "score": 1.0, + "content": "even when performing down-sampling of", + "type": "text" + }, + { + "bbox": [ + 276, + 162, + 290, + 173 + ], + "score": 0.87, + "content": "\\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 162, + 363, + 175 + ], + "score": 1.0, + "content": "to a resolution of", + "type": "text" + }, + { + "bbox": [ + 364, + 162, + 398, + 173 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 162, + 505, + 175 + ], + "score": 1.0, + "content": ". By learning a pixel-wise", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "correspondence between the LR and HR images, pix2pixHD is able to obtain satisfying results even", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 184, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 262, + 198 + ], + "score": 1.0, + "content": "when down-sampled to a resolution of", + "type": "text" + }, + { + "bbox": [ + 263, + 184, + 297, + 195 + ], + "score": 0.92, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 184, + 316, + 198 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + }, + { + "bbox": [ + 316, + 184, + 336, + 195 + ], + "score": 0.88, + "content": "\\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 184, + 506, + 198 + ], + "score": 1.0, + "content": "down-sampling). However, visually, their", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 504, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 504, + 207 + ], + "score": 1.0, + "content": "results appear less photo-realistic. Contrary to these previous works, we are able to obtain high-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 342, + 219 + ], + "score": 1.0, + "content": "quality results even when down-sampling to resolutions of", + "type": "text" + }, + { + "bbox": [ + 342, + 206, + 376, + 217 + ], + "score": 0.91, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 205, + 393, + 219 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 206, + 417, + 217 + ], + "score": 0.89, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 205, + 506, + 219 + ], + "score": 1.0, + "content": ". Finally, we generate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "multiple outputs for a given LR image using our multi-modal technique by perform style-mixing on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 189, + 240 + ], + "score": 1.0, + "content": "layers (4-7) with an", + "type": "text" + }, + { + "bbox": [ + 189, + 230, + 197, + 238 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "value of 0.5 with a randomly sampled w vector, which alters medium-level", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 239, + 399, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 399, + 252 + ], + "score": 1.0, + "content": "styles that mainly control facial features. 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We can then naturally interpolate between the two", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 319, + 401 + ], + "score": 1.0, + "content": "images by computing their intermediate latent code", + "type": "text" + }, + { + "bbox": [ + 319, + 389, + 420, + 402 + ], + "score": 0.91, + "content": "w ^ { \\prime } = \\lambda w _ { 1 } + ( 1 \\bar { - \\lambda } ) \\bar { w } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 390, + 437, + 401 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 437, + 390, + 486, + 401 + ], + "score": 0.91, + "content": "0 \\leq \\lambda \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 390, + 505, + 401 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 338, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 322, + 413 + ], + "score": 1.0, + "content": "generate the corresponding image using the new code", + "type": "text" + }, + { + "bbox": [ + 322, + 401, + 334, + 411 + ], + "score": 0.86, + "content": "w ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 401, + 338, + 413 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 108, + 425, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 504, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 504, + 438 + ], + "score": 1.0, + "content": "Inpainting Finally, we show the ability of our framework to reconstruct missing parts of an im-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "age using a simple, symmetric triangular mask. 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Method 90°个 Similarity ↓Runtime 70° 50° 30°
R&R 0.34 0.56 0.660.7 1.5
pSp 0.32 0.52 0.600.63 0.1
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Method↑ Similarity↓LPIPS↓MSERuntime
ALAE IDInvert0.06 0.180.32 0.220.15 0.060.207 0.032
W Encoder Naive W+0.350.23 0.190.06 0.040.064 0.064
0.49 0.560.170.030.105
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