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Section: Learning in NMF using dynamic logic algorithm. The learning process consists of estimating model parameters S and associating signals with concepts by maximizing the similarity L. Note that all possible combinations of signals and models are accounted for in expression (2) for L. This can be seen by expanding ... | Wikipedia - Neural modeling fields - Learning in NMF using dynamic logic algorithm | 218 | 1,083 | null |
The maximization of similarity L is done as follows. First, the unknown parameters {Sm} are randomly initialized. Then the association variables f(m|n) are computed, f ( m | n ) = r ( m ) l ( X β ( n | m ) ) β m β² = 1 M r ( m β² ) l ( X β ( n | m β² ) ) {\displaystyle f(m|n)={\frac {r(m)l({\vec {X}}(n|m))}{\sum _{m'=1}^{... | Wikipedia - Neural modeling fields - Learning in NMF using dynamic logic algorithm | 221 | 582 | null |
Equation for f(m|n) looks like the Bayes formula for a posteriori probabilities; if l(n|m) in the result of learning become conditional likelihoods, f(m|n) become Bayesian probabilities for signal n originating from object m. The dynamic logic of the NMF is defined as follows: d S β m d t = β n = 1 N f ( m | n ) β ln β‘... | Wikipedia - Neural modeling fields - Learning in NMF using dynamic logic algorithm | 242 | 551 | null |
The dynamic logic of the NMF is defined as follows: d S β m d t = β n = 1 N f ( m | n ) β ln β‘ l ( n | m ) β M β m β M β m β S β m {\displaystyle {\frac {d{\vec {S}}_{m}}{dt}}=\sum _{n=1}^{N}{f(m|n){\frac {\partial {\ln l(n|m)}}{\partial {{\vec {M}}_{m}}}}{\frac {\partial {{\vec {M}}_{m}}}{\partial {{\vec {S}}_{m}}}}}}... | Wikipedia - Neural modeling fields - Learning in NMF using dynamic logic algorithm | 437 | 771 | null |
d f ( m | n ) d t = f ( m | n ) β m β² = 1 M [ Ξ΄ m m β² β f ( m β² | n ) ] β ln β‘ l ( n | m β² ) β M β m β² β M β m β² β S β m β² d S β m β² d t {\displaystyle {\frac {df(m|n)}{dt}}=f(m|n)\sum _{m'=1}^{M}{[\delta _{mm'}-f(m'|n)]{\frac {\partial {\ln l(n|m')}}{\partial {{\vec {M}}_{m'}}}}}{\frac {\partial {{\vec {M}}_{m'}}}{\pa... | Wikipedia - Neural modeling fields - Learning in NMF using dynamic logic algorithm | 344 | 855 | null |
Section: Example of dynamic logic operations. Finding patterns below noise can be an exceedingly complex problem. If an exact pattern shape is not known and depends on unknown parameters, these parameters should be found by fitting the pattern model to the data. However, when the locations and orientations of patterns ... | Wikipedia - Neural modeling fields - Example of dynamic logic operations | 325 | 1,561 | null |
To apply NMF and dynamic logic to this problem one needs to develop parametric adaptive models of expected patterns. The models and conditional partial similarities for this case are described in details in: a uniform model for noise, Gaussian blobs for highly-fuzzy, poorly resolved patterns, and parabolic models for '... | Wikipedia - Neural modeling fields - Example of dynamic logic operations | 326 | 1,539 | null |
Section: Neural modeling fields hierarchical organization. Above, a single processing level in a hierarchical NMF system was described. At each level of hierarchy there are input signals from lower levels, models, similarity measures (L), emotions, which are defined as changes in similarity, and actions; actions includ... | Wikipedia - Neural modeling fields - Neural modeling fields hierarchical organization | 306 | 1,493 | null |
For example, consider a concept-model "chair." It has a "behavioral" purpose of initiating sitting behavior (if sitting is required by the body), this is the "bodily" purpose at the same hierarchical level. In addition, it has a "purely mental" purpose at a higher level in the hierarchy, a purpose of helping to recogni... | Wikipedia - Neural modeling fields - Neural modeling fields hierarchical organization | 322 | 1,549 | null |
Article: Neural network quantum states. Neural Network Quantum States (NQS or NNQS) is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body qu... | Wikipedia - Neural network quantum states - Summary | 338 | 1,161 | null |
Section: Learning the Ground-State Wave Function. One common application of NQS is to find an approximate representation of the ground state wave function of a given Hamiltonian H ^ {\displaystyle {\hat {H}}} . The learning procedure in this case consists in finding the best neural-network weights that minimize the var... | Wikipedia - Neural network quantum states - Learning the Ground-State Wave Function | 201 | 784 | null |
{\displaystyle E(W)=\langle \Psi ;W|{\hat {H}}|\Psi ;W\rangle .} Since, for a general artificial neural network, computing the expectation value is an exponentially costly operation in N {\displaystyle N} , stochastic techniques based, for example, on the Monte Carlo method are used to estimate E ( W ) {\displaystyle E... | Wikipedia - Neural network quantum states - Learning the Ground-State Wave Function | 297 | 828 | null |
More specifically, a set of M {\displaystyle M} samples S ( 1 ) , S ( 2 ) β¦ S ( M ) {\displaystyle S^{(1)},S^{(2)}\ldots S^{(M)}} , with S ( i ) = s 1 ( i ) β¦ s N ( i ) {\displaystyle S^{(i)}=s_{1}^{(i)}\ldots s_{N}^{(i)}} , is generated such that they are uniformly distributed according to the Born probability density... | Wikipedia - Neural network quantum states - Learning the Ground-State Wave Function | 321 | 767 | null |
E ( W ) β 1 M β i M E l o c ( S ( i ) ) . {\displaystyle E(W)\simeq {\frac {1}{M}}\sum _{i}^{M}E_{\mathrm {loc} }(S^{(i)}).} Similarly, it can be shown that the gradient of the energy with respect to the network weights W {\displaystyle W} is also approximated by a sample mean β E ( W ) β W k β 1 M β i M ( E l o c ( S ... | Wikipedia - Neural network quantum states - Learning the Ground-State Wave Function | 335 | 716 | null |
{\displaystyle E(W)\simeq {\frac {1}{M}}\sum _{i}^{M}E_{\mathrm {loc} }(S^{(i)}).} Similarly, it can be shown that the gradient of the energy with respect to the network weights W {\displaystyle W} is also approximated by a sample mean β E ( W ) β W k β 1 M β i M ( E l o c ( S ( i ) ) β E ( W ) ) O k β ( S ( i ) ) , {\... | Wikipedia - Neural network quantum states - Learning the Ground-State Wave Function | 353 | 841 | null |
Article: Normalization (machine learning). In machine learning, normalization is a statistical technique with various applications. There are two main forms of normalization, namely data normalization and activation normalization. Data normalization (or feature scaling) includes methods that rescale input data so that ... | Wikipedia - Normalization (machine learning) - Summary | 271 | 1,379 | null |
Section: Batch normalization. Batch normalization (BatchNorm) operates on the activations of a layer for each mini-batch. Consider a simple feedforward network, defined by chaining together modules: x ( 0 ) β¦ x ( 1 ) β¦ x ( 2 ) β¦ β― {\displaystyle x^{(0)}\mapsto x^{(1)}\mapsto x^{(2)}\mapsto \cdots } where each network m... | Wikipedia - Normalization (machine learning) - Batch normalization | 325 | 1,020 | null |
Concretely, suppose we have a batch of inputs x ( 1 ) ( 0 ) , x ( 2 ) ( 0 ) , β¦ , x ( B ) ( 0 ) {\displaystyle x_{(1)}^{(0)},x_{(2)}^{(0)},\dots ,x_{(B)}^{(0)}} , fed all at once into the network. We would obtain in the middle of the network some vectors: x ( 1 ) ( l ) , x ( 2 ) ( l ) , β¦ , x ( B ) ( l ) {\displaystyle... | Wikipedia - Normalization (machine learning) - Batch normalization | 433 | 892 | null |
We would obtain in the middle of the network some vectors: x ( 1 ) ( l ) , x ( 2 ) ( l ) , β¦ , x ( B ) ( l ) {\displaystyle x_{(1)}^{(l)},x_{(2)}^{(l)},\dots ,x_{(B)}^{(l)}} The BatchNorm module computes the coordinate-wise mean and variance of these vectors: ΞΌ i ( l ) = 1 B β b = 1 B x ( b ) , i ( l ) ( Ο i ( l ) ) 2 ... | Wikipedia - Normalization (machine learning) - Batch normalization | 375 | 852 | null |
In other words, we are considering the i {\displaystyle i} -th coordinate of each vector in the batch, and computing the mean and variance of these numbers. It then normalizes each coordinate to have zero mean and unit variance: x ^ ( b ) , i ( l ) = x ( b ) , i ( l ) β ΞΌ i ( l ) ( Ο i ( l ) ) 2 + Ο΅ {\displaystyle {\ha... | Wikipedia - Normalization (machine learning) - Batch normalization | 347 | 948 | null |
Section: Batch normalization > Special cases. The original paper recommended to only use BatchNorms after a linear transform, not after a nonlinear activation. That is, Ο ( B N ( W x + b ) ) {\displaystyle \phi (\mathrm {BN} (Wx+b))} , not B N ( Ο ( W x + b ) ) {\displaystyle \mathrm {BN} (\phi (Wx+b))} . Also, the bia... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 258 | 911 | null |
This is sometimes called Spatial BatchNorm, or BatchNorm2D, or per-channel BatchNorm. Concretely, suppose we have a 2-dimensional convolutional layer defined by: x h , w , c ( l ) = β h β² , w β² , c β² K h β² β h , w β² β w , c , c β² ( l ) x h β² , w β² , c β² ( l β 1 ) + b c ( l ) {\displaystyle x_{h,w,c}^{(l)}=\sum _{h',w',... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 316 | 687 | null |
K Ξ h , Ξ w , c , c β² ( l ) {\displaystyle K_{\Delta h,\Delta w,c,c'}^{(l)}} is a kernel tensor. Each channel c {\displaystyle c} corresponds to a kernel K h β² β h , w β² β w , c , c β² ( l ) {\displaystyle K_{h'-h,w'-w,c,c'}^{(l)}} , with indices Ξ h , Ξ w , c β² {\displaystyle \Delta h,\Delta w,c'} . b c ( l ) {\display... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 215 | 577 | null |
That is, it is applied once per kernel c {\displaystyle c} (equivalently, once per channel c {\displaystyle c} ), not per activation x h , w , c ( l + 1 ) {\displaystyle x_{h,w,c}^{(l+1)}} : ΞΌ c ( l ) = 1 B H W β b = 1 B β h = 1 H β w = 1 W x ( b ) , h , w , c ( l ) ( Ο c ( l ) ) 2 = 1 B H W β b = 1 B β h = 1 H β w = 1... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 343 | 606 | null |
That is, even though there are only B {\displaystyle B} data points in a batch, all B H W {\displaystyle BHW} outputs from the kernel in this batch are treated equally. Subsequently, normalization and the linear transform is also done per kernel: x ^ ( b ) , h , w , c ( l ) = x ( b ) , h , w , c ( l ) β ΞΌ c ( l ) ( Ο c... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 323 | 712 | null |
Subsequently, normalization and the linear transform is also done per kernel: x ^ ( b ) , h , w , c ( l ) = x ( b ) , h , w , c ( l ) β ΞΌ c ( l ) ( Ο c ( l ) ) 2 + Ο΅ y ( b ) , h , w , c ( l ) = Ξ³ c x ^ ( b ) , h , w , c ( l ) + Ξ² c {\displaystyle {\begin{aligned}{\hat {x}}_{(b),h,w,c}^{(l)}&={\frac {x_{(b),h,w,c}^{(l)}... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 334 | 763 | null |
The following is a Python implementation of BatchNorm for 2D convolutions: For multilayered recurrent neural networks (RNN), BatchNorm is usually applied only for the input-to-hidden part, not the hidden-to-hidden part. Let the hidden state of the l {\displaystyle l} -th layer at time t {\displaystyle t} be h t ( l ) {... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 277 | 700 | null |
The standard RNN, without normalization, satisfies h t ( l ) = Ο ( W ( l ) h t l β 1 + U ( l ) h t β 1 l + b ( l ) ) {\displaystyle h_{t}^{(l)}=\phi (W^{(l)}h_{t}^{l-1}+U^{(l)}h_{t-1}^{l}+b^{(l)})} where W ( l ) , U ( l ) , b ( l ) {\displaystyle W^{(l)},U^{(l)},b^{(l)}} are weights and biases, and Ο {\displaystyle \ph... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 323 | 715 | null |
Then frame-wise BatchNorm means normalizing over b {\displaystyle b} : ΞΌ t ( l ) = 1 B β b = 1 B h i , t ( l ) ( Ο t ( l ) ) 2 = 1 B β b = 1 B ( h t ( l ) β ΞΌ t ( l ) ) 2 {\displaystyle {\begin{aligned}\mu _{t}^{(l)}&={\frac {1}{B}}\sum _{b=1}^{B}h_{i,t}^{(l)}\\(\sigma _{t}^{(l)})^{2}&={\frac {1}{B}}\sum _{b=1}^{B}(h_{... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 339 | 634 | null |
_{t=1}^{T}h_{i,t}^{(l)}\\(\sigma ^{(l)})^{2}&={\frac {1}{BT}}\sum _{b=1}^{B}\sum _{t=1}^{T}(h_{t}^{(l)}-\mu ^{(l)})^{2}\end{aligned}}} Frame-wise BatchNorm is suited for causal tasks such as next-character prediction, where future frames are unavailable, forcing normalization per frame. Sequence-wise BatchNorm is suite... | Wikipedia - Normalization (machine learning) - Batch normalization > Special cases | 237 | 759 | null |
Section: Batch normalization > Improvements. BatchNorm has been very popular and there were many attempted improvements. Some examples include: ghost batching: randomly partition a batch into sub-batches and perform BatchNorm separately on each; weight decay on Ξ³ {\displaystyle \gamma } and Ξ² {\displaystyle \beta } ; a... | Wikipedia - Normalization (machine learning) - Batch normalization > Improvements | 348 | 1,200 | null |
Section: Layer normalization. Layer normalization (LayerNorm) is a popular alternative to BatchNorm. Unlike BatchNorm, which normalizes activations across the batch dimension for a given feature, LayerNorm normalizes across all the features within a single data sample. Compared to BatchNorm, LayerNorm's performance is ... | Wikipedia - Normalization (machine learning) - Layer normalization | 167 | 684 | null |
Similar to BatchNorm, learnable parameters Ξ³ {\displaystyle \gamma } (scale) and Ξ² {\displaystyle \beta } (shift) are applied. It is defined by: x i ^ = x i β ΞΌ Ο 2 + Ο΅ , y i = Ξ³ i x i ^ + Ξ² i {\displaystyle {\hat {x_{i}}}={\frac {x_{i}-\mu }{\sqrt {\sigma ^{2}+\epsilon }}},\quad y_{i}=\gamma _{i}{\hat {x_{i}}}+\beta _... | Wikipedia - Normalization (machine learning) - Layer normalization | 277 | 582 | null |
In the previous notation, we have: ΞΌ ( l ) = 1 H W C β h = 1 H β w = 1 W β c = 1 C x h , w , c ( l ) ( Ο ( l ) ) 2 = 1 H W C β h = 1 H β w = 1 W β c = 1 C ( x h , w , c ( l ) β ΞΌ ( l ) ) 2 x ^ h , w , c ( l ) = x ^ h , w , c ( l ) β ΞΌ ( l ) ( Ο ( l ) ) 2 + Ο΅ y h , w , c ( l ) = Ξ³ ( l ) x ^ h , w , c ( l ) + Ξ² ( l ) {\d... | Wikipedia - Normalization (machine learning) - Layer normalization > Examples | 333 | 559 | null |
{x}}_{h,w,c}^{(l)}&={\frac {{\hat {x}}_{h,w,c}^{(l)}-\mu ^{(l)}}{\sqrt {(\sigma ^{(l)})^{2}+\epsilon }}}\\y_{h,w,c}^{(l)}&=\gamma ^{(l)}{\hat {x}}_{h,w,c}^{(l)}+\beta ^{(l)}\end{aligned}}} Notice that the batch index b {\displaystyle b} is removed, while the channel index c {\displaystyle c} is added. In recurrent neur... | Wikipedia - Normalization (machine learning) - Layer normalization > Examples | 194 | 401 | null |
For example, if the hidden vector in an RNN at timestep t {\displaystyle t} is x ( t ) β R D {\displaystyle x^{(t)}\in \mathbb {R} ^{D}} , where D {\displaystyle D} is the dimension of the hidden vector, then LayerNorm will be applied with: x i ^ ( t ) = x i ( t ) β ΞΌ ( t ) ( Ο ( t ) ) 2 + Ο΅ , y i ( t ) = Ξ³ i x i ^ ( t... | Wikipedia - Normalization (machine learning) - Layer normalization > Examples | 344 | 678 | null |
Section: Layer normalization > Root mean square layer normalization. Root mean square layer normalization (RMSNorm) changes LayerNorm by: x i ^ = x i 1 D β i = 1 D x i 2 , y i = Ξ³ x i ^ + Ξ² {\displaystyle {\hat {x_{i}}}={\frac {x_{i}}{\sqrt {{\frac {1}{D}}\sum _{i=1}^{D}x_{i}^{2}}}},\quad y_{i}=\gamma {\hat {x_{i}}}+\b... | Wikipedia - Normalization (machine learning) - Layer normalization > Root mean square layer normalization | 177 | 416 | null |
Section: Weight normalization. Weight normalization (WeightNorm) is a technique inspired by BatchNorm that normalizes weight matrices in a neural network, rather than its activations. One example is spectral normalization, which divides weight matrices by their spectral norm. The spectral normalization is used in gener... | Wikipedia - Normalization (machine learning) - Weight normalization | 208 | 750 | null |
This is the eigenvector of W {\displaystyle W} with eigenvalue β W β s {\displaystyle \|W\|_{s}} . RETURN x β , β W x β β 2 {\displaystyle x^{*},\|Wx^{*}\|_{2}} By reassigning W i β W i β W i β s {\displaystyle W_{i}\leftarrow {\frac {W_{i}}{\|W_{i}\|_{s}}}} after each update of the discriminator, we can upper-bound β ... | Wikipedia - Normalization (machine learning) - Weight normalization | 291 | 689 | null |
Then, at step t + 1 {\displaystyle t+1} , use x i β ( t ) {\displaystyle x_{i}^{*}(t)} as the initial guess for the algorithm. Since W i ( t + 1 ) {\displaystyle W_{i}(t+1)} is very close to W i ( t ) {\displaystyle W_{i}(t)} , so is x i β ( t ) {\displaystyle x_{i}^{*}(t)} to x i β ( t + 1 ) {\displaystyle x_{i}^{*}(t... | Wikipedia - Normalization (machine learning) - Weight normalization | 153 | 359 | null |
Section: CNN-specific normalization > Response normalization. Local response normalization was used in AlexNet. It was applied in a convolutional layer, just after a nonlinear activation function. It was defined by: b x , y i = a x , y i ( k + Ξ± β j = max ( 0 , i β n / 2 ) min ( N β 1 , i + n / 2 ) ( a x , y j ) 2 ) Ξ² ... | Wikipedia - Normalization (machine learning) - CNN-specific normalization > Response normalization | 329 | 897 | null |
It was a variant of the earlier local contrast normalization. b x , y i = a x , y i ( k + Ξ± β j = max ( 0 , i β n / 2 ) min ( N β 1 , i + n / 2 ) ( a x , y j β a Β― x , y j ) 2 ) Ξ² {\displaystyle b_{x,y}^{i}={\frac {a_{x,y}^{i}}{\left(k+\alpha \sum _{j=\max(0,i-n/2)}^{\min(N-1,i+n/2)}\left(a_{x,y}^{j}-{\bar {a}}_{x,y}^{... | Wikipedia - Normalization (machine learning) - CNN-specific normalization > Response normalization | 326 | 802 | null |
It can be understood as the LayerNorm for CNN applied once per channel, or equivalently, as group normalization where each group consists of a single channel: ΞΌ c ( l ) = 1 H W β h = 1 H β w = 1 W x h , w , c ( l ) ( Ο c ( l ) ) 2 = 1 H W β h = 1 H β w = 1 W ( x h , w , c ( l ) β ΞΌ c ( l ) ) 2 x ^ h , w , c ( l ) = x ^... | Wikipedia - Normalization (machine learning) - CNN-specific normalization > Instance normalization | 341 | 653 | null |
{x}}_{h,w,c}^{(l)}&={\frac {{\hat {x}}_{h,w,c}^{(l)}-\mu _{c}^{(l)}}{\sqrt {(\sigma _{c}^{(l)})^{2}+\epsilon }}}\\y_{h,w,c}^{(l)}&=\gamma _{c}^{(l)}{\hat {x}}_{h,w,c}^{(l)}+\beta _{c}^{(l)}\end{aligned}}} | Wikipedia - Normalization (machine learning) - CNN-specific normalization > Instance normalization | 161 | 204 | null |
Section: CNN-specific normalization > Adaptive instance normalization. Adaptive instance normalization (AdaIN) is a variant of instance normalization, designed specifically for neural style transfer with CNNs, rather than just CNNs in general. In the AdaIN method of style transfer, we take a CNN and two input images, o... | Wikipedia - Normalization (machine learning) - CNN-specific normalization > Adaptive instance normalization | 290 | 1,024 | null |
Note that x ( l ) , style {\displaystyle x^{(l),{\text{ style}}}} itself remains unchanged. Explicitly, we have: y h , w , c ( l ) , content = Ο c ( l ) , style ( x h , w , c ( l ) , content β ΞΌ c ( l ) , content ( Ο c ( l ) , content ) 2 + Ο΅ ) + ΞΌ c ( l ) , style {\displaystyle {\begin{aligned}y_{h,w,c}^{(l),{\text{ c... | Wikipedia - Normalization (machine learning) - CNN-specific normalization > Adaptive instance normalization | 273 | 560 | null |
Section: Transformers. Some normalization methods were designed for use in transformers. The original 2017 transformer used the "post-LN" configuration for its LayerNorms. It was difficult to train, and required careful hyperparameter tuning and a "warm-up" in learning rate, where it starts small and gradually increase... | Wikipedia - Normalization (machine learning) - Transformers | 264 | 1,137 | null |
Article: Novelty detection. Novelty detection is the mechanism by which an intelligent organism is able to identify an incoming sensory pattern as being hitherto unknown. If the pattern is sufficiently salient or associated with a high positive or strong negative utility, it will be given computational resources for ef... | Wikipedia - Novelty detection - Summary | 217 | 1,164 | null |
Article: Overfitting. In mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more par... | Wikipedia - Overfitting - Summary | 344 | 1,727 | null |
As an extreme example, if the number of parameters is the same as or greater than the number of observations, then a model can perfectly predict the training data simply by memorizing the data in its entirety. (For an illustration, see Figure 2.) Such a model, though, will typically fail severely when making prediction... | Wikipedia - Overfitting - Summary | 293 | 1,419 | null |
Section: Statistical inference. In statistics, an inference is drawn from a statistical model, which has been selected via some procedure. Burnham & Anderson, in their much-cited text on model selection, argue that to avoid overfitting, we should adhere to the "Principle of Parsimony". The authors also state the follow... | Wikipedia - Overfitting - Statistical inference | 282 | 1,300 | null |
Section: Statistical inference > Regression. In regression analysis, overfitting occurs frequently. As an extreme example, if there are p variables in a linear regression with p data points, the fitted line can go exactly through every point. For logistic regression or Cox proportional hazards models, there are a varie... | Wikipedia - Overfitting - Statistical inference > Regression | 216 | 1,059 | null |
Section: Machine learning. Usually, a learning algorithm is trained using some set of "training data": exemplary situations for which the desired output is known. The goal is that the algorithm will also perform well on predicting the output when fed "validation data" that was not encountered during its training. Overf... | Wikipedia - Overfitting - Machine learning | 332 | 1,754 | null |
When comparing different types of models, complexity cannot be measured solely by counting how many parameters exist in each model; the expressivity of each parameter must be considered as well. For example, it is nontrivial to directly compare the complexity of a neural net (which can track curvilinear relationships) ... | Wikipedia - Overfitting - Machine learning | 343 | 1,742 | null |
Section: Machine learning > Consequences. The most obvious consequence of overfitting is poor performance on the validation dataset. Other negative consequences include: A function that is overfitted is likely to request more information about each item in the validation dataset than does the optimal function; gatherin... | Wikipedia - Overfitting - Machine learning > Consequences | 282 | 1,474 | null |
Section: Machine learning > Remedy. The optimal function usually needs verification on bigger or completely new datasets. There are, however, methods like minimum spanning tree or life-time of correlation that applies the dependence between correlation coefficients and time-series (window width). Whenever the window wi... | Wikipedia - Overfitting - Machine learning > Remedy | 153 | 861 | null |
Section: Underfitting. Underfitting is the inverse of overfitting, meaning that the statistical model or machine learning algorithm is too simplistic to accurately capture the patterns in the data. A sign of underfitting is that there is a high bias and low variance detected in the current model or algorithm used (the ... | Wikipedia - Overfitting - Underfitting | 334 | 1,614 | null |
Section: Underfitting > Resolving underfitting. There are multiple ways to deal with underfitting: Increase the complexity of the model: If the model is too simple, it may be necessary to increase its complexity by adding more features, increasing the number of parameters, or using a more flexible model. However, this ... | Wikipedia - Overfitting - Underfitting > Resolving underfitting | 297 | 1,536 | null |
Section: Paraphrase generation > Multiple sequence alignment. Barzilay and Lee proposed a method to generate paraphrases through the usage of monolingual parallel corpora, namely news articles covering the same event on the same day. Training consists of using multi-sequence alignment to generate sentence-level paraphr... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase generation > Multiple sequence alignment | 278 | 1,322 | null |
Section: Paraphrase generation > Phrase-based machine translation. Paraphrase can also be generated through the use of phrase-based translation as proposed by Bannard and Callison-Burch. The chief concept consists of aligning phrases in a pivot language to produce potential paraphrases in the original language. For exa... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase generation > Phrase-based machine translation | 301 | 1,088 | null |
Additionally, the sentence e 1 {\displaystyle e_{1}} is added as a prior to add context to the paraphrase. Thus the optimal paraphrase, e 2 ^ {\displaystyle {\hat {e_{2}}}} can be modeled as: e 2 ^ = arg max e 2 β e 1 Pr ( e 2 | e 1 , S ) = arg max e 2 β e 1 β f Pr ( e 2 | f , S ) Pr ( f | e 1 , S ) {\displaystyle {\ha... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase generation > Phrase-based machine translation | 344 | 800 | null |
Section: Paraphrase generation > Long short-term memory. There has been success in using long short-term memory (LSTM) models to generate paraphrases. In short, the model consists of an encoder and decoder component, both implemented using variations of a stacked residual LSTM. First, the encoding LSTM takes a one-hot ... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase generation > Long short-term memory | 197 | 865 | null |
Section: Paraphrase generation > Transformers. With the introduction of Transformer models, paraphrase generation approaches improved their ability to generate text by scaling neural network parameters and heavily parallelizing training through feed-forward layers. These models are so fluent in generating text that hum... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase generation > Transformers | 206 | 1,041 | null |
Section: Paraphrase recognition > Recursive autoencoders. Paraphrase recognition has been attempted by Socher et al through the use of recursive autoencoders. The main concept is to produce a vector representation of a sentence and its components by recursively using an autoencoder. The vector representations of paraph... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase recognition > Recursive autoencoders | 335 | 1,362 | null |
Given two sentences W 1 {\displaystyle W_{1}} and W 2 {\displaystyle W_{2}} of length 4 and 3 respectively, the autoencoders would produce 7 and 5 vector representations including the initial word embeddings. The euclidean distance is then taken between every combination of vectors in W 1 {\displaystyle W_{1}} and W 2 ... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase recognition > Recursive autoencoders | 264 | 954 | null |
Section: Paraphrase recognition > Skip-thought vectors. Skip-thought vectors are an attempt to create a vector representation of the semantic meaning of a sentence, similarly to the skip gram model. Skip-thought vectors are produced through the use of a skip-thought model which consists of three key components, an enco... | Wikipedia - Paraphrasing (computational linguistics) - Paraphrase recognition > Skip-thought vectors | 214 | 1,024 | null |
Section: Evaluation. Multiple methods can be used to evaluate paraphrases. Since paraphrase recognition can be posed as a classification problem, most standard evaluations metrics such as accuracy, f1 score, or an ROC curve do relatively well. However, there is difficulty calculating f1-scores due to trouble producing ... | Wikipedia - Paraphrasing (computational linguistics) - Evaluation | 323 | 1,651 | null |
Automated approaches to evaluation prove to be challenging as it is essentially a problem as difficult as paraphrase recognition. While originally used to evaluate machine translations, bilingual evaluation understudy (BLEU) has been used successfully to evaluate paraphrase generation models as well. However, paraphras... | Wikipedia - Paraphrasing (computational linguistics) - Evaluation | 344 | 1,604 | null |
Section: Noisy version ("Learning Parity with Noise"). In Learning Parity with Noise (LPN), the samples may contain some error. Instead of samples (x, Ζ(x)), the algorithm is provided with (x, y), where for random boolean b β { 0 , 1 } {\displaystyle b\in \{0,1\}} y = { f ( x ) , if b 1 β f ( x ) , otherwise {\displays... | Wikipedia - Parity learning - Noisy version ("Learning Parity with Noise") | 173 | 512 | null |
Section: Definition. Given a finite set Ξ£ of constant symbols and a countable set X of variable symbols disjoint from Ξ£, a pattern is a finite non-empty string of symbols from Ξ£βͺX. The length of a pattern p, denoted by |p|, is just the number of its symbols. The set of all patterns containing exactly n distinct variabl... | Wikipedia - Pattern language (formal languages) - Definition | 341 | 1,175 | null |
For example, using the constants Ξ£ = { 0, 1 } and the variables X = { x, y, z, ... }, the pattern 0x10xx1 βP1 and xxy βP2 has length 7 and 3, respectively. An instance of the former pattern is 00z100z0z1 and 01z101z1z1, it is obtained by the substitution that maps x to 0z and to 1z, respectively, and each other symbol ... | Wikipedia - Pattern language (formal languages) - Definition | 225 | 699 | null |
Section: Properties. The problem of deciding whether s β L(p) for an arbitrary string s β Ξ£+ and pattern p is NP-complete (see picture), and so is hence the problem of deciding p β€ q for arbitrary patterns p, q. The class of pattern languages is not closed under ... union: e.g. for Ξ£ = {0,1} as above, L(01)βͺL(10) is no... | Wikipedia - Pattern language (formal languages) - Properties | 350 | 1,085 | null |
Section: Location in the Chomsky hierarchy. In a refined Chomsky hierarchy, the class of pattern languages is a proper superclass and subclass of the singleton and the indexed languages, respectively, but incomparable to the language classes in between; due to the latter, the pattern language class is not explicitly sh... | Wikipedia - Pattern language (formal languages) - Location in the Chomsky hierarchy | 169 | 812 | null |
Each singleton language is trivially a pattern language, generated by a pattern without variables. Each pattern language can be produced by an indexed grammar: For example, using Ξ£ = { a, b, c } and X = { x, y }, the pattern a x b y c x a y b is generated by a grammar with nonterminal symbols N = { Sx, Sy, S } βͺ X, ter... | Wikipedia - Pattern language (formal languages) - Location in the Chomsky hierarchy | 333 | 817 | null |
Each pattern language can be produced by an indexed grammar: For example, using Ξ£ = { a, b, c } and X = { x, y }, the pattern a x b y c x a y b is generated by a grammar with nonterminal symbols N = { Sx, Sy, S } βͺ X, terminal symbols T = Ξ£, index symbols F = { ax, bx, cx, ay, by, cy }, start symbol Sx, and the followi... | Wikipedia - Pattern language (formal languages) - Location in the Chomsky hierarchy | 367 | 810 | null |
Article: Pattern recognition. Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their primary function is to distinguish and create e... | Wikipedia - Pattern recognition - Summary | 325 | 1,809 | null |
Pattern recognition is a more general problem that encompasses other types of output as well. Other examples are regression, which assigns a real-valued output to each input; sequence labeling, which assigns a class to each member of a sequence of values (for example, part of speech tagging, which assigns a part of spe... | Wikipedia - Pattern recognition - Summary | 192 | 1,016 | null |
Section: Overview. A modern definition of pattern recognition is: The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories. Patte... | Wikipedia - Pattern recognition - Overview | 320 | 1,642 | null |
Sometimes different terms are used to describe the corresponding supervised and unsupervised learning procedures for the same type of output. The unsupervised equivalent of classification is normally known as clustering, based on the common perception of the task as involving no training data to speak of, and of groupi... | Wikipedia - Pattern recognition - Overview | 232 | 1,225 | null |
These feature vectors can be seen as defining points in an appropriate multidimensional space, and methods for manipulating vectors in vector spaces can be correspondingly applied to them, such as computing the dot product or the angle between two vectors. Features typically are either categorical (also known as nomina... | Wikipedia - Pattern recognition - Overview | 261 | 1,016 | null |
Section: Overview > Probabilistic classifiers. Many common pattern recognition algorithms are probabilistic in nature, in that they use statistical inference to find the best label for a given instance. Unlike other algorithms, which simply output a "best" label, often probabilistic algorithms also output a probability... | Wikipedia - Pattern recognition - Overview > Probabilistic classifiers | 303 | 1,531 | null |
Section: Overview > Number of important feature variables. Feature selection algorithms attempt to directly prune out redundant or irrelevant features. A general introduction to feature selection which summarizes approaches and challenges, has been given. The complexity of feature-selection is, because of its non-monot... | Wikipedia - Pattern recognition - Overview > Number of important feature variables | 263 | 1,395 | null |
Section: Problem statement. The problem of pattern recognition can be stated as follows: Given an unknown function g : X β Y {\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}} (the ground truth) that maps input instances x β X {\displaystyle {\boldsymbol {x}}\in {\mathcal {X}}} to output labels y β Y {\displays... | Wikipedia - Pattern recognition - Problem statement | 326 | 927 | null |
(For example, if the problem is filtering spam, then x i {\displaystyle {\boldsymbol {x}}_{i}} is some representation of an email and y {\displaystyle y} is either "spam" or "non-spam"). In order for this to be a well-defined problem, "approximates as closely as possible" needs to be defined rigorously. In decision the... | Wikipedia - Pattern recognition - Problem statement | 331 | 1,306 | null |
For example, in the case of classification, the simple zero-one loss function is often sufficient. This corresponds simply to assigning a loss of 1 to any incorrect labeling and implies that the optimal classifier minimizes the error rate on independent test data (i.e. counting up the fraction of instances that the lea... | Wikipedia - Pattern recognition - Problem statement | 337 | 1,238 | null |
In a discriminative approach to the problem, f is estimated directly. In a generative approach, however, the inverse probability p ( x | l a b e l ) {\displaystyle p({{\boldsymbol {x}}|{\rm {label}}})} is instead estimated and combined with the prior probability p ( l a b e l | ΞΈ ) {\displaystyle p({\rm {label}}|{\bold... | Wikipedia - Pattern recognition - Problem statement | 173 | 483 | null |
In a generative approach, however, the inverse probability p ( x | l a b e l ) {\displaystyle p({{\boldsymbol {x}}|{\rm {label}}})} is instead estimated and combined with the prior probability p ( l a b e l | ΞΈ ) {\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})} using Bayes' rule, as follows: p ( l a b e l | x ,... | Wikipedia - Pattern recognition - Problem statement | 393 | 940 | null |
{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\sum _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})}}.} When the labels are continuously distributed (e.g... | Wikipedia - Pattern recognition - Problem statement | 429 | 937 | null |
{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\int _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})\operatorname {d} L}}.} The value of ΞΈ {\displaystyle ... | Wikipedia - Pattern recognition - Problem statement | 313 | 959 | null |
In a Bayesian context, the regularization procedure can be viewed as placing a prior probability p ( ΞΈ ) {\displaystyle p({\boldsymbol {\theta }})} on different values of ΞΈ {\displaystyle {\boldsymbol {\theta }}} . Mathematically: ΞΈ β = arg β‘ max ΞΈ p ( ΞΈ | D ) {\displaystyle {\boldsymbol {\theta }}^{*}=\arg \max _{\bol... | Wikipedia - Pattern recognition - Problem statement | 277 | 753 | null |
Mathematically: ΞΈ β = arg β‘ max ΞΈ p ( ΞΈ | D ) {\displaystyle {\boldsymbol {\theta }}^{*}=\arg \max _{\boldsymbol {\theta }}p({\boldsymbol {\theta }}|\mathbf {D} )} where ΞΈ β {\displaystyle {\boldsymbol {\theta }}^{*}} is the value used for ΞΈ {\displaystyle {\boldsymbol {\theta }}} in the subsequent evaluation procedure... | Wikipedia - Pattern recognition - Problem statement | 442 | 1,147 | null |
{\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )=\left[\prod _{i=1}^{n}p(y_{i}|{\boldsymbol {x}}_{i},{\boldsymbol {\theta }})\right]p({\boldsymbol {\theta }}).} In the Bayesian approach to this problem, instead of choosing a single parameter vector ΞΈ β {\displaystyle {\boldsymbol {\theta }}^{*}} , the probability... | Wikipedia - Pattern recognition - Problem statement | 329 | 804 | null |
Section: Problem statement > Frequentist or Bayesian approach to pattern recognition. The first pattern classifier β the linear discriminant presented by Fisher β was developed in the frequentist tradition. The frequentist approach entails that the model parameters are considered unknown, but objective. The parameters ... | Wikipedia - Pattern recognition - Problem statement > Frequentist or Bayesian approach to pattern recognition | 326 | 1,398 | null |
Section: Uses. Within medical science, pattern recognition is the basis for computer-aided diagnosis (CAD) systems. CAD describes a procedure that supports the doctor's interpretations and findings. Other typical applications of pattern recognition techniques are automatic speech recognition, speaker identification, cl... | Wikipedia - Pattern recognition - Uses | 348 | 1,897 | null |
Section: Design. Perceiver is designed without modality-specific elements. For example, it does not have elements specialized to handle images, or text, or audio. Further it can handle multiple correlated input streams of heterogeneous types. It uses a small set of latent units that forms an attention bottleneck throug... | Wikipedia - Perceiver - Design | 168 | 855 | null |
Section: Design > Perceiver IO. Perceiver IO can flexibly query the model's latent space to produce outputs of arbitrary size and semantics. It achieves results on tasks with structured output spaces, such as natural language and visual understanding, StarCraft II, and multi-tasking. Perceiver IO matches a Transformer-... | Wikipedia - Perceiver - Design > Perceiver IO | 177 | 860 | null |
Article: PHerc. Paris. 4. PHerc. Paris. 4 is a carbonized scroll of papyrus, dating to the 1st century BC to the 1st century AD. Part of a corpus known as the Herculaneum papyri, it was buried by hot-ash in the Roman city of Herculaneum during the eruption of Mount Vesuvius in 79 AD. It was subsequently discovered in e... | Wikipedia - PHerc. Paris. 4 - Summary | 184 | 787 | null |
Section: Background and provenance. The Villa of the Papyri was buried during the eruption of Vesuvius in 79 AD, subjecting the scrolls to temperatures of 310β320 Β°C, compacting them and converting them to charcoal. The first scrolls were uncovered in 1752, with subsequent excavations uncovering more scrolls. There wer... | Wikipedia - PHerc. Paris. 4 - Background and provenance | 266 | 1,264 | null |
Section: Unscrolling and reading. The 20th century yielded progress in the readings of Herculaneum texts utilizing microscopes, digital photography and multispectral filters approaching the usage infrared spectroscopy to gain better clarity of the texts. In 2015, PHerc. Paris. 1 and PHerc. Paris. 4 were studied side by... | Wikipedia - PHerc. Paris. 4 - Unscrolling and reading | 314 | 1,343 | null |
The scans were completed at a resolution of 4-8 ΞΌm per voxel. The Vesuvius Challenge raised US$1 million, with an objective of clear readings of the scroll and the future aim of reading other carbonized, sealed fragments of the Herculaneum corpus, with a distant idea towards excavating more portions of the Villa of the... | Wikipedia - PHerc. Paris. 4 - Unscrolling and reading | 338 | 1,441 | null |
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