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+# LEARNING WORLD GRAPH DECOMPOSITIONS TO ACCELERATE REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Efficiently learning to solve tasks in complex environments is a key challenge for reinforcement learning (RL) agents. We propose to decompose a complex environment using a task-agnostic world graphs, an abstraction that accelerates learning by enabling agents to focus exploration on a subspace of the environment. The nodes of a world graph are important waypoint states and edges represent feasible traversals between them. Our framework has two learning phases: 1) identifying world graph nodes and edges by training a binary recurrent variational autoencoder (VAE) on trajectory data and 2) a hierarchical RL framework that leverages structural and connectivity knowledge from the learned world graph to bias exploration towards task-relevant waypoints and regions. We thoroughly evaluate our approach on a suite of challenging maze tasks and show that using world graphs significantly accelerates RL, achieving higher reward and faster learning.
+
+# 1 INTRODUCTION
+
+Many real-world applications, e.g., self-driving cars and in-home robotics, require an autonomous agent to execute different tasks within a single environment that features, e.g. high-dimensional state space, complex world dynamics or structured layouts. In these settings, model-free reinforcement learning (RL) agents often struggle to learn efficiently, requiring a large amount of experience collections to converge to optimal behaviors. Intuitively, an agent could learn more efficiently by focusing its exploration in task-relevant regions, if it has knowledge of the high-level structure of the environment.
+
+We propose a method to 1) learn and 2) use an environment decomposition in the form of a world graph, a task-agnostic abstraction. World graph nodes are waypoint states, a set of salient states that can summarize agent trajectories and provide meaningful starting points for efficient exploration (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018; Ghosh et al., 2018). The directed and weighted world graph edges characterize feasible traversals among the waypoints. To leverage the world graph, we model hierarchical RL (HRL) agents where a high-level policy chooses a waypoint state as a goal to guide exploration towards task-relevant regions, and a low-level policy strives to reach the chosen goals.
+
+Our framework consists of two phases. In the task-agnostic phase, we obtain world graphs by training a recurrent variational auto-encoder (VAE) (Chung et al., 2015; Gregor et al., 2015; Kingma & Welling, 2013) with binary latent variables (Nalisnick & Smyth, 2016) over trajectories collected using a random walk policy (Ha & Schmidhuber, 2018) and a curiosity-driven goal-conditioned policy (Ghosh et al., 2018; Nair et al., 2018). World graph nodes are states that are most frequently selected by the binary latent variables, while edges are inferred from empirical transition statistics between neighboring waypoints. In the task-specific phase, taking advantage of the learned world graph for structured exploration, we efficiently train an HRL model (Taylor & Stone, 2009).
+
+In summary, our main contributions are:
+
+• A task-agnostic unsupervised approach to learn world graphs, using a recurrent VAE with binary latent variables and a curiosity-driven goal-conditioned policy. • An HRL scheme for the task-specific phase that features multi-goal selection (Wide-thenNarrow) and navigation via world graph traversal.
+
+![](images/87e8c713c76547993153ec99d35d40aae11b5121b86240dad4d5ac65a6d03b12.jpg)  
+Figure 1: Top Left: overall pipeline of our 2-phase framework. Top Right (world graph discovery): a subgraph exemplifies traversal between waypoint states (in blue), see Section 3 for more details. Bottom (Hierarhical $R L$ ): an example rollout from our proposed HRL policy with Wide-then-Narrow Manager instructions and world graph traversals, solving a challenging Door-Key task, see Section 4 for more details.
+
+• Empirical evaluations on multiple tasks in complex 2D grid worlds to validate that our framework produces descriptive world graphs and significantly improves both sample efficiency and final performance on these tasks over baselines, especially thanks to transfer learning from the unsupervised phase and world graph traversal.
+
+# 2 RELATED WORK
+
+An understanding of the environment and its dynamics is essential for effective planning and control in model-based RL. For example, a robotics agent often locates or navigates by interpreting a map (Lowry et al., 2015; Thrun, 1998; Angeli et al., 2008). Our exploration strategy draws inspiration from active localization, where robots are actively guided to investigate unfamiliar regions (Fox et al., 1998; Li et al., 2016). Besides mapping, recent works (Azar et al., 2019; Ha & Schmidhuber, 2018; Guo et al., 2018) learn to represent the world with generative latent states (Tian & Gong, 2017; Haarnoja et al., 2018; Racanière et al., 2017). If the latent dynamics are also extrapolated, the latent states can assist planning (Mnih et al., 2016a; Hafner et al., 2018) or model-based RL (Gregor & Besse, 2018; Kaiser et al., 2019).
+
+While also aiming to model the world, we approach this as abstracting both the structure and dynamics of the environment in a graph representation, where nodes are states from the environment and edges encode actionable efficient transitions between nodes. Existing works (Metzen, 2013; Mannor et al., 2004; Eysenbach et al., 2019; Entezari et al., 2010) have shown benefits of such graph abstractions but typically select nodes only subject to a good coverage the observed state space. Instead, we identify a parsimonious subset of states that can summarize trajectories and provide more useful intermediate landmarks, i.e. waypoints, for navigating complex environments.
+
+Our method for estimating waypoint states can be viewed as performing automatic (sub)goal discovery. Subgoal and subpolicy learning are two major approaches to identify a set of temporally-extended actions, “skills”, that allow agents to efficiently learn to solve complex tasks. Subpolicy learning identifies policies useful to solve RL tasks, such as option-based methods (Daniel et al., 2016; Bacon et al., 2017) and subtask segmentations (Pertsch et al., 2019; Kipf et al., 2018). Subgoal learning, on the other hand, identifies “important states” to reach ( ¸Sim¸sek et al., 2005).
+
+Previous works consider various definitions of “important” states: frequently visited states during successful task completions (Digney, 1998; McGovern & Barto, 2001), states introducing the most novel information (Goyal et al., 2019), bottleneck states connecting densely-populated regions (Chen et al., 2007; ¸Sim¸sek et al., 2005), or environment-specific heuristics (Ecoffet et al., 2019). Our work draws intuition from unsupervised temporal segmentation (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018) and imitation learning (Abbeel & $\mathrm { N g }$ , 2004; Hussein et al., 2017). We define “important” states (waypoints) as the most critical states in recovering action sequences generated by some agent, which indicates that these states contain the richest information about the executed policy (Azar et al., 2019).
+
+![](images/0b8b33e4d41c1c614e19d9ac24f4cdb5614a9211348b71ee40716b3df882e83c.jpg)  
+Figure 2: Our recurrent latent model with differentiable binary latent units to identify waypoint states. A prior network (left) learns the state-conditioned prior in Beta distribution, $p _ { \psi } ( z _ { t } | s _ { t } ) { = } \mathrm { B e t a } ( \alpha _ { t } , \beta _ { t } )$ . An inference encoder learns an approximate posterior in HardKuma distribution inferred from the state-action sequence input, $q _ { \phi } ( z _ { t } | \mathbf { a } , z ) { = } \mathrm { H a r d } \hat { \mathrm { K } } \mathrm { u m a } ( \tilde { \alpha _ { t } } , \mathbf { \hat { 1 } } )$ . A generation network $p _ { \theta }$ reconstructs $\textbf { \em a }$ from $\{ s _ { t } | z _ { t } = 1 \}$ .
+
+# 3 LEARNING WORLD GRAPHS
+
+We propose a method for learning a world graph $\mathcal { G } _ { w }$ , a task-agnostic abstraction of an environment that captures its high-level structure and dynamics. In this work, the primary use of world graphs is to accelerate reinforcement learning of downstream tasks. The nodes of $\mathcal { G } _ { w }$ , denoted by a set of waypoints states $s _ { p } \in \mathcal { V } _ { p }$ , are generically “important” for accomplishing tasks within the environment, and therefore useful as starting points for exploration. Our method identifies such waypoint states from interactions with the environment. In addition, we embed feasible transitions between nearby waypoint states as the edges of $\mathcal { G } _ { w }$ .
+
+In this work, we define important states in the context of learning $\mathcal { G } _ { w }$ (see Section 2 for alternative definitions). That is, we wish to discover a small set of states that, when used as world graph nodes, concisely summarize the structure and dynamics of the environment. Below, we describe 1) how to collect state-action trajectories and an unsupervised learning objective to identify world graph nodes, and 2) how the graph’s edges (i.e., how to transition between nodes) are formed from trajectories.
+
+# 3.1 WAYPOINT STATE IDENTIFICATION
+
+The structure and dynamics of an environment are implicit in the state-action trajectories observed during exploration. To identify world graph nodes from such data, we train a recurrent variational autoencoder (VAE) that, given a sequence of state-action pairs, identifies a subset of the states in the sequence from which the full action sequence can be reconstructed (Figure 2). In particular, the VAE infers binary latent variables that controls whether each state in the sequence is used by the generative decoder, i.e., whether a state is “important” or not.
+
+Binary Latent VAE The VAE consists of an inference, a generative and a prior network. These are structured as follows: the input to the inference network $q _ { \phi }$ is a trajectory of state-action pairs observed from the environment ${ \tau } = \{ ( s _ { t } , a _ { t } ) \} _ { t = 0 } ^ { T }$ , with ${ \pmb s } = \{ { \boldsymbol s } _ { t } \} _ { t = 0 } ^ { T }$ and $\pmb { a } { = } \{ a _ { t } \} _ { t = 0 } ^ { T }$ denoting the state and action sequences respectively. The output of the inference network is the approximated posterior over a sequence ${ z } = \{ z _ { t } \} _ { t = 0 } ^ { \bar { T } }$ of binary latent variables, denoted as $\varphi _ { \phi } ( \boldsymbol { z } | \boldsymbol { a } , \boldsymbol { s } )$ . The generative network $p _ { \theta }$ computes a distribution over the full action sequence $\textbf { \em a }$ using the masked state sequence, where $s _ { t }$ is masked if $z _ { t } { = } 0$ (we fix $z _ { 0 } { = } z _ { T } { = } 1$ during training), denoted as $p _ { \theta } ( { \pmb a } | { \pmb s } , z )$ .
+
+Finally, a state-conditioned $p _ { \psi } ( z _ { t } | s _ { t } )$ given by the prior network $p _ { \psi }$ for each $s _ { t }$ encodes the empirical average probability that state $s _ { t }$ is activated for reconstruction. This choice encourages inference to select within a consistent subset of states for use in action reconstruction. In particular, the waypoint states $\nu _ { p }$ are chosen as the states with the largest prior means and during training, once every few iterations, $\nu _ { p }$ is updated based on the current prior network.
+
+<table><tr><td>Algorithm 1: Identifying waypoint states Vp and learning a goal-conditioned policy g Result: Waypoint states Vp and a goal-conditioned policy π g</td></tr><tr><td>Initialize network parameters for the recurrent variational inference model V Initialize network parameters for the goal-conditioned policy g Initialize Vp with the initial position of the agent,i.e.Vp = {so = (1,1)}</td></tr><tr><td>while VAE reconstruction error has not converged do</td></tr><tr><td>forn←1toNdo Sample random waypoint sp ∈ Vp</td></tr><tr><td>Navigate agent to sp and perform T-step rollout using a randow walk policy: T𝑛 ←{(s0= Sp,ao),.,(sT,ar)}</td></tr><tr><td>gn←ST Navigate agent to Sp and perform T-step rollout using Tg with goal gn:</td></tr><tr><td>Tπ ←{(s= Sp,ao),.,(sT,ar)}at~πg(-st,9n) Re-label πg rewards with action reconstruction error as curiosity bonus:</td></tr><tr><td>rπ←{1st+1=n-λ·p(at|s,z)}=0</td></tr><tr><td>end</td></tr><tr><td>Perform policy gradient update of πg using T&quot; and rπ</td></tr><tr><td>Update V using T and T</td></tr><tr><td>Update Vp as set of states with largest prior mean αs αs+βs</td></tr></table>
+
+Objective Formally, we optimize the VAE using the following evidence lower bound (ELBO):
+
+$$
+\mathrm { E L B O } = \mathbb { E } _ { q _ { \phi } ( z | a , s ) } \left[ \log p _ { \theta } ( a | s , z ) \right] - D _ { \mathrm { K L } } \left( q _ { \phi } ( z | a , s ) | p _ { \psi } ( z | s ) \right) .
+$$
+
+To ensure differentiablity, we apply a continuous relaxation over the discrete $z _ { t }$ . We use the Beta distribution $p _ { \psi } ( z _ { t } ) = \mathrm { B e t a } ( \alpha _ { t } , \beta _ { t } )$ for the prior and the Hard Kumaraswamy distribution $q _ { \psi } ( z _ { t } | { a } , { z } ) = \mathrm { H a r d K u m a } ( \tilde { \alpha } _ { t } , \tilde { \beta } _ { t } )$ for the approximate posterior, which resembles the Beta distribution but is outside the exponential family (Bastings et al., 2019). This choice allows us to sample 0s and 1s without sacrificing differentiability, accomplished via the stretch-and-rectify procedure (Bastings et al., 2019; Louizos et al., 2017) and the reparametrization trick (Kingma & Welling, 2013). Lastly, to prevent the trivial solution of using all states for reconstruction, we use a secondary objective $\mathcal { L } _ { 0 }$ to regularize the $L _ { 0 }$ norm of $_ z$ at a targeted value $\mu _ { 0 }$ (Louizos et al., 2017; Bastings et al., 2019), the desired number of selected states out of $T$ steps, e.g. for when $T = 2 5$ , we set $\mu _ { 0 } = 5$ , meaning ideally 5 out of 25 states are activated for action reconstruction. Another term $\mathcal { L } _ { T }$ to encourage temporal separation between selected states by targeting the number of $0 / 1$ switches among $_ z$ at $2 \mu _ { 0 }$ :
+
+$$
+\mathcal { L } _ { 0 } = \Big | \Big | \mathbb { E } _ { q _ { \phi } ( z | s , a ) } [ \| z \| _ { 0 } ] - \mu _ { 0 } \Big | \Big | ^ { 2 } , \quad \mathcal { L } _ { T } = \Bigg | \Bigg | \mathbb { E } _ { q _ { \phi } ( z | s , a ) } \left[ \sum _ { t = 0 } ^ { T } \mathbb { 1 } [ z _ { t } \neq z _ { t + 1 } ] \right] - 2 \mu _ { 0 } \Bigg | \Bigg | ^ { 2 } .
+$$
+
+See Appendix A for details on training the VAE with binary $z _ { t }$ , including integration of the Hard Kumaraswamy distribution and how to regularize the statistics of $_ z$ .
+
+# 3.2 EXPLORATION FOR WORLD GRAPH DISCOVERY
+
+Naturally, the latent structure learned by the VAE depends on the trajectories used to train it. Hence, collecting a rich set of trajectories is crucial. Here, we propose a strategy to bootstrap a useful set of trajectories by alternately exploring the environment based on the current iteration’s $\nu _ { p }$ and updating the VAE and $\nu _ { p }$ , repeating this cycle until the action reconstruction accuracy plateaus (Algorithm 1).
+
+During exploration, we use action replay to navigate the agent to a state drawn from the current iteration’s $\nu _ { p }$ . Although resetting via action replay assumes our underneath environment to be deterministic, in cases where this resetting strategy is infeasible, it may be modified so long as to allow the exploration starting points to expand as the agent discovers more of its environment. For each such starting point, we collect two rollouts. In the first rollout, we perform a random walk to explore the nearby region. In the second rollout, we perform actions using a goal-conditioned policy $\pi _ { g }$ (GCP), setting the final state reached by the random walk as the goal. Both rollouts are used for trianing the VAE and the latter is also used for training $\pi _ { g }$ .
+
+![](images/0c4516ced121df0c03431d831806e1817ee32a043d24758d20a0950c4f69de1d.jpg)  
+Figure 3: Left: a standard Feudal Network. Right: using Wide-then-Narrow goals. The Manager first outputs a waypoint state as the wide goal $g ^ { w }$ , then attends to a closer-up area around $g ^ { w }$ to narrow down the final goal $g ^ { n }$ .
+
+GCP provides a venue to integrate intrinsic motivation, such as curiosity (Burda et al., 2018; Achiam & Sastry, 2017; Pathak et al., 2017; Azar et al., 2019) to generate more diverse rollouts. Specifically, we use the action reconstruction error of the VAE as an intrinsic reward signal when training $\pi _ { g }$ . This choice of curioisty also prevents the VAE from collapsing to the simple behaviors of a vanilla $\pi _ { g }$ .
+
+# 3.3 EDGE FORMATION
+
+The final stage is to construct the edges of $\mathcal { G } _ { w }$ , which should ideally capture the environment dynamics, i.e. how to transition between waypoint states. Once VAE training is complete and $\nu _ { p }$ is fixed, we collect random walk rollouts from each of the waypoints $s _ { p } \in \mathcal { V } _ { p }$ to estimate the underlying adjacency matrix (Biggs, 1993). More precisely, we claim a directed edge $s _ { p } \to s _ { q }$ if there exists a random walk trajectory from $s _ { p }$ to $s _ { q }$ that does not intersect a third waypoint. We also consider paths taken by $\pi _ { g }$ (starting at $s _ { p }$ and setting $s _ { q }$ as the goal) and keep the shortest observed path from $s _ { p }$ to $s _ { q }$ as a world graph edge transition. We use the action sequence length of the edge transition between adjacent waypoints as the weight of the edge. As shown experimentally, a key benefit of our approach is the ability to plan over $\mathcal { G } _ { w }$ . To navigate from one waypoint to another, we can use dynamic programming (Sutton, 1998; Feng et al., 2004) to output the optimal traversal of the graph.
+
+# 4 ACCELERATING REINFORCEMENT LEARNING WITH WORLD GRAPHS
+
+World graphs present a high-level, task-agnostic abstraction of the environment through waypoints and feasible transition routes between them. A key example of world graph applications for taskspecific RL is structured exploration: instead of exploring the entire environment, RL agents can use world graphs to quickly identify task-relevant regions and bias low-level exploration to these regions. Our framework to leverage world graphs for structured exploration consists of two parts:
+
+1. Hierarchical RL wherein the high-level policy selects subgoals from $\nu _ { p }$ .   
+2. Traversals using world graph edges.
+
+# 4.1 HIERARCHICAL RL OVER WORLD GRAPHS
+
+Formally, an RL agent learning to solve a task is formulated as a Markov Decision Process: at time $t$ , the agent is in a state $s _ { t }$ , executes an action $a _ { t }$ via a policy $\pi ( a _ { t } | s _ { t } )$ and receives a rewards $r _ { t }$ . The agent’s goal is to maximize its cumulative expected return $\begin{array} { r } { R = \mathbb { E } _ { ( s _ { t } , a _ { t } ) \sim \pi , p , p _ { 0 } } \left[ \sum _ { t \geq 0 } \gamma ^ { t } r _ { t } \right] } \end{array}$ , where $p ( s _ { t + 1 } | s _ { t } , a _ { t } ) , p _ { 0 } ( s _ { 0 } )$ are the transition and initial state distributions.
+
+To incorporate world graphs with RL, we use a hierarchical approach based on the Feudal Network (FN) (Dayan $\&$ Hinton, 1993; Vezhnevets et al., 2017), depicted in Figure 3. A standard FN
+
+# Task Description
+
+Task MultiGoal
+
+# Environment Characteristics
+
+Balls are located randomly, dense reward.
+
+# MultiGoal-Sparse
+
+Collect randomly spawned balls, each ball gives $+ 1$ reward. To end an episode, the agent has to exit at a designated point. Agents receive a single reward $r \leq 1$ proportional to the number of balls collected upon exiting.
+
+MultiGoalStochastic Door-Key
+
+Balls are located randomly, sparse reward.
+
+Spawn lava blocks at random locations each time step that immediately terminates the episode if stepped on. Agent has to pick up a key to open a door (reward $+ 1 \AA$ and reach the exit point on the other side (reward $+ 1$ ).
+
+Stochastic environment. Multiple objects: lava and balls are randomly located, dense reward.   
+Walls, door and key are located randomly. Agents have additional actions: pick and toggle.
+
+Table 1: An overview of tasks used to evaluate the benefit of using world graphs. Visualizations can be found in Appendix D.
+
+decomposes the policy of the agent into two separate policies that receive distinct streams of reward: a high-level policy (“Manager”) learns to propose subgoals; a low-level policy (“Worker”) receives subgoals from the Manager as inputs and is rewarded for taking actions in the environment that reach the subgoals. The Manager receives the environment reward defined by the task and therefore must learn to emit subgoals that lead to task completion. The Manager and Worker do not share weights and operate at different temporal resolutions: the Manager only outputs a new subgoal if either the Worker reaches the chosen one or a subgoal horizon $c$ is exceeded.
+
+For all our experiments, policies are trained using advantage actor-critic (A2C), an on-policy RL algorithm (Wu & Tian, 2016; Pane et al., 2016; Mnih et al., 2016b). To ease optimization, the feature extraction layers of the Manager and Worker that encode $s _ { t }$ are initialized with the corresponding layers from $\pi _ { g }$ , the GCP learned during world graph discovery phase. More details are in Appendix B.
+
+# 4.2 WIDE-THEN-NARROW GOALS AND WORLD GRAPHS
+
+To incorporate the world graph, we introduce a Manager policy that factorizes subgoal selection as follows: a wide policy $\pi ^ { w } ( g _ { t } ^ { w } | s _ { t } )$ selects a waypoint state as the wide goal $g ^ { w } \in \mathcal { V } _ { p }$ , and a narrow policy $\pi ^ { n } ( g _ { t } ^ { n } | s _ { t } , g _ { t } ^ { w } )$ selects a state within a local neighborhood of $g _ { t } ^ { w }$ , i.e. its $\epsilon$ -net (Mahadevan $\&$ Maggioni, 2007), as the narrow goal $g ^ { n } \in \{ s : \mathcal { D } ( s , \bar { g } _ { t } ^ { w } ) \leq \epsilon \}$ . The Worker policy $\pi ^ { \mathrm { w o r k e r } } ( a _ { t } | s _ { t } , g _ { t } ^ { n } , g _ { t } ^ { \bar { w } } )$ chooses the action taken by the agent given the current state and the wide and narrow goals from the Manager. A visual illustration is in Figure 4 and training details in Appendix C.2.
+
+# 4.3 WORLD GRAPH TRAVERSAL
+
+The wide-then-narrow subgoal format simplifies the search space for the Manager policy. Using waypoints as wide goals also makes it possible to leverage the edges of the world graph for planning and executing the planned traversals. This process breaks down as follows:
+
+1. When to Traverse: When the agent encounters a waypoint state $s _ { t } \in \mathcal V _ { p }$ , a “traversal” is initiated if $s _ { t }$ has a feasible connection in $\mathcal { G } _ { w }$ to the active wide goal $g _ { t } ^ { w }$ . 2. Planning: Upon triggering a traversal, the optimal traversal route from the initiating state to $g _ { t } ^ { w }$ is estimated from the $\mathcal { G } _ { w }$ edge weights using classic dynamic programming planning (Sutton, 1998; Feng et al., 2004). This yields a sequence of intermediate waypoint states. 3. Execution: Execution of graph traversals depends on the nature of the environment. If deterministic, the agent simply follows the action sequences given by the edges of the traversal. Otherwise, the agent uses the pretrained $\mathbf { G C P } \pi _ { g }$ to sequentially reach each of the intermediate waypoint states along the traversal (we fine-tune $\pi _ { g }$ in parallel where applicable). If the agent fails to reach the next waypoint state within a certain time limit, it stops its current pursuit and a new $( g ^ { w } , g ^ { n } )$ pair is received from the Manager.
+
+World graph traversal allows the Manager to assign task-relevant wide goals $g ^ { w }$ that can be far away from the agent yet still reachable, which consequentially accelerates learning by focusing exploration around the task-relevant region near $g ^ { w }$ .
+
+# 5 EXPERIMENTAL VALIDATION
+
+We now assess each component of our framework on a set of challenging 2D grid worlds. Our ablation studies demonstrate the following benefits of our framework:
+
+Table 2: On a variety of tasks and environment setups, we evaluate RL models trained with GCP $\pi _ { g }$ initialization, with $\mathcal { G } _ { w }$ world graph travresal, and with both. All models on the right are equipped with WN. Left are baselines for additional comparison. We report final rewards for MultiGoal tasks and success rates for Door-Key are reported. If no result reported, the agent failed to solve the task.   
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Size</td><td rowspan="2">A2C</td><td rowspan="2">FN+πg init</td><td colspan="3">Ours</td></tr><tr><td>+πg-init</td><td>+ 9w-traversal</td><td>+πg-init+Gw-traversal</td></tr><tr><td rowspan="3">MultiGoal</td><td>Small</td><td>2.04±0.05</td><td>2.93±0.74</td><td>5.25±0.13</td><td>3.92±0.22</td><td>5.05±0.03</td></tr><tr><td>Medium</td><td>=</td><td>=</td><td>5.15±0.11</td><td>2.56±0.09</td><td>3.00±0.90</td></tr><tr><td>Larger</td><td>=</td><td>=</td><td>-</td><td>2.18±0.12</td><td>2.72±0.59</td></tr><tr><td rowspan="3">MultiGoal-Sparse</td><td>Small</td><td>=</td><td>=</td><td>0.39±0.09</td><td>0.24±0.04</td><td>0.42±0.07</td></tr><tr><td>Medium</td><td></td><td></td><td></td><td>0.20±0.04</td><td>0.25±0.03</td></tr><tr><td>Larger</td><td></td><td></td><td></td><td>0.16±0.22</td><td>0.26±0.11</td></tr><tr><td rowspan="3">MultiGoal-Stochastic</td><td>Small</td><td>1.38±1.20</td><td>1.93±0.16</td><td>3.06±0.31</td><td></td><td>2.92±0.45</td></tr><tr><td>Medium</td><td></td><td>=</td><td>2.99±0.12</td><td>2.42±0.24</td><td>2.64±0.14</td></tr><tr><td>Larger</td><td>=</td><td></td><td>=</td><td>=</td><td>0.60±0.12</td></tr><tr><td rowspan="3">Door-Key</td><td>Small</td><td></td><td>=</td><td>0.99±0.00</td><td>0.37±0.15</td><td>0.92±0.02</td></tr><tr><td>Medium</td><td></td><td></td><td>0.56±0.02</td><td></td><td>0.76±0.06</td></tr><tr><td>Larger</td><td></td><td></td><td>=</td><td></td><td>0.26±0.19</td></tr></table>
+
+Table 3: Comparing learned $\nu _ { p }$ versus random $\mathcal { V } _ { \mathrm { r a n d } }$ as wide subgoals on large mazes, all trained with $\pi _ { g }$ initialization and graph traversal. $\nu _ { p }$ generally is superior in terms of performance and consistency. We report final rewards for MultiGoal tasks and success rates for Door-Key are reported.   
+
+<table><tr><td>Waypoint type</td><td>MultiGoal</td><td>MultiGoal-Sparse</td><td>MultiGoal-Stochastic</td><td>Door-Key</td></tr><tr><td>Learned</td><td>2.72±0.59</td><td>0.26±0.11</td><td>0.60±0.12</td><td>0.26±0.19</td></tr><tr><td>Random</td><td>2.30±0.49</td><td>0.19±0.11</td><td>0.41±0.25</td><td>0.27±0.40</td></tr></table>
+
+• It improves sample efficiency and performance over the baseline HRL model. • It benefits tasks varying in envirionment scale, task type, reward structure, and stochasticity. • The identified waypoints provide superior world representations for solving downstream tasks, as compared to graphs using randomly selected states as nodes.
+
+Implementation details, snippets of the tasks and mazes are in Appendix C-D.
+
+# 5.1 ABLATION STUDIES ON 2D GRID WORLDS
+
+For our ablation studies, we construct 2D grid worlds of increasing sizes (small, medium and large) along with challenging tasks with different reward structures, levels of stochasticity and logic (summarized in Table 1). In all tasks, every action taken by the agent receives a negative reward penalty. We follow a rigorous evaluation protocol (Wu et al., 2017; Ostrovski et al., 2017; Henderson et al., 2018): each experiment is repeated with 3 training seeds. 10 additional validation seeds are used to pick the model with the best reward performance. This model is then tested on 100 testing seeds. We report mean reward and standard deviation.
+
+We ablate each of the following components in our framework and compare against non-hierarchical (A2C) and hierarchical baselines (FN):
+
+1. initializing the feature extraction layers of the Manager and Worker from $\pi _ { g }$ ,   
+2. applying Wide-then-Narrow Manager (WN) goal instruction, and   
+3. allowing the Worker to traverse along $\mathcal { G } _ { w }$ .
+
+Results are shown in Table 2. In sum, each component improves performance over the baselines.
+
+Wide and narrow goals Using two goal types is a highly effective way to structure the Manager instructions and enables the Worker to differentiate the transition and local task-solving phases. We note that for small MultiGoal, agents do not benefit much from $\mathcal { G } _ { w }$ traversal: it can rely solely on the guidance from WN goals to master both phases. However with increasing maze size, the Worker struggles to master traversals on its own and thus fails solving the tasks.
+
+World Graph Traversal As conjectured in Section 4.3, the performance gain of our framework can be explained by the larger range and more targeted exploration strategy. In addition, the Worker does not have to learn long distance transitions with the aid of $\mathcal { G } _ { w }$ traversals. Figure 4 confirms that $\mathcal { G } _ { w }$ traversal speeds up convergence and its effect becomes more evident with larger mazes. Note that the graph learning stage only need 2.4K iterations to converge. Even when taking these additional environment interactions into account, $\mathcal { G } _ { w }$ traversal still exhibits superior sample efficiency, not to mention that the graph is shared among all tasks. Moreover, solving Door-Key involves a complex combination of sub-tasks: find and pick up the key, reach and open the door and finally exit. With limited reward feedback, this is particularly difficult to learn. The ability to traverse along $\mathcal { G } _ { w }$ enables longer-horizon planning on top of the waypoints, thanks to which the agents boost the success rate on medium Door-Key from $0 . 5 6 { \pm } 0 . 0 2$ to $0 . 7 5 { \pm } 0 . 0 6 $ .
+
+![](images/eeccd173a66cc7f083254caca6b3d6fe078ddac17830e5c24ef8e6d8706c4b0b.jpg)  
+Figure 4: Validation performance during training (mean and standard-deviation of reward, 3 seeds) for MultiGoal. Left: Comparing $\nu _ { p }$ and $\mathcal { V } _ { \mathrm { r a n d } }$ , with or without traversal, all models use WN and $\pi _ { g }$ initialization. We see that 1) traversal speeds up convergence, 2) $\mathcal { V } _ { \mathrm { r a n d } }$ gives higher variance and slightly worse performance than $\nu _ { p }$ . Right: comparing with or without $\pi _ { g }$ initialization on $\nu _ { p }$ , all models use WN. We see that initializing the task-specific phase with the task-agnostic goal-conditioned policy significantly boosts learning.
+
+Benefits of Learned Waypoints To highlight the benefit of establishing the waypoints learned by the VAE as nodes for $\mathcal { G } _ { w }$ , we compare against results using a $\mathcal { G } _ { w }$ constructed around randomly selected states $( \nu _ { \mathrm { r a n d } } )$ . The edges of the random-node graph are formed in the same way as described in Section 3.3 and its feature extractor is also initialized from $\pi _ { g }$ . Although granting knowledge acquired during the unsupervised phase to $\mathcal { V } _ { \mathrm { r a n d } }$ is unfair to $\nu _ { p }$ , deploying both initialization and traversal while only varying $\mathcal { V } _ { \mathrm { r a n d } }$ and $\nu _ { p }$ isolates the effect from the nodes to the best extent. The comparative results (in Table 3, learning curves for MultiGoal in Figure 4) suggest $\nu _ { p }$ generally outperforms $\mathcal { V } _ { \mathrm { r a n d } }$ . Door-Key is the only task in which the two matches. However, $\mathcal { V } _ { \mathrm { r a n d } }$ exhibits a large variance, implying that certain sets of random states can be suitable for this task, but using learned waypoints gives strong performance more consistently.
+
+Initialization with GCP Initializing the weights of the Worker and Manager feature extractors from $\pi _ { g }$ (learned during the task-agnostic phase) consistently benefits learning.In fact, we observe that models starting from scratch fail on almost all tasks within the maximal number of training iterations, unless coupled with $\mathcal { G } _ { w }$ traversal, which is still inferior to using $\pi _ { g }$ -initialization. Particularly, for the small MultiGoal-Stochastic environment, there is a high chance that a lava square blocks traversal; therefore, without the environment knowledge from $\pi _ { g }$ transferred by weight initialization, the interference created by the episode-terminating lava prevents the agent from learning the task.
+
+# 6 CONCLUSION
+
+We have shown that world graphs are powerful environment abstractions, which, in particular, are capable of accelerating reinforcement learning. Future works may extend their applications to more challenging RL setups, such as real-world multi-task learning and navigation. It is also interesting to generalize the proposed framework to learn dynamic world graphs for evolving environments, and applying world graphs to multi-agent problems, where agents become part of the world graphs of other agents.
+
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+# A RECURRENT VAE WITH DIFFERENTIABLE BINARY LATENT VARIABLES
+
+As illustrated in the main text, the main objective for the recurrent VAE is the following evidence lower bound with derivation:
+
+$$
+\begin{array} { r l } & { \log p ( a | s ) = \log \int p ( a | s , z ) d z } \\ & { \qquad = \log \int p ( a | s , z ) p ( z | s ) \frac { q ( z | a , s ) } { q ( z | a , s ) } d z } \\ & { \qquad = \log \int p ( a | s , z ) \frac { p ( z | s ) } { q ( z | a , s ) } q ( z | a , s ) d z } \\ & { \qquad \geq \mathbb { E } _ { q ( z | a , s ) } [ \log p ( a | s , z ) - \log \frac { q ( z | a , s ) } { p ( z | s ) } ] } \\ & { \qquad = \mathbb { E } _ { q ( z | a , s ) } [ \log p ( a | s , z ) ] - D _ { \mathrm { K L } } ( q ( z | a , s ) | | p ( z | s ) ) } \end{array}
+$$
+
+The inference network $q _ { \psi }$ takes in the trajectories of state-action pairs $\tau$ and at each time step approximates the posterior of the corresponding latent variable $z _ { t }$ . The prior network $p _ { \psi }$ takes the state $s _ { t }$ at each time step and outputs the state-conditioned prior $p _ { \psi } ( s _ { t } )$ . We choose Beta as the prior distribution and the Hard Kuma as the approximated posterior to relax the discrete latent variables to continuous surrogates.
+
+The Kuma distribution $\mathrm { K u m a } ( \alpha , \beta )$ highly resembles the Beta Distribution in shape but does not come from the exponential family. Similar to Beta, the Kuma distribution also ranges from bimodal (when $\alpha \approx \beta )$ to unimodal $( \alpha / \beta \to 0$ or $\alpha / \beta \to \infty )$ ). Also, when $\alpha = 1$ or $\beta = 1$ , $\operatorname { K u m a } ( \alpha , \beta ) = \operatorname { B e t a } ( \alpha , \beta )$ . We observe empirically better performance when we fix $\beta = 1$ for the Kuma approximated posterior. One major advantage of the Kuma distribution is its simple Cumulative Distribution Function (CDF):
+
+$$
+F _ { \mathrm { K u m a } } ( x , \alpha , \beta ) = ( 1 - ( 1 - x ^ { \alpha } ) ) ^ { \beta } .
+$$
+
+It is therefore amendable to the reparametrization trick (Kingma & Welling, 2013; Rezende et al., 2014; Maddison et al., 2016) by sampling from uniform distribution $u \sim \mathcal { U } ( 0 , 1 )$ :
+
+$$
+z = F _ { \mathrm { K u m a } } ^ { - 1 } ( u ; \alpha , \beta ) \sim \mathrm { K u m a } ( \alpha , \beta ) .
+$$
+
+Lastly, the KL-divergence between the Kuma and Beta distributions can be approximated in closed form (Nalisnick & Smyth, 2016):
+
+$$
+\begin{array} { l } { \displaystyle { { \cal D } _ { \mathrm { K L } } ( \mathrm { K u m a } ( a , b ) | \mathrm { B e t a } ( \alpha , \beta ) ) = \frac { a - \alpha } { a } \left( - \gamma - \Psi ( b ) - \frac { 1 } { b } \right) } } \\ { \displaystyle { \phantom { \frac { b - a } { b - a } ( \mathrm { K u m a } ( a , b ) + \log \mathrm { B e t a } ( \alpha , \beta ) - \frac { b - 1 } { b } + ( \beta - 1 ) b \sum _ { m = 1 } ^ { \infty } \frac { 1 } { m + a b } \mathrm { B e t a } \left( \frac { m } { a } , b \right) , } } } \end{array}
+$$
+
+where $\Psi$ is the Digamma function, $\gamma$ the Euler constant, and the approximation uses the first few terms of the Taylor series expansion. We take the first 5 terms here.
+
+Next, we make the Kuma distribution “hard” by following the steps in Bastings et al. (2019). First stretch the support to $( r = 0 - \epsilon _ { 1 } , l = 1 + \epsilon _ { 2 }$ ), $\epsilon _ { 1 } , \epsilon _ { 2 } > 0$ , and the resulting CDF distribution takes the form:
+
+$$
+F _ { S } ( z ) = F _ { \mathrm { K u m a } } \left( { \frac { z - l } { r - l } } ; \alpha , \beta \right) .
+$$
+
+Then, the non-eligible probabilities for 0’s and 1’s are attained by rectifying all samples below 0 to 0 and above 1 to 1, and other value as it is, that is
+
+$$
+P ( z = 0 ) = F _ { \mathrm { K u m a } } \left( \frac { - l } { r - l } ; \alpha , \beta \right) , \quad P ( z = 1 ) = 1 - F _ { \mathrm { K u m a } } \left( \frac { 1 - l } { r - l } ; \alpha , \beta \right) .
+$$
+
+Lastly, we impose two additional regularization terms ${ \mathcal { L } } _ { \prime }$ and $\mathcal { L } _ { T }$ on the approximated posteriors. As described in the main text, ${ \mathcal { L } } _ { \prime }$ prevents the model from selecting all states to reconstruct $\{ a _ { t } \} _ { 0 } ^ { T - 1 }$ by restraining the expected $L _ { 0 }$ norm of $z = \left( z _ { 1 } \cdot \cdot \cdot z _ { T - 1 } \right)$ to approximately be at a targeted value $\mu _ { 0 }$ (Louizos et al., 2017; Bastings et al., 2019). In other words, this objective adds the constraint that there should be $\mu _ { 0 }$ of activated $z _ { t } = 1$ given a sequence of length $T$ . The other term $\mathcal { L } _ { T }$ encourages temporally isolated activation of $z _ { t }$ , meaning the number of transition between 0 and 1 among $z _ { t }$ ’s should roughly be $2 \mu _ { 0 }$ . Note that both expectations in Equation 2 have closed forms for HardKuma.
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { 0 } = \left\| \mathbb { E } _ { q ( \boldsymbol { z } | \boldsymbol { s } , \boldsymbol { a } ) } \left[ \left\| \boldsymbol { z } \right\| _ { 0 } \right] - \mu _ { 0 } \right\| ^ { 2 } , \mathrm { w h e r e } } \\ { \displaystyle \mathbb { E } _ { q ( \boldsymbol { z } | \boldsymbol { s } , \boldsymbol { a } ) } \left[ \left\| \boldsymbol { z } \right\| _ { 0 } \right] = \sum _ { t = 1 } ^ { T } \mathbb { E } _ { q ( \boldsymbol { z } _ { t } | \boldsymbol { s } , \boldsymbol { a } ) } \left[ \mathbb { 1 } _ { \boldsymbol { z } _ { t } \neq 0 } \right] } \\ { \displaystyle \qquad = \sum _ { t = 1 } ^ { T } 1 - p \left( \boldsymbol { z } _ { t } = 0 \right) = \sum _ { t = 1 } ^ { T } 1 - F _ { \mathrm { K u m a } } \left( \frac { - l } { r - l } ; \alpha _ { t } , \beta _ { t } \right) , } \end{array}
+$$
+
+$$
+\mathbb { E } _ { q ( z | s , a ) } [ \sum _ { t = 1 } ^ { T - 1 } \mathbb { 1 } _ { z _ { t } \neq z _ { t + 1 } } ] = \sum _ { t = 1 } ^ { T - 1 } p \left( z _ { t } = 0 \right) \left( 1 - p \left( z _ { t + 1 } = 0 \right) \right) + \left( 1 - p \left( z _ { t } = 0 \right) \right) p \left( z _ { t + 1 } = 0 \right) .
+$$
+
+Lagrangian Relaxation. The overall optimization objective consists of action sequence reconstruction, KL-divergence between the posterior and prior, $\mathcal { L } _ { 0 }$ and $\mathcal { L } _ { T }$ (Equation 12). We tune the objective weights $\lambda _ { i }$ using Lagrangian relaxation (Higgins et al., 2017; Bastings et al., 2019; Bertsekas, 1999), treating $\lambda _ { i }$ ’s as learnable parameters and performing alternative optimization between $\lambda _ { i }$ ’s and the model parameters. We observe that as long as their initialization is within a reasonable range, $\lambda _ { i }$ ’s converge to a local optimum:
+
+$$
+\operatorname* { m a x } _ { \{ \lambda _ { 1 } , 2 , 3 \} } \operatorname* { m i n } _ { \substack { \left\{ \theta , \phi , \psi \right\} } } - \mathbb { E } _ { q _ { \psi } ( z | a , s ) } \left[ \log p _ { \theta } ( a | s , z ) \right] + \lambda _ { 1 } D _ { \mathrm { K L } } \left( q _ { \phi } ( z | a , s ) | p _ { \psi } ( z | s ) \right) + \lambda _ { 2 } \mathcal { L } _ { 0 } + \lambda _ { 3 } \mathcal { L } _ { T } .
+$$
+
+We observe this approach to produce efficient and stable mini-batch training.
+
+# B GOAL-CONDITIONED POLICY INITIALIZATION FOR HRL
+
+Optimizing composite neural networks like HRL (Co-Reyes et al., 2018) is sensitive to weight initialization (Mishkin & Matas, 2015; Le et al., 2015), due to its complexity and lack of clear supervision at various levels. Therefore, taking inspiration from prevailing pre-training procedures in computer vision (Russakovsky et al., 2015; Donahue et al., 2014) and NLP (Devlin et al., 2018; Radford et al., 2019), we take advantage of the weights learned by $\pi _ { g }$ during world graph discovery when initializing the Worker and Manager policies for downstream HRL, as $\pi _ { g }$ has already implicitly embodied much environment dynamics information.
+
+More specifically, we extract the weights of the feature extractor, i.e. the state encoder, and use them as the initial weights for the state encoders of the HRL policies. Our empirical results demonstrate that such weight initialization consistently improves performance and validates the value of skill/knowledge transfer from GCP (Taylor & Stone, 2009; Barreto et al., 2017).
+
+# C ADDITIONAL IMPLEMENTATION DETAILS
+
+Model code folder including all architecture details is shared in comment.
+
+# C.1 HYPERPARAMETERS FOR VAE TRAINING
+
+Our models are optimized with Adam (Kingma & Ba, 2014) using mini-batches of size 128, thus spawning 128 asynchronous agents to explore. We use an initial learning rate of 0.0001, with $\bar { \epsilon } = 0 . 0 0 \bar { 1 } , \beta _ { 1 } = \bar { 0 } . 9 , \beta _ { 2 } = 0 . 9 9 \bar { 9 }$ ; gradients are clipped to 40 for inference and generation nets. For HardKuma, we set $l = - 0 . 1$ and $r = 1 . 1$ . The maximum sequence length for BiLSTM is 25. The total number of training iterations is 3600 and model usually converges around 2400 iterations. We train the prior, inference, and generation networks end-to-end.
+
+We initialize $\lambda _ { i }$ ’s (see Lagrangian Relaxation) to be $\lambda _ { 1 } = 0 . 0 1$ (KL-divergence), $\begin{array} { r } { , \lambda _ { 2 } = 0 . 0 6 ( \mathcal { L } _ { 0 } ) . } \end{array}$ $\lambda _ { 3 } = 0 . 0 2 ( \mathcal { L } _ { T } )$ . After each update of the latent model, we update $\lambda _ { i }$ ’s, whose initial learning rate is 0.0005, by maximizing the original objective in a similar way as using Lagrangian Multiplier. At the end of optimization, $\lambda _ { i }$ ’s converge to locally optimal values. For example, with the medium maze, $\lambda _ { 1 } = 0 . 0 6 7$ for the KL-term, $\lambda _ { 2 } = 0 . 0 7 0$ for the $\mathcal { L } _ { 0 }$ and $\lambda _ { 3 } = 0 . 0 5 1$ for the $\mathcal { L } _ { T }$ term. The total number of waypoints $| \nu _ { p } |$ is set to be $2 0 \%$ of the size of the full state space.
+
+# C.2 TRAINING HRL MODELS
+
+The procedure of the Manager and the Worker in sending/receiving orders using either traversal paths among $\nu _ { p }$ from replay buffer for deterministic environments or with $\pi _ { g }$ for stochastic ones follows:
+
+1. The Manager gives a wide-narrow subgoal pair $( g _ { w } , g _ { n } )$ .   
+2. The agent takes action based on the Worker policy $\pi ^ { \omega }$ conditioned on $( g _ { w } , g _ { n } )$ and reaches a new state $s ^ { \prime }$ . If $s ^ { \prime } \in \mathcal { V } _ { p }$ , $g _ { w }$ has not yet met, and there exists a valid path basing on the edge paths from the world graph $s ^ { \prime } \to g _ { w }$ , agent then either follows replay actions or $\pi _ { g }$ to reach $g _ { w }$ . If $\pi _ { g }$ still does not reach desired destination in a certain steps, then stop the agent wherever it stands; also $\pi _ { g }$ can be finetuned here.   
+3. The Worker receives positive reward for reaching $g _ { w }$ for the first time.   
+4. If agent reaches $g _ { n }$ , the Worker also receives positive rewards and terminates this horizon.   
+5. The Worker receives negative for every action taken except for during traversal; the Manager receives negative reward for every action taken including traversal.   
+6. When either $g _ { n }$ is reached or the maximum time step for this horizon is met, the Manager renews its subgoal pair.
+
+The training of the Worker policy $\pi ^ { \omega }$ follows the same A2C algorithm as $\pi _ { g }$
+
+The training of the Manager policy $\pi ^ { m }$ also follows a similar procedure but as it operates at a lower temporal resolution, its value function regresses against the $t _ { m }$ -step discounted reward where $t _ { m }$ covers all actions and rewards generated from the Worker.
+
+When using the Wide-then-Narrow instruction, the policy gradient for the Manager policy $\pi _ { m }$ becomes:
+
+$\begin{array} { r } { \boldsymbol { \mathbb { E } } _ { ( s _ { t } , a _ { t } ) \sim \pi , p , p _ { 0 } } \left[ A _ { m , t } \nabla \log \left( \pi ^ { \omega } \left( g _ { w , t } | s _ { t } \right) \pi ^ { n } \left( g _ { n , t } | s _ { t } , g _ { w , t } , s _ { w , t } \right) \right) \right] + \nabla \left[ \mathcal { H } \left( \pi ^ { \omega } \right) + \mathcal { H } \left( \pi ^ { n } ( \cdot | g _ { w , t } ) \right) \right] , } \end{array}$ where $A _ { m , t }$ is the Manager’s advantage at time $t$ . Also, for Manager, as the size of the action space scales linearly with $| S |$ , the exact entropy for the $\pi ^ { m }$ can easily become intractable. Essentially there are $O$ $^ { \prime } \left( | \mathcal { V } | \times \left( N ^ { 2 } \right) \right)$ possible actions. To calculate the entropy exactly, all of them has to be summed, making it easily computationally intractable:
+
+$$
+\mathcal { H } = \sum _ { w \in \mathcal { V } } \sum _ { w _ { n } \in s _ { w } } \pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \pi ^ { \omega } ( w | s _ { t } ) \log { \nabla \pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \pi ^ { \omega } ( w | s _ { t } ) } .
+$$
+
+Thus in practice we resort to an effective alternative $\mathcal { H } \left( \pi ^ { \omega } \right) + \mathcal { H } \left( \pi ^ { n } ( \cdot | g _ { w , t } ) \right)$ .
+
+Psuedo-code for Manager training is in Algorithm 2.
+
+# C.3 HYPERPARAMETERS FOR HRL
+
+For training the HRL policies, we inherit most hyperparameters from those used when training $\pi _ { g }$ , as the Manager and the Worker both share similar architectures with $\pi _ { g }$ . The hyperparameters used when training $\pi _ { g }$ follow those from Shang et al. (2019). Because the tasks used in HRL experiments are more difficult than the generic goal-reaching task, we set the maximal number of training iterations to 100K abd training is stopped early if model performance reaches a plateau. The rollout steps for each iteration is 60. Hyperparameters specific to HRL are the horizon $c = 2 0$ and the size of the Manager’s local attention range (that is, the neighborhood around $g ^ { w }$ within which $g ^ { n }$ is selected), which are $N = 5$ for small and medium mazes, and $N = 7$ for the large maze.
+
+Algorithm 2: Training of $\pi ^ { m }$ for HRL models   
+
+<table><tr><td>Clear gradients dθ ←O; while t &lt;= tmax or episode not terminated do Simulate under current policy πm,t-1, πω,t-1; if the Worker has met the previous subgoal or exceeded the horizon c then</td><td>Reset the set of time steps where πm,t omits a new subgoal Sm = {} and tm = 0.;</td></tr><tr><td colspan="2">Sample a new subgoal gm,t from πm,t; end</td></tr><tr><td>Zm,t = fLsTM(CNN(sm,t,sv),hm,tm),Vm,t = fu(zm,t),Tt = fp(2m,t) ;</td><td></td></tr><tr><td>Sm= SmU {tm} and tm =t;</td><td></td></tr><tr><td></td><td></td></tr><tr><td colspan="2">O, if terminal</td></tr><tr><td>Vtmax+1, otherwise</td><td></td></tr><tr><td>for t = tmax,...1 do R←rt+γR;</td><td></td></tr><tr><td>if t ∈ Sm then</td><td></td></tr><tr><td>Am,t ←R-Vm,t;</td><td></td></tr><tr><td></td><td></td></tr><tr><td>Accumulate gradients from value loss: dθ ← d0 + 入</td><td></td></tr><tr><td></td><td></td></tr><tr><td>Accumulate policy gradients with entropy regularization:</td><td>80</td></tr><tr><td></td><td></td></tr><tr><td>d0 ← d0+ VlogTm,t(gm,t)Am,t + βVH(πm,t);</td><td></td></tr><tr><td></td><td></td></tr><tr><td>end</td><td></td></tr><tr><td>end</td><td></td></tr></table>
+
+# D 2D GRID WORLD VISUALIZATIONS
+
+![](images/e873103b674e3b139b1dde9c9a9cc1ebfed8b9e0f0f4150e66c98c1ba56795b4.jpg)  
+Figure 5: Visualization of the 2D grid environments in our experiments, along with the learned waypoints in blue.
+
+![](images/86812f5f54acb2108da0bc58155e8700f4575fcfaa3ed83f6a9417d89e4ae0b7.jpg)  
+Figure 6: Visualization of tasks in our experiments.

parse/train/H1VyHY9gg/H1VyHY9gg.md

@@ -0,0 +1,313 @@
+# DATA NOISING AS SMOOTHING IN NEURAL NETWORK LANGUAGE MODELS
+
+Ziang Xie, Sida I. Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, Andrew Y. Ng ´
+
+Computer Science Department, Stanford University zxie,sidaw,danilevy,anie,ang @cs.stanford.edu, jiweil,jurafsky @stanford.edu
+
+# ABSTRACT
+
+Data noising is an effective technique for regularizing neural network models. While noising is widely adopted in application domains such as vision and speech, commonly used noising primitives have not been developed for discrete sequencelevel settings such as language modeling. In this paper, we derive a connection between input noising in neural network language models and smoothing in $n$ - gram models. Using this connection, we draw upon ideas from smoothing to develop effective noising schemes. We demonstrate performance gains when applying the proposed schemes to language modeling and machine translation. Finally, we provide empirical analysis validating the relationship between noising and smoothing.
+
+# 1 INTRODUCTION
+
+Language models are a crucial component in many domains, such as autocompletion, machine translation, and speech recognition. A key challenge when performing estimation in language modeling is the data sparsity problem: due to large vocabulary sizes and the exponential number of possible contexts, the majority of possible sequences are rarely or never observed, even for very short subsequences.
+
+In other application domains, data augmentation has been key to improving the performance of neural network models in the face of insufficient data. In computer vision, for example, there exist well-established primitives for synthesizing additional image data, such as by rescaling or applying affine distortions to images (LeCun et al., 1998; Krizhevsky et al., 2012). Similarly, in speech recognition adding a background audio track or applying small shifts along the time dimension has been shown to yield significant gains, especially in noisy settings (Deng et al., 2000; Hannun et al., 2014). However, widely-adopted noising primitives have not yet been developed for neural network language models.
+
+Classic $n$ -gram models of language cope with rare and unseen sequences by using smoothing methods, such as interpolation or absolute discounting (Chen & Goodman, 1996). Neural network models, however, have no notion of discrete counts, and instead use distributed representations to combat the curse of dimensionality (Bengio et al., 2003). Despite the effectiveness of distributed representations, overfitting due to data sparsity remains an issue. Existing regularization methods, however, are typically applied to weights or hidden units within the network (Srivastava et al., 2014; Le et al., 2015) instead of directly considering the input data.
+
+In this work, we consider noising primitives as a form of data augmentation for recurrent neural network-based language models. By examining the expected pseudocounts from applying the noising schemes, we draw connections between noising and linear interpolation smoothing. Using this connection, we then derive noising schemes that are analogues of more advanced smoothing methods. We demonstrate the effectiveness of these schemes for regularization through experiments on language modeling and machine translation. Finally, we validate our theoretical claims by examining the empirical effects of noising.
+
+# 2 RELATED WORK
+
+Our work can be viewed as a form of data augmentation, for which to the best of our knowledge there exists no widely adopted schemes in language modeling with neural networks. Classical regularization methods such as $L _ { 2 }$ -regularization are typically applied to the model parameters, while dropout is applied to activations which can be along the forward as well as the recurrent directions (Zaremba et al., 2014; Semeniuta et al., 2016; Gal, 2015). Others have introduced methods for recurrent neural networks encouraging the hidden activations to remain stable in norm, or constraining the recurrent weight matrix to have eigenvalues close to one (Krueger & Memisevic, 2015; Arjovsky et al., 2015; Le et al., 2015). These methods, however, all consider weights and hidden units instead of the input data, and are motivated by the vanishing and exploding gradient problem.
+
+Feature noising has been demonstrated to be effective for structured prediction tasks, and has been interpreted as an explicit regularizer (Wang et al., 2013). Additionally, Wager et al. (2014) show that noising can inject appropriate generative assumptions into discriminative models to reduce their generalization error, but do not consider sequence models (Wager et al., 2016).
+
+The technique of randomly zero-masking input word embeddings for learning sentence representations has been proposed by Iyyer et al. (2015), Kumar et al. (2015), and Dai & Le (2015), and adopted by others such as Bowman et al. (2015). However, to the best of our knowledge, no analysis has been provided besides reasoning that zeroing embeddings may result in a model ensembling effect similar to that in standard dropout. This analysis is applicable to classification tasks involving sum-of-embeddings or bag-of-words models, but does not capture sequence-level effects. Bengio et al. (2015) also make an empirical observation that the method of randomly replacing words with fixed probability with a draw from the uniform distribution improved performance slightly for an image captioning task; however, they do not examine why performance improved.
+
+# 3 METHOD
+
+# 3.1 PRELIMINARIES
+
+We consider language models where given a sequence of indices $X = ( x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { T } )$ , over the vocabulary $V$ , we model
+
+$$
+p ( X ) = \prod _ { t = 1 } ^ { T } p ( x _ { t } | x _ { < t } )
+$$
+
+In $n$ -gram models, it is not feasible to model the full context $x _ { < t }$ for large $t$ due to the exponential number of possible histories. Recurrent neural network (RNN) language models can (in theory) model longer dependencies, since they operate over distributed hidden states instead of modeling an exponential number of discrete counts (Bengio et al., 2003; Mikolov, 2012).
+
+An $L$ -layer recurrent neural network is modeled as $h _ { t } ^ { ( l ) } = f _ { \theta } ( h _ { t - 1 } ^ { ( l ) } , h _ { t } ^ { ( l - 1 ) } )$ , where $l$ denotes the layer index, $h ^ { ( 0 ) }$ contains the one-hot encoding of $X$ , and in its simplest form $f _ { \theta }$ applies an affine transformation followed by a nonlinearity. In this work, we use RNNs with a more complex form of $f _ { \theta }$ , namely long short-term memory (LSTM) units (Hochreiter & Schmidhuber, 1997), which have been shown to ease training and allow RNNs to capture longer dependencies. The output distribution over the vocabulary V at time t is pθ(xt|x<t) = softmax(gθ(h(L)t )), where g : R|h| → R|V | a pplies an affine transformation. The RNN is then trained by minimizing over its parameters $\theta$ the sequence cross-entropy loss $\begin{array} { r } { \ell ( \theta ) = - \sum _ { t } \log p _ { \theta } ( x _ { t } | x _ { < t } ) } \end{array}$ , thus maximizing the likelihood $p _ { \theta } ( X )$ .
+
+As an extension, we also consider encoder-decoder or sequence-to-sequence (Cho et al., 2014; Sutskever et al., 2014) models where given an input sequence $X$ and output sequence $Y$ of length $T _ { Y }$ , we model
+
+$$
+p ( Y | X ) = \prod _ { t = 1 } ^ { T _ { Y } } p ( y _ { t } | X , y _ { < t } ) .
+$$
+
+and minimize the loss $\begin{array} { r } { \ell ( \theta ) = - \sum _ { t } \log p _ { \theta } ( y _ { t } | X , y _ { < t } ) } \end{array}$ . This setting can also be seen as conditional language modeling, and encompasses tasks such as machine translation, where $X$ is a source lan
+
+guage sequence and $Y$ a target language sequence, as well as language modeling, where $Y$ is the given sequence and $X$ is the empty sequence.
+
+# 3.2 SMOOTHING AND NOISING
+
+Recall that for a given context length $l$ , an $n$ -gram model of order $l + 1$ is optimal under the loglikelihood criterion. Hence in the case where an RNN with finite context achieves near the lowest possible cross-entropy loss, it behaves like an $n$ -gram model.
+
+Like $n$ -gram models, RNNs are trained using maximum likelihood, and can easily overfit (Zaremba et al., 2014). While generic regularization methods such $L _ { 2 }$ -regularization and dropout are effective, they do not take advantage of specific properties of sequence modeling. In order to understand sequence-specific regularization, it is helpful to examine $n$ -gram language models, whose properties are well-understood.
+
+Smoothing for $n$ -gram models When modeling $p ( x _ { t } | x _ { < t } )$ , the maximum likelihood estimate $c ( x _ { < t } , x _ { t } ) \bar { / } c ( x _ { < t } )$ based on empirical counts puts zero probability on unseen sequences, and thus smoothing is crucial for obtaining good estimates. In particular, we consider interpolation, which performs a weighted average between higher and lower order models. The idea is that when there are not enough observations of the full sequence, observations of subsequences can help us obtain better estimates.1 For example, in a bigram model, $p _ { \mathrm { i n t e r p } } ( x _ { t } | x _ { t - 1 } ) = \bar { \lambda p } ( x _ { t } | x _ { t - 1 } ) + ( \bar { 1 - \lambda } ) p ( x _ { t } )$ , where $0 \leq \lambda \leq 1$ .
+
+Noising for RNN models We would like to apply well-understood smoothing methods such as interpolation to RNNs, which are also trained using maximum likelihood. Unfortunately, RNN models have no notion of counts, and we cannot directly apply one of the usual smoothing methods. In this section, we consider two simple noising schemes which we proceed to show correspond to smoothing methods. Since we can noise the data while training an RNN, we can then incorporate well-understood generative assumptions that are known to be helpful in the domain. First consider the following two noising schemes:
+
+• unigram noising For each $x _ { i }$ in $x _ { < t }$ , with probability $\gamma$ replace $x _ { i }$ with a sample from the unigram frequency distribution. • blank noising For each $x _ { i }$ in $x _ { < t }$ , with probability $\gamma$ replace $x _ { i }$ with a placeholder token “ ”.
+
+While blank noising can be seen as a way to avoid overfitting on specific contexts, we will see that both schemes are related to smoothing, and that unigram noising provides a path to analogues of more advanced smoothing methods.
+
+# 3.3 NOISING AS SMOOTHING
+
+We now consider the maximum likelihood estimate of $n$ -gram probabilities estimated using the pseudocounts of the noised data. By examining these estimates, we draw a connection between linear interpolation smoothing and noising.
+
+Unigram noising as interpolation To start, we consider the simplest case of bigram probabilities. Let $c ( x )$ denote the count of a token $x$ in the original data, and let $c _ { \gamma } ( x ) \ { \stackrel { \mathrm { d e f } } { = } } \ \mathbb { E } _ { \tilde { x } } \left[ c ( { \tilde { x } } ) \right]$ be the expected count of $x$ under the unigram noising scheme. We then have
+
+$$
+\begin{array} { r l } & { p _ { \gamma } ( x _ { t } | x _ { t - 1 } ) = \frac { c _ { \gamma } ( x _ { t - 1 } , x _ { t } ) } { c _ { \gamma } ( x _ { t - 1 } ) } } \\ & { \qquad = [ ( 1 - \gamma ) c ( x _ { t - 1 } , x _ { t } ) + \gamma p ( x _ { t - 1 } ) c ( x _ { t } ) ] / c ( x _ { t - 1 } ) } \\ & { \qquad = ( 1 - \gamma ) p ( x _ { t } | x _ { t - 1 } ) + \gamma p ( x _ { t } ) , } \end{array}
+$$
+
+where $c _ { \gamma } ( x ) = c ( x )$ since our proposal distribution $q ( x )$ is the unigram distribution, and the last line follows since $c ( x _ { t - 1 } ) / p ( x _ { t - 1 } ) = c ( x _ { t } ) / p ( x _ { t } )$ is equal to the total number of tokens in the training
+
+set. Thus we see that the noised data has pseudocounts corresponding to interpolation or a mixture of different order $n$ -gram models with fixed weighting.
+
+More generally, let $\tilde { { \boldsymbol { x } } } _ { < t }$ be noised tokens from $\tilde { x }$ . We consider the expected prediction under noise
+
+$$
+\begin{array} { r } { p _ { \gamma } ( x _ { t } | x _ { < t } ) = \mathbb { E } _ { \tilde { x } _ { < t } } \left[ p ( x _ { t } | \tilde { x } _ { < t } ) \right] \ ~ } \\ { = \sum _ { J } \underbrace { \pi ( | J | ) } _ { p ( | J | \mathrm { s w a p s } ) } \ \sum _ { x _ { K } } \underbrace { p ( x _ { t } | x _ { J } , x _ { K } ) } _ { p ( x _ { t } | \mathrm { n o i s e d c o n t e x t } ) } \prod _ { z \in x _ { K } } \underbrace { p ( z ) } _ { p ( \mathrm { d r a w i n g } \ z ) } } \end{array}
+$$
+
+where the mixture coefficients are $\pi ( | J | ) ~ = ~ ( 1 - \gamma ) ^ { | J | } \gamma ^ { t - 1 - | J | }$ with $\begin{array} { r } { \sum _ { J } \pi ( | J | ) ~ = ~ 1 } \end{array}$ . $J \subseteq$ $\{ 1 , 2 , \ldots , t - 1 \}$ denotes the set of indices whose corresponding tokens are left unchanged, and $K$ the set of indices that were replaced.
+
+Blank noising as interpolation Next we consider the blank noising scheme and show that it corresponds to interpolation as well. This also serves as an alternative explanation for the gains that other related work have found with the “word-dropout” idea (Kumar et al., 2015; Dai & Le, 2015; Bowman et al., 2015). As before, we do not noise the token being predicted $x _ { t }$ . Let $\tilde { { \boldsymbol { x } } } _ { < t }$ denote the random variable where each of its tokens is replaced by “ ” with probability $\gamma$ , and let $x J$ denote the sequence with indices $J$ unchanged, and the rest replaced by $\underline { { \boldsymbol { \cdot } } } \underline { { \boldsymbol { \mathit { \Pi } } } } ,$ . To make a prediction, we use the expected probability over different noisings of the context
+
+$$
+p _ { \gamma } ( x _ { t } | x _ { < t } ) = \mathbb { E } _ { \tilde { x } _ { < t } } \left[ p ( x _ { t } | \tilde { x } _ { < t } ) \right] = \sum _ { J } \underbrace { \pi ( | J | ) } _ { p ( | J | \mathrm { s w a p s } ) } \underbrace { p ( x _ { t } | x _ { J } ) } _ { p ( x _ { t } | \mathrm { n o i s e d c o n t e x t } ) } ,
+$$
+
+where $J \subseteq \{ 1 , 2 , \dots , t - 1 \}$ , which is also a mixture of the unnoised probabilities over subsequences of the current context. For example, in the case of trigrams, we have
+
+$$
+p _ { \gamma } ( x _ { 3 } | x _ { 1 } , x _ { 2 } ) = \pi ( 2 ) p ( x _ { 3 } | x _ { 1 } , x _ { 2 } ) + \pi ( 1 ) p ( x _ { 3 } | x _ { 1 } , \ldots ) + \pi ( 1 ) p ( x _ { 3 } | \ldots , x _ { 2 } ) + \pi ( 0 ) p ( x _ { 3 } | \ldots , \ldots )
+$$
+
+where the mixture coefficient $\pi ( i ) = ( 1 - \gamma ) ^ { i } \gamma ^ { 2 - i }$ .
+
+# 3.4 BORROWING TECHNIQUES
+
+With the connection between noising and smoothing in place, we now consider how we can improve the two components of the noising scheme by considering:
+
+1. Adaptively computing noising probability $\gamma$ to reflect our confidence about a particular input subsequence.   
+2. Selecting a proposal distribution $q ( x )$ that is less naive than the unigram distribution by leveraging higher order $n$ -gram statistics.
+
+Noising Probability Although it simplifies analysis, there is no reason why we should choose fixed $\gamma$ ; we now consider defining an adaptive $\gamma ( \boldsymbol { x } _ { 1 : t } )$ which depends on the input sequence. Consider the following bigrams:
+
+“and the”
+
+“Humpty Dumpty”
+
+The first bigram is one of the most common in English corpora; its probability is hence well estimated and should not be interpolated with lower order distributions. In expectation, however, using fixed $\gamma _ { 0 }$ when noising results in the same lower order interpolation weight $\pi _ { \gamma _ { 0 } }$ for common as well as rare bigrams. Intuitively, we should define $\gamma ( \boldsymbol { x } _ { 1 : t } )$ such that commonly seen bigrams are less likely to be noised.
+
+The second bigram, “Humpty Dumpty,” is relatively uncommon, as are its constituent unigrams. However, it forms what Brown et al. (1992) term a “sticky pair”: the unigram “Dumpty” almost always follows the unigram “Humpty”, and similarly, “Humpty” almost always precedes “Dumpty”. For pairs with high mutual information, we wish to avoid backing off from the bigram to the unigram distribution.
+
+<table><tr><td>Noised</td><td>γ(x1:2)</td><td>q(x）</td><td>Analogue</td></tr><tr><td>x1</td><td>20</td><td>q(“_&quot;)=1</td><td>interpolation</td></tr><tr><td>x1</td><td>70</td><td>unigram</td><td>interpolation</td></tr><tr><td>x1</td><td>20N1+(x1,.)/c(x1)</td><td>unigram</td><td>absolute discounting</td></tr><tr><td>x1,x2</td><td>20N1+(x1,·)/c(x1)</td><td>q(x) x Ni+(•,x)</td><td>Kneser-Ney</td></tr></table>
+
+Table 1: Noising schemes Example noising schemes and their bigram smoothing analogues. Here we consider the bigram probability $p ( x _ { 1 } , x _ { 2 } ) \ : = \ : p ( x _ { 2 } \vert x _ { 1 } ) p ( x _ { 1 } )$ . Notation: $\gamma ( \boldsymbol { x } _ { 1 : t } )$ denotes the noising probability for a given input sequence $x _ { 1 : t }$ , $q ( x )$ denotes the proposal distribution, and $N _ { 1 + } ( x , \bullet )$ denotes the number of distinct bigrams in the training set where $x$ is the first unigram. In all but the last case we only noise the context $x _ { 1 }$ and not the target prediction $x _ { 2 }$ .
+
+Let $N _ { 1 + } ( x _ { 1 } , \bullet ) \ { \stackrel { \mathrm { d e f } } { = } } \ | \{ x _ { 2 } : c ( x _ { 1 } , x _ { 2 } ) > 0 \} |$ be the number of distinct continutions following $x _ { 1 }$ , or equivalently the number of bigram types beginning with $x _ { 1 }$ (Chen $\&$ Goodman, 1996). From the above intuitions, we arrive at the absolute discounting noising probability
+
+$$
+\gamma _ { \mathrm { A D } } ( x _ { 1 } ) = \gamma _ { 0 } \frac { N _ { 1 + } ( x _ { 1 } , \bullet ) } { \sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) }
+$$
+
+where for $0 \leq \gamma _ { 0 } \leq 1$ we have $0 \leq \gamma _ { \mathrm { A D } } \leq 1$ , though in practice we can also clip larger noising probabilities to 1. Note that this encourages noising of unigrams that precede many possible other tokens while discouraging noising of common unigrams, since if we ignore the final token, $\begin{array} { r } { \sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) = c ( x _ { 1 } ) } \end{array}$ .
+
+Proposal Distribution While choosing the unigram distribution as the proposal distribution $q ( x )$ preserves unigram frequencies, by borrowing from the smoothing literature we find another distribution performs better. We again begin with two motivating examples:
+
+“San Francisco”
+
+“New York”
+
+Both bigrams appear frequently in text corpora. As a direct consequence, the unigrams “Francisco” and “York” also appear frequently. However, since “Francisco” and “York” typically follow “San” and “New”, respectively, they should not have high probability in the proposal distribution as they might if we use unigram frequencies (Chen $\&$ Goodman, 1996). Instead, it would be better to increase the proposal probability of unigrams with diverse histories, or more precisely unigrams that complete a large number of bigram types. Thus instead of drawing from the unigram distribution, we consider drawing from
+
+$$
+q ( x ) \propto N _ { 1 + } ( \bullet , x )
+$$
+
+Note that we now noise the prediction $x _ { t }$ in addition to the context $x _ { 1 : t - 1 }$ . Combining this new proposal distribution with the discounted $\gamma _ { \mathrm { A D } } ( x _ { 1 } )$ from the previous section, we obtain the noising analogue of Kneser-Ney smoothing.
+
+Table 1 summarizes the discussed noising schemes.
+
+# 3.5 TRAINING AND TESTING
+
+During training, noising is performed per batch and is done online such that each epoch of training sees a different noised version of the training data. At test time, to match the training objective we should sample multiple corrupted versions of the test data, then average the predictions (Srivastava et al., 2014). In practice, however, we find that simply using the maximum likelihood (uncorrupted) input sequence works well; evaluation runtime remains unchanged.
+
+# 3.6 EXTENSIONS
+
+The schemes described are for the language model setting. To extend them to the sequence-tosequence or encoder-decoder setting, we noise both $x _ { < t }$ as well as $y _ { < t }$ . While in the decoder we
+
+Table 2: Single-model perplexity on Penn Treebank with different noising schemes. We also compare to the variational method of Gal (2015), who also train LSTM models with the same hidden dimension. Note that performing Monte Carlo dropout at test time is significantly more expensive than our approach, where test time is unchanged.   
+
+<table><tr><td>Noising scheme</td><td>Validation</td><td>Test</td></tr><tr><td colspan="2">Medium models (512 hidden size)</td><td></td></tr><tr><td>none (dropout only) blank</td><td>84.3</td><td>80.4</td></tr><tr><td></td><td>82.7</td><td>78.8</td></tr><tr><td>unigram</td><td>83.1</td><td>80.1</td></tr><tr><td>bigram Kneser-Ney</td><td>79.9</td><td>76.9</td></tr><tr><td colspan="2">Large models (1500 hidden size)</td><td></td></tr><tr><td rowspan="3">none (dropout only) blank unigram bigram Kneser-Ney</td><td>81.6</td><td>77.5</td></tr><tr><td>79.4</td><td>75.5</td></tr><tr><td>79.4 76.2</td><td>76.1 73.4</td></tr><tr><td>Zaremba et al. (2014)</td><td>82.2</td><td></td></tr><tr><td>Gal (2015) variational dropout (tied weights)</td><td></td><td>78.4</td></tr><tr><td>Gal (2015) (untied weights,Monte Carlo)</td><td>77.3 丨</td><td>75.0 73.4</td></tr></table>
+
+Table 3: Perplexity on Text8 with different noising schemes.   
+
+<table><tr><td>Noising scheme</td><td>Validation</td><td>Test</td></tr><tr><td>none</td><td>94.3</td><td>123.6</td></tr><tr><td>blank</td><td>85.0</td><td>110.7</td></tr><tr><td>unigram</td><td>85.2</td><td>111.3</td></tr><tr><td>bigram Kneser-Ney</td><td>84.5</td><td>110.6</td></tr></table>
+
+have $y _ { < t }$ and $y _ { t }$ as analogues to language model context and target prediction, it is unclear whether noising $x _ { < t }$ should be beneficial. Empirically, however, we find this to be the case (Table 4).
+
+# 4 EXPERIMENTS
+
+# 4.1 LANGUAGE MODELING
+
+Penn Treebank We train networks for word-level language modeling on the Penn Treebank dataset, using the standard preprocessed splits with a 10K size vocabulary (Mikolov, 2012). The PTB dataset contains $9 2 9 \mathrm { k }$ training tokens, 73k validation tokens, and ${ 8 2 } \mathrm { k }$ test tokens. Following Zaremba et al. (2014), we use minibatches of size 20 and unroll for 35 time steps when performing backpropagation through time. All models have two hidden layers and use LSTM units. Weights are initialized uniformly in the range $[ - 0 . 1 , 0 . 1 ]$ . We consider models with hidden sizes of 512 and 1500.
+
+We train using stochastic gradient descent with an initial learning rate of 1.0, clipping the gradient if its norm exceeds 5.0. When the validation cross entropy does not decrease after a training epoch, we halve the learning rate. We anneal the learning rate 8 times before stopping training, and pick the model with the lowest perplexity on the validation set.
+
+For regularization, we apply feed-forward dropout (Pham et al., 2014) in combination with our noising schemes. We report results in Table 2 for the best setting of the dropout rate (which we find to match the settings reported in Zaremba et al. (2014)) as well as the best setting of noising
+
+![](images/a74791f89bc2ae43fdc9a304f47b23fda338e2c01b613335fe659ced2c4f1495.jpg)  
+Figure 1: Example training and validation curves for an unnoised model and model regularized using the bigram Kneser-Ney noising scheme.
+
+<table><tr><td>Scheme</td><td>Perplexity</td><td>BLEU</td></tr><tr><td>dropout, no noising</td><td>8.84</td><td>24.6</td></tr><tr><td>blank noising</td><td>8.28</td><td>25.3 (+0.7)</td></tr><tr><td>unigram noising</td><td>8.15</td><td>25.5 (+0.9)</td></tr><tr><td>bigram Kneser-Ney</td><td>7.92</td><td>26.0 (+1.4)</td></tr><tr><td>source only</td><td>8.74</td><td>24.8 (+0.2)</td></tr><tr><td>target only</td><td>8.14</td><td>25.6 (+1.0)</td></tr></table>
+
+Table 4: Perplexities and BLEU scores for machine translation task. Results for bigram KN noising on only the source sequence and only the target sequence are given as well.
+
+probability $\gamma _ { 0 }$ on the validation set.2 Figure 1 shows the training and validation perplexity curves for a noised versus an unnoised run.
+
+Our large models match the state-of-the-art regularization method for single model performance on this task. In particular, we find that picking $\gamma _ { \mathrm { A D } } ( x _ { 1 } )$ and $q ( x )$ corresponding to Kneser-Ney smoothing yields significant gains in validation perplexity, both for the medium and large size models. Recent work (Merity et al., 2016; Zilly et al., 2016) has also achieved impressive results on this task by proposing different architectures which are orthogonal to our data augmentation schemes.
+
+Text8 In order to determine whether noising remains effective with a larger dataset, we perform experiments on the Text8 corpus3. The first 90M characters are used for training, the next 5M for validation, and the final 5M for testing, resulting in $1 5 . 3 \mathbf { M }$ training tokens, 848K validation tokens, and 855K test tokens. We preprocess the data by mapping all words which appear 10 or fewer times to the unknown token, resulting in a 42K size vocabulary. Other parameter settings are the same as described in the Penn Treebank experiments, besides that only models with hidden size 512 are considered, and noising is not combined with feed-forward dropout. Results are given in Table 3.
+
+# 4.2 MACHINE TRANSLATION
+
+For our machine translation experiments we consider the English-German machine translation track of IWSLT $2 0 1 5 ^ { 4 }$ . The IWSLT 2015 corpus consists of sentence-aligned subtitles of TED and TEDx talks. The training set contains roughly 190K sentence pairs with 5.4M tokens. Following Luong & Manning (2015), we use TED tst2012 as a validation set and report BLEU score results (Papineni et al., 2002) on tst2014. We limit the vocabulary to the top 50K most frequent words for each language.
+
+![](images/c52239f55d9409031ab76a2c2d420a3bbb70c906f6f5db3a48bd4677b262db5b.jpg)  
+Figure 2: Perplexity with noising on Penn Treebank while varying the value of $\gamma _ { 0 }$ . Using discounting to scale $\gamma _ { 0 }$ (yielding $\gamma _ { \mathrm { A D . } }$ ) maintains gains for a range of values of noising probability, which is not true for the unscaled case.
+
+![](images/66596b6cf7d5d61afc0773e432d36885d11d8c1b6493b1dddbc53657d86ae861.jpg)  
+Figure 3: Mean KL-divergence over validation set between softmax distributions of noised and unnoised models and lower order distributions. Noised model distributions are closer to the uniform and unigram frequency distributions.
+
+We train a two-layer LSTM encoder-decoder network (Sutskever et al., 2014; Cho et al., 2014) with 512 hidden units in each layer. The decoder uses an attention mechanism (Bahdanau et al., 2014) with the dot alignment function (Luong et al., 2015). The initial learning rate is 1.0 and we start halving the learning rate when the relative difference in perplexity on the validation set between two consecutive epochs is less than $1 \%$ . We follow training protocols as described in Sutskever et al. (2014): (a) LSTM parameters and word embeddings are initialized from a uniform distribution between $[ - 0 . 1 , 0 . 1 ]$ , (b) inputs are reversed, (c) batch size is set to 128, (d) gradient clipping is performed when the norm exceeds a threshold of 5. We set hidden unit dropout rate to 0.2 across all settings as suggested in Luong et al. (2015). We compare unigram, blank, and bigram Kneser-Ney noising. Noising rate $\gamma$ is selected on the validation set.
+
+Results are shown in Table 4. We observe performance gains for both blank noising and unigram noising, giving roughly $+ 0 . 7$ BLEU score on the test set. The proposed bigram Kneser-Ney noising scheme gives an additional performance boost of $+ 0 . 5 – 0 . 7$ on top of the blank noising and unigram noising models, yielding a total gain of $+ 1 . 4$ BLEU.
+
+# 5 DISCUSSION
+
+# 5.1 SCALING $\gamma$ VIA DISCOUNTING
+
+We now examine whether discounting has the desired effect of noising subsequences according to their uncertainty. If we consider the discounting
+
+$$
+\gamma _ { \mathrm { A D } } ( x _ { 1 } ) = \gamma _ { 0 } \frac { N _ { 1 + } ( x _ { 1 } , \bullet ) } { c ( x _ { 1 } ) }
+$$
+
+we observe that the denominator $c ( x _ { 1 } )$ can dominate than the numerator $N _ { 1 + } ( x _ { 1 } , \bullet )$ . Common tokens are often noised infrequently when discounting is used to rescale the noising probability, while rare tokens are noised comparatively much more frequently, where in the extreme case when a token appears exactly once, we have $\gamma _ { \mathrm { A D } } ~ = ~ \gamma _ { 0 }$ . Due to word frequencies following a Zipfian power law distribution, however, common tokens constitute the majority of most texts, and thus discounting leads to significantly less noising.
+
+We compare the performance of models trained with a fixed $\gamma _ { 0 }$ versus a $\gamma _ { 0 }$ rescaled using discounting. As shown in Figure 2, bigram discounting leads to gains in perplexity for a much broader range of $\gamma _ { 0 }$ . Thus the discounting ratio seems to effectively capture the “right” tokens to noise.
+
+<table><tr><td>Noising</td><td>Bigrams</td><td>Trigrams</td></tr><tr><td>none (dropout only)</td><td>2881</td><td>381</td></tr><tr><td>blank noising</td><td>2760</td><td>372</td></tr><tr><td>unigram noising</td><td>2612</td><td>365</td></tr></table>
+
+Table 5: Perplexity of last unigram for unseen bigrams and trigrams in Penn Treebank validation set. We compare noised and unnoised models with noising probabilities chosen such that models have near-identical perplexity on full validation set.
+
+# 5.2 NOISED VERSUS UNNOISED MODELS
+
+Smoothed distributions In order to validate that data noising for RNN models has a similar effect to that of smoothing counts in $n$ -gram models, we consider three models trained with unigram noising as described in Section 4.1 on the Penn Treebank corpus with $\gamma = 0$ (no noising), $\gamma = 0 . 1$ , and $\gamma = 0 . 2 5$ . Using the trained models, we measure the Kullback-Leibler divergence $D _ { \mathrm { K L } } ( p \Vert q ) =$ $\textstyle \sum _ { i } p _ { i } \log ( p _ { i } / q _ { i } )$ over the validation set between the predicted softmax distributions, $\hat { p }$ , and the uniform distribution as well as the unigram frequency distribution. We then take the mean KL divergence over all tokens in the validation set.
+
+Recall that in interpolation smoothing, a weighted combination of higher and lower order $n$ -gram models is used. As seen in Figure 3, the softmax distributions of noised models are significantly closer to the lower order frequency distributions than unnoised models, in particular in the case of the unigram distribution, thus validating our analysis in Section 3.3.
+
+Unseen $n$ -grams Smoothing is most beneficial for increasing the probability of unobserved sequences. To measure whether noising has a similar effect, we consider bigrams and trigrams in the validation set that do not appear in the training set. For these unseen bigrams (15062 occurrences) and trigrams (43051 occurrences), we measure the perplexity for noised and unnoised models with near-identical perplexity on the full set. As expected, noising yields lower perplexity for these unseen instances.
+
+# 6 CONCLUSION
+
+In this work, we show that data noising is effective for regularizing neural network-based sequence models. By deriving a correspondence between noising and smoothing, we are able to adapt advanced smoothing methods for $n$ -gram models to the neural network setting, thereby incorporating well-understood generative assumptions of language. Possible applications include exploring noising for improving performance in low resource settings, or examining how these techniques generalize to sequence modeling in other domains.
+
+# ACKNOWLEDGMENTS
+
+We thank Will Monroe for feedback on a draft of this paper, Anand Avati for help running experiments, and Jimmy Wu for computing support. We also thank the developers of Theano (Theano Development Team, 2016) and Tensorflow (Abadi et al., 2016). Some GPUs used in this work were donated by NVIDIA Corporation. ZX, SW, and JL were supported by an NDSEG Fellowship, NSERC PGS-D Fellowship, and Facebook Fellowship, respectively. This project was funded in part by DARPA MUSE award FA8750-15-C-0242 AFRL/RIKF.
+
+# REFERENCES
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+# A SKETCH OF NOISING ALGORITHM
+
+We provide pseudocode of the noising algorithm corresponding to bigram Kneser-Ney smoothing for $n$ -grams (In the case of sequence-to-sequence tasks, we estimate the count-based parameters separately for source and target). To simplify, we assume a batch size of one. The noising algorithm is applied to each data batch during training. No noising is applied at test time.
+
+<table><tr><td>Algorithm 1 Bigram KN noising (Language modeling setting)</td></tr><tr><td>Require counts c(x), number of distinct continuations N1+(x,·), proposal distribution q(x） N1+(.,x) Inputs X,Y batch of unnoised data indices, scaling factor γo</td></tr><tr><td>procedure NOISEBGKN(X, Y) &gt;X=(x1,...,xt),Y=(x2,...,xt+1)</td></tr><tr><td>X,Y←X,Y</td></tr><tr><td>for j = 1,...,t do γ ←γ0N1+(xj,·)/c(xj)</td></tr><tr><td>if ~ Bernoulli(~) then</td></tr><tr><td>xj~ Categorical(q) Updates X</td></tr><tr><td>yj~ Categorical(q)</td></tr><tr><td>end if</td></tr><tr><td>end for</td></tr><tr><td>return X, Y Run training iteration with noised batch</td></tr><tr><td>end procedure</td></tr></table>

parse/train/H1VyHY9gg/H1VyHY9gg_content_list.json

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+    {
+        "type": "text",
+        "text": "DATA NOISING AS SMOOTHING IN NEURAL NETWORK LANGUAGE MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ziang Xie, Sida I. Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, Andrew Y. Ng ´ ",
+        "bbox": [
+            184,
+            170,
+            802,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Computer Science Department, Stanford University zxie,sidaw,danilevy,anie,ang @cs.stanford.edu, jiweil,jurafsky @stanford.edu ",
+        "bbox": [
+            186,
+            185,
+            637,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Data noising is an effective technique for regularizing neural network models. While noising is widely adopted in application domains such as vision and speech, commonly used noising primitives have not been developed for discrete sequencelevel settings such as language modeling. In this paper, we derive a connection between input noising in neural network language models and smoothing in $n$ - gram models. Using this connection, we draw upon ideas from smoothing to develop effective noising schemes. We demonstrate performance gains when applying the proposed schemes to language modeling and machine translation. Finally, we provide empirical analysis validating the relationship between noising and smoothing. ",
+        "bbox": [
+            233,
+            301,
+            764,
+            441
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            488,
+            336,
+            503
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Language models are a crucial component in many domains, such as autocompletion, machine translation, and speech recognition. A key challenge when performing estimation in language modeling is the data sparsity problem: due to large vocabulary sizes and the exponential number of possible contexts, the majority of possible sequences are rarely or never observed, even for very short subsequences. ",
+        "bbox": [
+            174,
+            526,
+            823,
+            595
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In other application domains, data augmentation has been key to improving the performance of neural network models in the face of insufficient data. In computer vision, for example, there exist well-established primitives for synthesizing additional image data, such as by rescaling or applying affine distortions to images (LeCun et al., 1998; Krizhevsky et al., 2012). Similarly, in speech recognition adding a background audio track or applying small shifts along the time dimension has been shown to yield significant gains, especially in noisy settings (Deng et al., 2000; Hannun et al., 2014). However, widely-adopted noising primitives have not yet been developed for neural network language models. ",
+        "bbox": [
+            174,
+            603,
+            825,
+            714
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Classic $n$ -gram models of language cope with rare and unseen sequences by using smoothing methods, such as interpolation or absolute discounting (Chen & Goodman, 1996). Neural network models, however, have no notion of discrete counts, and instead use distributed representations to combat the curse of dimensionality (Bengio et al., 2003). Despite the effectiveness of distributed representations, overfitting due to data sparsity remains an issue. Existing regularization methods, however, are typically applied to weights or hidden units within the network (Srivastava et al., 2014; Le et al., 2015) instead of directly considering the input data. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we consider noising primitives as a form of data augmentation for recurrent neural network-based language models. By examining the expected pseudocounts from applying the noising schemes, we draw connections between noising and linear interpolation smoothing. Using this connection, we then derive noising schemes that are analogues of more advanced smoothing methods. We demonstrate the effectiveness of these schemes for regularization through experiments on language modeling and machine translation. Finally, we validate our theoretical claims by examining the empirical effects of noising. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            344,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work can be viewed as a form of data augmentation, for which to the best of our knowledge there exists no widely adopted schemes in language modeling with neural networks. Classical regularization methods such as $L _ { 2 }$ -regularization are typically applied to the model parameters, while dropout is applied to activations which can be along the forward as well as the recurrent directions (Zaremba et al., 2014; Semeniuta et al., 2016; Gal, 2015). Others have introduced methods for recurrent neural networks encouraging the hidden activations to remain stable in norm, or constraining the recurrent weight matrix to have eigenvalues close to one (Krueger & Memisevic, 2015; Arjovsky et al., 2015; Le et al., 2015). These methods, however, all consider weights and hidden units instead of the input data, and are motivated by the vanishing and exploding gradient problem. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            261
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Feature noising has been demonstrated to be effective for structured prediction tasks, and has been interpreted as an explicit regularizer (Wang et al., 2013). Additionally, Wager et al. (2014) show that noising can inject appropriate generative assumptions into discriminative models to reduce their generalization error, but do not consider sequence models (Wager et al., 2016). ",
+        "bbox": [
+            174,
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+    {
+        "type": "text",
+        "text": "The technique of randomly zero-masking input word embeddings for learning sentence representations has been proposed by Iyyer et al. (2015), Kumar et al. (2015), and Dai & Le (2015), and adopted by others such as Bowman et al. (2015). However, to the best of our knowledge, no analysis has been provided besides reasoning that zeroing embeddings may result in a model ensembling effect similar to that in standard dropout. This analysis is applicable to classification tasks involving sum-of-embeddings or bag-of-words models, but does not capture sequence-level effects. Bengio et al. (2015) also make an empirical observation that the method of randomly replacing words with fixed probability with a draw from the uniform distribution improved performance slightly for an image captioning task; however, they do not examine why performance improved. ",
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 PRELIMINARIES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We consider language models where given a sequence of indices $X = ( x _ { 1 } , x _ { 2 } , \\cdot \\cdot \\cdot , x _ { T } )$ , over the vocabulary $V$ , we model ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/c0d2f832e5e905d46da1dbe6fa5ecbb8e2bc7af43c18208d5774ce81ef3d5ebd.jpg",
+        "text": "$$\np ( X ) = \\prod _ { t = 1 } ^ { T } p ( x _ { t } | x _ { < t } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In $n$ -gram models, it is not feasible to model the full context $x _ { < t }$ for large $t$ due to the exponential number of possible histories. Recurrent neural network (RNN) language models can (in theory) model longer dependencies, since they operate over distributed hidden states instead of modeling an exponential number of discrete counts (Bengio et al., 2003; Mikolov, 2012). ",
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+    {
+        "type": "text",
+        "text": "An $L$ -layer recurrent neural network is modeled as $h _ { t } ^ { ( l ) } = f _ { \\theta } ( h _ { t - 1 } ^ { ( l ) } , h _ { t } ^ { ( l - 1 ) } )$ , where $l$ denotes the layer index, $h ^ { ( 0 ) }$ contains the one-hot encoding of $X$ , and in its simplest form $f _ { \\theta }$ applies an affine transformation followed by a nonlinearity. In this work, we use RNNs with a more complex form of $f _ { \\theta }$ , namely long short-term memory (LSTM) units (Hochreiter & Schmidhuber, 1997), which have been shown to ease training and allow RNNs to capture longer dependencies. The output distribution over the vocabulary V at time t is pθ(xt|x<t) = softmax(gθ(h(L)t )), where g : R|h| → R|V | a pplies an affine transformation. The RNN is then trained by minimizing over its parameters $\\theta$ the sequence cross-entropy loss $\\begin{array} { r } { \\ell ( \\theta ) = - \\sum _ { t } \\log p _ { \\theta } ( x _ { t } | x _ { < t } ) } \\end{array}$ , thus maximizing the likelihood $p _ { \\theta } ( X )$ . ",
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+    {
+        "type": "text",
+        "text": "As an extension, we also consider encoder-decoder or sequence-to-sequence (Cho et al., 2014; Sutskever et al., 2014) models where given an input sequence $X$ and output sequence $Y$ of length $T _ { Y }$ , we model ",
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+    {
+        "type": "equation",
+        "img_path": "images/834c59cfefc4fe594b900da8062028f031b5e5d81427473bdf5d97f117e3da52.jpg",
+        "text": "$$\np ( Y | X ) = \\prod _ { t = 1 } ^ { T _ { Y } } p ( y _ { t } | X , y _ { < t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and minimize the loss $\\begin{array} { r } { \\ell ( \\theta ) = - \\sum _ { t } \\log p _ { \\theta } ( y _ { t } | X , y _ { < t } ) } \\end{array}$ . This setting can also be seen as conditional language modeling, and encompasses tasks such as machine translation, where $X$ is a source lan",
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+    {
+        "type": "text",
+        "text": "guage sequence and $Y$ a target language sequence, as well as language modeling, where $Y$ is the given sequence and $X$ is the empty sequence. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.2 SMOOTHING AND NOISING ",
+        "text_level": 1,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recall that for a given context length $l$ , an $n$ -gram model of order $l + 1$ is optimal under the loglikelihood criterion. Hence in the case where an RNN with finite context achieves near the lowest possible cross-entropy loss, it behaves like an $n$ -gram model. ",
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+    {
+        "type": "text",
+        "text": "Like $n$ -gram models, RNNs are trained using maximum likelihood, and can easily overfit (Zaremba et al., 2014). While generic regularization methods such $L _ { 2 }$ -regularization and dropout are effective, they do not take advantage of specific properties of sequence modeling. In order to understand sequence-specific regularization, it is helpful to examine $n$ -gram language models, whose properties are well-understood. ",
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+    {
+        "type": "text",
+        "text": "Smoothing for $n$ -gram models When modeling $p ( x _ { t } | x _ { < t } )$ , the maximum likelihood estimate $c ( x _ { < t } , x _ { t } ) \\bar { / } c ( x _ { < t } )$ based on empirical counts puts zero probability on unseen sequences, and thus smoothing is crucial for obtaining good estimates. In particular, we consider interpolation, which performs a weighted average between higher and lower order models. The idea is that when there are not enough observations of the full sequence, observations of subsequences can help us obtain better estimates.1 For example, in a bigram model, $p _ { \\mathrm { i n t e r p } } ( x _ { t } | x _ { t - 1 } ) = \\bar { \\lambda p } ( x _ { t } | x _ { t - 1 } ) + ( \\bar { 1 - \\lambda } ) p ( x _ { t } )$ , where $0 \\leq \\lambda \\leq 1$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Noising for RNN models We would like to apply well-understood smoothing methods such as interpolation to RNNs, which are also trained using maximum likelihood. Unfortunately, RNN models have no notion of counts, and we cannot directly apply one of the usual smoothing methods. In this section, we consider two simple noising schemes which we proceed to show correspond to smoothing methods. Since we can noise the data while training an RNN, we can then incorporate well-understood generative assumptions that are known to be helpful in the domain. First consider the following two noising schemes: ",
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+    {
+        "type": "text",
+        "text": "• unigram noising For each $x _ { i }$ in $x _ { < t }$ , with probability $\\gamma$ replace $x _ { i }$ with a sample from the unigram frequency distribution. • blank noising For each $x _ { i }$ in $x _ { < t }$ , with probability $\\gamma$ replace $x _ { i }$ with a placeholder token “ ”. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While blank noising can be seen as a way to avoid overfitting on specific contexts, we will see that both schemes are related to smoothing, and that unigram noising provides a path to analogues of more advanced smoothing methods. ",
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+    {
+        "type": "text",
+        "text": "3.3 NOISING AS SMOOTHING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We now consider the maximum likelihood estimate of $n$ -gram probabilities estimated using the pseudocounts of the noised data. By examining these estimates, we draw a connection between linear interpolation smoothing and noising. ",
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+    {
+        "type": "text",
+        "text": "Unigram noising as interpolation To start, we consider the simplest case of bigram probabilities. Let $c ( x )$ denote the count of a token $x$ in the original data, and let $c _ { \\gamma } ( x ) \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ \\mathbb { E } _ { \\tilde { x } } \\left[ c ( { \\tilde { x } } ) \\right]$ be the expected count of $x$ under the unigram noising scheme. We then have ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/241793d87fdc726ea54893ba7935fd412bfe09d457a908aa91f9d87a533783c2.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p _ { \\gamma } ( x _ { t } | x _ { t - 1 } ) = \\frac { c _ { \\gamma } ( x _ { t - 1 } , x _ { t } ) } { c _ { \\gamma } ( x _ { t - 1 } ) } } \\\\ & { \\qquad = [ ( 1 - \\gamma ) c ( x _ { t - 1 } , x _ { t } ) + \\gamma p ( x _ { t - 1 } ) c ( x _ { t } ) ] / c ( x _ { t - 1 } ) } \\\\ & { \\qquad = ( 1 - \\gamma ) p ( x _ { t } | x _ { t - 1 } ) + \\gamma p ( x _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $c _ { \\gamma } ( x ) = c ( x )$ since our proposal distribution $q ( x )$ is the unigram distribution, and the last line follows since $c ( x _ { t - 1 } ) / p ( x _ { t - 1 } ) = c ( x _ { t } ) / p ( x _ { t } )$ is equal to the total number of tokens in the training ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "set. Thus we see that the noised data has pseudocounts corresponding to interpolation or a mixture of different order $n$ -gram models with fixed weighting. ",
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+    {
+        "type": "text",
+        "text": "More generally, let $\\tilde { { \\boldsymbol { x } } } _ { < t }$ be noised tokens from $\\tilde { x }$ . We consider the expected prediction under noise ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/de7b60cba45caaa79680718a35bbb43c39f7887ef6a098b1853e5b18ec840038.jpg",
+        "text": "$$\n\\begin{array} { r } { p _ { \\gamma } ( x _ { t } | x _ { < t } ) = \\mathbb { E } _ { \\tilde { x } _ { < t } } \\left[ p ( x _ { t } | \\tilde { x } _ { < t } ) \\right] \\ ~ } \\\\ { = \\sum _ { J } \\underbrace { \\pi ( | J | ) } _ { p ( | J | \\mathrm { s w a p s } ) } \\ \\sum _ { x _ { K } } \\underbrace { p ( x _ { t } | x _ { J } , x _ { K } ) } _ { p ( x _ { t } | \\mathrm { n o i s e d c o n t e x t } ) } \\prod _ { z \\in x _ { K } } \\underbrace { p ( z ) } _ { p ( \\mathrm { d r a w i n g } \\ z ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the mixture coefficients are $\\pi ( | J | ) ~ = ~ ( 1 - \\gamma ) ^ { | J | } \\gamma ^ { t - 1 - | J | }$ with $\\begin{array} { r } { \\sum _ { J } \\pi ( | J | ) ~ = ~ 1 } \\end{array}$ . $J \\subseteq$ $\\{ 1 , 2 , \\ldots , t - 1 \\}$ denotes the set of indices whose corresponding tokens are left unchanged, and $K$ the set of indices that were replaced. ",
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+    {
+        "type": "text",
+        "text": "Blank noising as interpolation Next we consider the blank noising scheme and show that it corresponds to interpolation as well. This also serves as an alternative explanation for the gains that other related work have found with the “word-dropout” idea (Kumar et al., 2015; Dai & Le, 2015; Bowman et al., 2015). As before, we do not noise the token being predicted $x _ { t }$ . Let $\\tilde { { \\boldsymbol { x } } } _ { < t }$ denote the random variable where each of its tokens is replaced by “ ” with probability $\\gamma$ , and let $x J$ denote the sequence with indices $J$ unchanged, and the rest replaced by $\\underline { { \\boldsymbol { \\cdot } } } \\underline { { \\boldsymbol { \\mathit { \\Pi } } } } ,$ . To make a prediction, we use the expected probability over different noisings of the context ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3c8fc28e5495d21db5e81a7c6979205ac30377154a8f74339b29127c348490f2.jpg",
+        "text": "$$\np _ { \\gamma } ( x _ { t } | x _ { < t } ) = \\mathbb { E } _ { \\tilde { x } _ { < t } } \\left[ p ( x _ { t } | \\tilde { x } _ { < t } ) \\right] = \\sum _ { J } \\underbrace { \\pi ( | J | ) } _ { p ( | J | \\mathrm { s w a p s } ) } \\underbrace { p ( x _ { t } | x _ { J } ) } _ { p ( x _ { t } | \\mathrm { n o i s e d c o n t e x t } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $J \\subseteq \\{ 1 , 2 , \\dots , t - 1 \\}$ , which is also a mixture of the unnoised probabilities over subsequences of the current context. For example, in the case of trigrams, we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7583458ebcb163fe0982ab8ec7910e0b6b8b0de45f514d5325cf4736095f41f9.jpg",
+        "text": "$$\np _ { \\gamma } ( x _ { 3 } | x _ { 1 } , x _ { 2 } ) = \\pi ( 2 ) p ( x _ { 3 } | x _ { 1 } , x _ { 2 } ) + \\pi ( 1 ) p ( x _ { 3 } | x _ { 1 } , \\ldots ) + \\pi ( 1 ) p ( x _ { 3 } | \\ldots , x _ { 2 } ) + \\pi ( 0 ) p ( x _ { 3 } | \\ldots , \\ldots )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the mixture coefficient $\\pi ( i ) = ( 1 - \\gamma ) ^ { i } \\gamma ^ { 2 - i }$ . ",
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+    {
+        "type": "text",
+        "text": "3.4 BORROWING TECHNIQUES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With the connection between noising and smoothing in place, we now consider how we can improve the two components of the noising scheme by considering: ",
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+    {
+        "type": "text",
+        "text": "1. Adaptively computing noising probability $\\gamma$ to reflect our confidence about a particular input subsequence.   \n2. Selecting a proposal distribution $q ( x )$ that is less naive than the unigram distribution by leveraging higher order $n$ -gram statistics. ",
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+    {
+        "type": "text",
+        "text": "Noising Probability Although it simplifies analysis, there is no reason why we should choose fixed $\\gamma$ ; we now consider defining an adaptive $\\gamma ( \\boldsymbol { x } _ { 1 : t } )$ which depends on the input sequence. Consider the following bigrams: ",
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+    {
+        "type": "text",
+        "text": "“and the” ",
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+    {
+        "type": "text",
+        "text": "“Humpty Dumpty” ",
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+    },
+    {
+        "type": "text",
+        "text": "The first bigram is one of the most common in English corpora; its probability is hence well estimated and should not be interpolated with lower order distributions. In expectation, however, using fixed $\\gamma _ { 0 }$ when noising results in the same lower order interpolation weight $\\pi _ { \\gamma _ { 0 } }$ for common as well as rare bigrams. Intuitively, we should define $\\gamma ( \\boldsymbol { x } _ { 1 : t } )$ such that commonly seen bigrams are less likely to be noised. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The second bigram, “Humpty Dumpty,” is relatively uncommon, as are its constituent unigrams. However, it forms what Brown et al. (1992) term a “sticky pair”: the unigram “Dumpty” almost always follows the unigram “Humpty”, and similarly, “Humpty” almost always precedes “Dumpty”. For pairs with high mutual information, we wish to avoid backing off from the bigram to the unigram distribution. ",
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+    {
+        "type": "table",
+        "img_path": "images/e13c6137b1c48c7239d1e1784e9ec03752dd675aa62c34ce23e0d0c44644ff61.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Noised</td><td>γ(x1:2)</td><td>q(x）</td><td>Analogue</td></tr><tr><td>x1</td><td>20</td><td>q(“_&quot;)=1</td><td>interpolation</td></tr><tr><td>x1</td><td>70</td><td>unigram</td><td>interpolation</td></tr><tr><td>x1</td><td>20N1+(x1,.)/c(x1)</td><td>unigram</td><td>absolute discounting</td></tr><tr><td>x1,x2</td><td>20N1+(x1,·)/c(x1)</td><td>q(x) x Ni+(•,x)</td><td>Kneser-Ney</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Noising schemes Example noising schemes and their bigram smoothing analogues. Here we consider the bigram probability $p ( x _ { 1 } , x _ { 2 } ) \\ : = \\ : p ( x _ { 2 } \\vert x _ { 1 } ) p ( x _ { 1 } )$ . Notation: $\\gamma ( \\boldsymbol { x } _ { 1 : t } )$ denotes the noising probability for a given input sequence $x _ { 1 : t }$ , $q ( x )$ denotes the proposal distribution, and $N _ { 1 + } ( x , \\bullet )$ denotes the number of distinct bigrams in the training set where $x$ is the first unigram. In all but the last case we only noise the context $x _ { 1 }$ and not the target prediction $x _ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "Let $N _ { 1 + } ( x _ { 1 } , \\bullet ) \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ | \\{ x _ { 2 } : c ( x _ { 1 } , x _ { 2 } ) > 0 \\} |$ be the number of distinct continutions following $x _ { 1 }$ , or equivalently the number of bigram types beginning with $x _ { 1 }$ (Chen $\\&$ Goodman, 1996). From the above intuitions, we arrive at the absolute discounting noising probability ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/103e5fafdc04d38a9dbcd28816e24601025f95aa7ddfec353e823ad8955d335b.jpg",
+        "text": "$$\n\\gamma _ { \\mathrm { A D } } ( x _ { 1 } ) = \\gamma _ { 0 } \\frac { N _ { 1 + } ( x _ { 1 } , \\bullet ) } { \\sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where for $0 \\leq \\gamma _ { 0 } \\leq 1$ we have $0 \\leq \\gamma _ { \\mathrm { A D } } \\leq 1$ , though in practice we can also clip larger noising probabilities to 1. Note that this encourages noising of unigrams that precede many possible other tokens while discouraging noising of common unigrams, since if we ignore the final token, $\\begin{array} { r } { \\sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) = c ( x _ { 1 } ) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "Proposal Distribution While choosing the unigram distribution as the proposal distribution $q ( x )$ preserves unigram frequencies, by borrowing from the smoothing literature we find another distribution performs better. We again begin with two motivating examples: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "“San Francisco” ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "“New York” ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Both bigrams appear frequently in text corpora. As a direct consequence, the unigrams “Francisco” and “York” also appear frequently. However, since “Francisco” and “York” typically follow “San” and “New”, respectively, they should not have high probability in the proposal distribution as they might if we use unigram frequencies (Chen $\\&$ Goodman, 1996). Instead, it would be better to increase the proposal probability of unigrams with diverse histories, or more precisely unigrams that complete a large number of bigram types. Thus instead of drawing from the unigram distribution, we consider drawing from ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/980c4872295f29ca37308b17006c5554cd89b053a47ad326e301ec6e7b60fe18.jpg",
+        "text": "$$\nq ( x ) \\propto N _ { 1 + } ( \\bullet , x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            652,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that we now noise the prediction $x _ { t }$ in addition to the context $x _ { 1 : t - 1 }$ . Combining this new proposal distribution with the discounted $\\gamma _ { \\mathrm { A D } } ( x _ { 1 } )$ from the previous section, we obtain the noising analogue of Kneser-Ney smoothing. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the discussed noising schemes. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.5 TRAINING AND TESTING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "During training, noising is performed per batch and is done online such that each epoch of training sees a different noised version of the training data. At test time, to match the training objective we should sample multiple corrupted versions of the test data, then average the predictions (Srivastava et al., 2014). In practice, however, we find that simply using the maximum likelihood (uncorrupted) input sequence works well; evaluation runtime remains unchanged. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.6 EXTENSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The schemes described are for the language model setting. To extend them to the sequence-tosequence or encoder-decoder setting, we noise both $x _ { < t }$ as well as $y _ { < t }$ . While in the decoder we ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/433cc1e793e26806f3efe06406a2893eefd2a4b30dcac6547f09e55dd8656961.jpg",
+        "table_caption": [
+            "Table 2: Single-model perplexity on Penn Treebank with different noising schemes. We also compare to the variational method of Gal (2015), who also train LSTM models with the same hidden dimension. Note that performing Monte Carlo dropout at test time is significantly more expensive than our approach, where test time is unchanged. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Noising scheme</td><td>Validation</td><td>Test</td></tr><tr><td colspan=\"2\">Medium models (512 hidden size)</td><td></td></tr><tr><td>none (dropout only) blank</td><td>84.3</td><td>80.4</td></tr><tr><td></td><td>82.7</td><td>78.8</td></tr><tr><td>unigram</td><td>83.1</td><td>80.1</td></tr><tr><td>bigram Kneser-Ney</td><td>79.9</td><td>76.9</td></tr><tr><td colspan=\"2\">Large models (1500 hidden size)</td><td></td></tr><tr><td rowspan=\"3\">none (dropout only) blank unigram bigram Kneser-Ney</td><td>81.6</td><td>77.5</td></tr><tr><td>79.4</td><td>75.5</td></tr><tr><td>79.4 76.2</td><td>76.1 73.4</td></tr><tr><td>Zaremba et al. (2014)</td><td>82.2</td><td></td></tr><tr><td>Gal (2015) variational dropout (tied weights)</td><td></td><td>78.4</td></tr><tr><td>Gal (2015) (untied weights,Monte Carlo)</td><td>77.3 丨</td><td>75.0 73.4</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/8bf8a47b4b2c61b0c81790044b83acf9981a8eb13ac5bc06ab7e338f3f44d3b6.jpg",
+        "table_caption": [
+            "Table 3: Perplexity on Text8 with different noising schemes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Noising scheme</td><td>Validation</td><td>Test</td></tr><tr><td>none</td><td>94.3</td><td>123.6</td></tr><tr><td>blank</td><td>85.0</td><td>110.7</td></tr><tr><td>unigram</td><td>85.2</td><td>111.3</td></tr><tr><td>bigram Kneser-Ney</td><td>84.5</td><td>110.6</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "have $y _ { < t }$ and $y _ { t }$ as analogues to language model context and target prediction, it is unclear whether noising $x _ { < t }$ should be beneficial. Empirically, however, we find this to be the case (Table 4). ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "4.1 LANGUAGE MODELING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Penn Treebank We train networks for word-level language modeling on the Penn Treebank dataset, using the standard preprocessed splits with a 10K size vocabulary (Mikolov, 2012). The PTB dataset contains $9 2 9 \\mathrm { k }$ training tokens, 73k validation tokens, and ${ 8 2 } \\mathrm { k }$ test tokens. Following Zaremba et al. (2014), we use minibatches of size 20 and unroll for 35 time steps when performing backpropagation through time. All models have two hidden layers and use LSTM units. Weights are initialized uniformly in the range $[ - 0 . 1 , 0 . 1 ]$ . We consider models with hidden sizes of 512 and 1500. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We train using stochastic gradient descent with an initial learning rate of 1.0, clipping the gradient if its norm exceeds 5.0. When the validation cross entropy does not decrease after a training epoch, we halve the learning rate. We anneal the learning rate 8 times before stopping training, and pick the model with the lowest perplexity on the validation set. ",
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+    },
+    {
+        "type": "text",
+        "text": "For regularization, we apply feed-forward dropout (Pham et al., 2014) in combination with our noising schemes. We report results in Table 2 for the best setting of the dropout rate (which we find to match the settings reported in Zaremba et al. (2014)) as well as the best setting of noising ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/a74791f89bc2ae43fdc9a304f47b23fda338e2c01b613335fe659ced2c4f1495.jpg",
+        "image_caption": [
+            "Figure 1: Example training and validation curves for an unnoised model and model regularized using the bigram Kneser-Ney noising scheme. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/85a9023a7951293ef108ad65feeb0560267842bb762db52ddd9b9e963097c913.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Scheme</td><td>Perplexity</td><td>BLEU</td></tr><tr><td>dropout, no noising</td><td>8.84</td><td>24.6</td></tr><tr><td>blank noising</td><td>8.28</td><td>25.3 (+0.7)</td></tr><tr><td>unigram noising</td><td>8.15</td><td>25.5 (+0.9)</td></tr><tr><td>bigram Kneser-Ney</td><td>7.92</td><td>26.0 (+1.4)</td></tr><tr><td>source only</td><td>8.74</td><td>24.8 (+0.2)</td></tr><tr><td>target only</td><td>8.14</td><td>25.6 (+1.0)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Perplexities and BLEU scores for machine translation task. Results for bigram KN noising on only the source sequence and only the target sequence are given as well. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "probability $\\gamma _ { 0 }$ on the validation set.2 Figure 1 shows the training and validation perplexity curves for a noised versus an unnoised run. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our large models match the state-of-the-art regularization method for single model performance on this task. In particular, we find that picking $\\gamma _ { \\mathrm { A D } } ( x _ { 1 } )$ and $q ( x )$ corresponding to Kneser-Ney smoothing yields significant gains in validation perplexity, both for the medium and large size models. Recent work (Merity et al., 2016; Zilly et al., 2016) has also achieved impressive results on this task by proposing different architectures which are orthogonal to our data augmentation schemes. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Text8 In order to determine whether noising remains effective with a larger dataset, we perform experiments on the Text8 corpus3. The first 90M characters are used for training, the next 5M for validation, and the final 5M for testing, resulting in $1 5 . 3 \\mathbf { M }$ training tokens, 848K validation tokens, and 855K test tokens. We preprocess the data by mapping all words which appear 10 or fewer times to the unknown token, resulting in a 42K size vocabulary. Other parameter settings are the same as described in the Penn Treebank experiments, besides that only models with hidden size 512 are considered, and noising is not combined with feed-forward dropout. Results are given in Table 3. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
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+            758,
+            388,
+            772
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For our machine translation experiments we consider the English-German machine translation track of IWSLT $2 0 1 5 ^ { 4 }$ . The IWSLT 2015 corpus consists of sentence-aligned subtitles of TED and TEDx talks. The training set contains roughly 190K sentence pairs with 5.4M tokens. Following Luong & Manning (2015), we use TED tst2012 as a validation set and report BLEU score results (Papineni et al., 2002) on tst2014. We limit the vocabulary to the top 50K most frequent words for each language. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/c52239f55d9409031ab76a2c2d420a3bbb70c906f6f5db3a48bd4677b262db5b.jpg",
+        "image_caption": [
+            "Figure 2: Perplexity with noising on Penn Treebank while varying the value of $\\gamma _ { 0 }$ . Using discounting to scale $\\gamma _ { 0 }$ (yielding $\\gamma _ { \\mathrm { A D . } }$ ) maintains gains for a range of values of noising probability, which is not true for the unscaled case. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            103,
+            454,
+            255
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/66596b6cf7d5d61afc0773e432d36885d11d8c1b6493b1dddbc53657d86ae861.jpg",
+        "image_caption": [
+            "Figure 3: Mean KL-divergence over validation set between softmax distributions of noised and unnoised models and lower order distributions. Noised model distributions are closer to the uniform and unigram frequency distributions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We train a two-layer LSTM encoder-decoder network (Sutskever et al., 2014; Cho et al., 2014) with 512 hidden units in each layer. The decoder uses an attention mechanism (Bahdanau et al., 2014) with the dot alignment function (Luong et al., 2015). The initial learning rate is 1.0 and we start halving the learning rate when the relative difference in perplexity on the validation set between two consecutive epochs is less than $1 \\%$ . We follow training protocols as described in Sutskever et al. (2014): (a) LSTM parameters and word embeddings are initialized from a uniform distribution between $[ - 0 . 1 , 0 . 1 ]$ , (b) inputs are reversed, (c) batch size is set to 128, (d) gradient clipping is performed when the norm exceeds a threshold of 5. We set hidden unit dropout rate to 0.2 across all settings as suggested in Luong et al. (2015). We compare unigram, blank, and bigram Kneser-Ney noising. Noising rate $\\gamma$ is selected on the validation set. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results are shown in Table 4. We observe performance gains for both blank noising and unigram noising, giving roughly $+ 0 . 7$ BLEU score on the test set. The proposed bigram Kneser-Ney noising scheme gives an additional performance boost of $+ 0 . 5 – 0 . 7$ on top of the blank noising and unigram noising models, yielding a total gain of $+ 1 . 4$ BLEU. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 SCALING $\\gamma$ VIA DISCOUNTING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            672,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We now examine whether discounting has the desired effect of noising subsequences according to their uncertainty. If we consider the discounting ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/47c0323ecd79554a5f1c5395c87014c0e560225c8e275b0bb72df7cb9486da72.jpg",
+        "text": "$$\n\\gamma _ { \\mathrm { A D } } ( x _ { 1 } ) = \\gamma _ { 0 } \\frac { N _ { 1 + } ( x _ { 1 } , \\bullet ) } { c ( x _ { 1 } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "we observe that the denominator $c ( x _ { 1 } )$ can dominate than the numerator $N _ { 1 + } ( x _ { 1 } , \\bullet )$ . Common tokens are often noised infrequently when discounting is used to rescale the noising probability, while rare tokens are noised comparatively much more frequently, where in the extreme case when a token appears exactly once, we have $\\gamma _ { \\mathrm { A D } } ~ = ~ \\gamma _ { 0 }$ . Due to word frequencies following a Zipfian power law distribution, however, common tokens constitute the majority of most texts, and thus discounting leads to significantly less noising. ",
+        "bbox": [
+            173,
+            790,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare the performance of models trained with a fixed $\\gamma _ { 0 }$ versus a $\\gamma _ { 0 }$ rescaled using discounting. As shown in Figure 2, bigram discounting leads to gains in perplexity for a much broader range of $\\gamma _ { 0 }$ . Thus the discounting ratio seems to effectively capture the “right” tokens to noise. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/3aaade1cbb46aa669668f26488ab20ff05510266f5c3d019e55c0935efcf7dbc.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Noising</td><td>Bigrams</td><td>Trigrams</td></tr><tr><td>none (dropout only)</td><td>2881</td><td>381</td></tr><tr><td>blank noising</td><td>2760</td><td>372</td></tr><tr><td>unigram noising</td><td>2612</td><td>365</td></tr></table>",
+        "bbox": [
+            346,
+            101,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Perplexity of last unigram for unseen bigrams and trigrams in Penn Treebank validation set. We compare noised and unnoised models with noising probabilities chosen such that models have near-identical perplexity on full validation set. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 NOISED VERSUS UNNOISED MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            261,
+            473,
+            275
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Smoothed distributions In order to validate that data noising for RNN models has a similar effect to that of smoothing counts in $n$ -gram models, we consider three models trained with unigram noising as described in Section 4.1 on the Penn Treebank corpus with $\\gamma = 0$ (no noising), $\\gamma = 0 . 1$ , and $\\gamma = 0 . 2 5$ . Using the trained models, we measure the Kullback-Leibler divergence $D _ { \\mathrm { K L } } ( p \\Vert q ) =$ $\\textstyle \\sum _ { i } p _ { i } \\log ( p _ { i } / q _ { i } )$ over the validation set between the predicted softmax distributions, $\\hat { p }$ , and the uniform distribution as well as the unigram frequency distribution. We then take the mean KL divergence over all tokens in the validation set. ",
+        "bbox": [
+            174,
+            287,
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+            386
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Recall that in interpolation smoothing, a weighted combination of higher and lower order $n$ -gram models is used. As seen in Figure 3, the softmax distributions of noised models are significantly closer to the lower order frequency distributions than unnoised models, in particular in the case of the unigram distribution, thus validating our analysis in Section 3.3. ",
+        "bbox": [
+            174,
+            392,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Unseen $n$ -grams Smoothing is most beneficial for increasing the probability of unobserved sequences. To measure whether noising has a similar effect, we consider bigrams and trigrams in the validation set that do not appear in the training set. For these unseen bigrams (15062 occurrences) and trigrams (43051 occurrences), we measure the perplexity for noised and unnoised models with near-identical perplexity on the full set. As expected, noising yields lower perplexity for these unseen instances. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            574,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we show that data noising is effective for regularizing neural network-based sequence models. By deriving a correspondence between noising and smoothing, we are able to adapt advanced smoothing methods for $n$ -gram models to the neural network setting, thereby incorporating well-understood generative assumptions of language. Possible applications include exploring noising for improving performance in low resource settings, or examining how these techniques generalize to sequence modeling in other domains. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+            717,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Will Monroe for feedback on a draft of this paper, Anand Avati for help running experiments, and Jimmy Wu for computing support. We also thank the developers of Theano (Theano Development Team, 2016) and Tensorflow (Abadi et al., 2016). Some GPUs used in this work were donated by NVIDIA Corporation. ZX, SW, and JL were supported by an NDSEG Fellowship, NSERC PGS-D Fellowship, and Facebook Fellowship, respectively. This project was funded in part by DARPA MUSE award FA8750-15-C-0242 AFRL/RIKF. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
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+        "text_level": 1,
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+            176,
+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We provide pseudocode of the noising algorithm corresponding to bigram Kneser-Ney smoothing for $n$ -grams (In the case of sequence-to-sequence tasks, we estimate the count-based parameters separately for source and target). To simplify, we assume a batch size of one. The noising algorithm is applied to each data batch during training. No noising is applied at test time. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            190
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/ba25aa9492fc996e662fbbadff1ac3507e55932b0ceb6cf05e1b211c9da77e81.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm 1 Bigram KN noising (Language modeling setting)</td></tr><tr><td>Require counts c(x), number of distinct continuations N1+(x,·), proposal distribution q(x） N1+(.,x) Inputs X,Y batch of unnoised data indices, scaling factor γo</td></tr><tr><td>procedure NOISEBGKN(X, Y) &gt;X=(x1,...,xt),Y=(x2,...,xt+1)</td></tr><tr><td>X,Y←X,Y</td></tr><tr><td>for j = 1,...,t do γ ←γ0N1+(xj,·)/c(xj)</td></tr><tr><td>if ~ Bernoulli(~) then</td></tr><tr><td>xj~ Categorical(q) Updates X</td></tr><tr><td>yj~ Categorical(q)</td></tr><tr><td>end if</td></tr><tr><td>end for</td></tr><tr><td>return X, Y Run training iteration with noised batch</td></tr><tr><td>end procedure</td></tr></table>",
+        "bbox": [
+            173,
+            202,
+            826,
+            444
+        ],
+        "page_idx": 11
+    }
+]

parse/train/H1xPR3NtPB/H1xPR3NtPB.md

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+# ARE PRE-TRAINED LANGUAGE MODELS AWARE OFPHRASES? SIMPLE BUT STRONG BASELINES FORGRAMMAR INDUCTION
+
+Taeuk $\mathbf { K i m ^ { 1 } }$ , Jihun Choi1, Daniel Edmiston2 & Sang-goo Lee1   
+1Dept. of Computer Science and Engineering, Seoul National University, Seoul, Korea   
+2Dept. of Linguistics, University of Chicago, Chicago, IL, USA   
+{taeuk,jhchoi,sglee}@europa.snu.ac.kr, danedmiston@uchicago.edu
+
+# ABSTRACT
+
+With the recent success and popularity of pre-trained language models (LMs) in natural language processing, there has been a rise in efforts to understand their inner workings. In line with such interest, we propose a novel method that assists us in investigating the extent to which pre-trained LMs capture the syntactic notion of constituency. Our method provides an effective way of extracting constituency trees from the pre-trained LMs without training. In addition, we report intriguing findings in the induced trees, including the fact that some pre-trained LMs outperform other approaches in correctly demarcating adverb phrases in sentences.
+
+# 1 INTRODUCTION
+
+Grammar induction, which is closely related to unsupervised parsing and latent tree learning, allows one to associate syntactic trees, i.e., constituency and dependency trees, with sentences. As grammar induction essentially assumes no supervision from gold-standard syntactic trees, the existing approaches for this task mainly rely on unsupervised objectives, such as language modeling (Shen et al., 2018b; 2019; Kim et al., 2019a;b) and cloze-style word prediction (Drozdov et al., 2019) to train their task-oriented models. On the other hand, there is a trend in the natural language processing (NLP) community of leveraging pre-trained language models (LMs), e.g., ELMo (Peters et al., 2018) and BERT (Devlin et al., 2019), as a means of acquiring contextualized word representations. These representations have proven to be surprisingly effective, playing key roles in recent improvements in various models for diverse NLP tasks.
+
+In this paper, inspired by the fact that the training objectives of both the approaches for grammar induction and for training LMs are identical, namely, (masked) language modeling, we investigate whether pre-trained LMs can also be utilized for grammar induction/unsupervised parsing, especially without training. Specifically, we focus on extracting constituency trees from pre-trained LMs without fine-tuning or introducing another task-specific module, at least one of which is usually required in other cases where representations from pre-trained LMs are employed. This restriction provides us with some advantages: (1) it enables us to derive strong baselines for grammar induction with reduced time and space complexity, offering a chance to reexamine the current status of existing grammar induction methods, (2) it facilitates an analysis on how much and what kind of syntactic information each pre-trained LM contains in its intermediate representations and attention distributions in terms of phrase-structure grammar, and (3) it allows us to easily inject biases into our framework, for instance, to encourage the right-skewness of the induced trees, resulting in performance gains in English unsupervised parsing.
+
+First, we briefly mention related work (§2). Then, we introduce the intuition behind our proposal in detail (§3), which is motivated by our observation that we can cluster words in a sentence according to the similarity of their attention distributions over words in the sentence. Based on this intuition, we define a straightforward yet effective method $( \ S 4 )$ of drawing constituency trees directly from pretrained LMs with no fine-tuning or addition of task-specific parts, instead resorting to the concept of Syntactic Distance (Shen et al., 2018a;b). Then, we conduct experiments (§5) on the induced constituency trees, discovering some intriguing phenomena. Moreover, we analyze the pre-trained
+
+LMs and constituency trees from various points of view, including looking into which layer(s) of the LMs is considered to be sensitive to phrase information (§6).
+
+To summarize, our contributions in this work are as follows:
+
+• By investigating the attention distributions from Transformer-based pre-trained LMs, we show that there is evidence to suggest that several attention heads of the LMs exhibit syntactic structure akin to constituency grammar. Inspired by the above observation, we propose a method that facilitates the derivation of constituency trees from pre-trained LMs without training. We also demonstrate that the induced trees can serve as a strong baseline for English grammar induction. We inspect, in view of our framework, what type of syntactic knowledge the pre-trained LMs capture, discovering interesting facts, e.g., that some pre-trained LMs are more aware of adverb phrases than other approaches.
+
+# 2 RELATED WORK
+
+Grammar induction is a task whose goal is to infer from sequential data grammars which generalize, and are able to account for unseen data (Lari & Young (1990); Clark (2001); Klein & Manning (2002; 2004), to name a few). Traditionally, this was done by learning explicit grammar rules (e.g., context free rewrite rules), though more recent methods employ neural networks to learn such rules implicitly, focusing more on the induced grammars’ ability to generate or parse sequences.
+
+Specifically, Shen et al. (2018b) proposed Parsing-Reading-Predict Network (PRPN) where the concept of Syntactic Distance is first introduced. They devised a neural model for language modeling where the model is encouraged to recognize syntactic structure. The authors also probed the possibility of inducing constituency trees without access to gold-standard trees by adopting an algorithm that recursively splits a sequence of words into two parts, the split point being determined according to correlated syntactic distances; the point having the biggest distance becomes the first target of division. Shen et al. (2019) presented a model called Ordered Neurons (ON), which is a revised version of LSTM (Long Short-Term Memory, Hochreiter & Schmidhuber (1997)) which reflects the hierarchical biases of natural language and can be used to compute syntactic distances. Shen et al. (2018a) trained a supervised parser relying on the concept of syntactic distance.
+
+Other studies include Drozdov et al. (2019), who trained deep inside-outside recursive autoencoders (DIORA) to derive syntactic trees in an exhaustive way with the aid of the inside-outside algorithm, and Kim et al. (2019a) who proposed Compound Probabilistic Context-Free Grammars (compound PCFG), showing that neural PCFG models are capable of producing promising unsupervised parsing results. Li et al. (2019) proved that an ensemble of unsupervised parsing models can be beneficial, while Shi et al. (2019) utilized additional training signals from pictures related with input text. Dyer et al. (2016) proposed Recurrent Neural Network Grammars (RNNG) for both language modeling and parsing, and Kim et al. (2019b) suggested an unsupervised variant of the RNNG. There also exists another line of research on task-specific latent tree learning (Yogatama et al., 2017; Choi et al., 2018; Havrylov et al., 2019; Maillard et al., 2019). The goal here is not to construct linguistically plausible trees, but to induce trees fitted to improving target performance. Naturally, the induced performance-based trees need not resemble linguistically plausible trees, and some studies (Williams et al., 2018a; Nangia & Bowman, 2018) examined the apparent fact that performance-based and lingusitically plausible trees bear little resemblance to one another.
+
+Concerning pre-trained language models (Peters et al. (2018); Devlin et al. (2019); Radford et al. (2019); Yang et al. (2019); Liu et al. (2019b), inter alia)—particularly those employing a Transformer architecture (Vaswani et al., 2017)—these have proven to be helpful for diverse NLP downstream tasks. In spite of this, there is no vivid picture for explaining what particular factors contribute to performance gains, even though some recent work has attempted to shed light on this question. In detail, one group of studies (Raganato & Tiedemann (2018); Clark et al. (2019); Ethayarajh (2019); Hao et al. (2019); Voita et al. (2019), inter alia) has focused on dissecting the intermediate representations and attention distributions of the pre-trained LMs, while the another group of publications (Marecek & Rosa (2018); Goldberg (2019); Hewitt & Manning (2019); Liu et al. (2019a); Rosa & ˇ Marecek (2019), to name a few) delve into the question of the existence of syntactic knowledge in ˇ
+
+![](images/6b59545393c8be32c7e4710ab90ef69ecbf77cabd130491c54d540a4d216ac52.jpg)  
+Figure 1: Self-attention heatmaps from two different pre-trained LMs. (Left) A heatmap for the average of attention distributions from the 7th layer of the XLNet-base (Yang et al., 2019) model given the sample sentence. (Right) A heatmap for the average of attention distributions from the 9th layer of the BERT-base (Devlin et al., 2019) model given another sample sentence. We can easily spot the chunks of words on the two heatmaps that are correlated with the constituents of the input sentences, e.g., (Left) ‘the price of plastics’, ‘took off in 1987’, ‘Quantum Chemical Corp.’, (Right) ‘when price increases can be sustained’, and ‘he remarks’.
+
+Transformer-based models. Particularly, Marecek & Rosa (2019) proposed an algorithm for extract- ˇ ing constituency trees from Transformers trained for machine translation, which is similar to our approach.
+
+# 3 MOTIVATION
+
+As pioneers in the literature have pointed out, the multi-head self-attention mechanism (Vaswani et al., 2017) is a key component in Transformer-based language models, and it seems this mechanism empowers the models to capture certain semantic and syntactic information existing in natural language. Among a diverse set of knowledge they may capture, in this work we concentrate on phrase-structure grammar by seeking to extract constituency trees directly from their attention information and intermediate weights.
+
+In preliminary experiments, where we manually visualize and investigate the intermediate representations and attention distributions of several pre-trained LMs given input, we have found some evidence which suggests that the pre-trained LMs exhibit syntactic structure akin to constituency grammar in some degree. Specifically, we have noticed some patterns which are often displayed in self-attention heatmaps as explicit horizontal lines, or groups of rectangles of various sizes. As an attention distribution of a word in an input sentence corresponds to a row in a heatmap matrix, we can say that the appearance of these patterns indicates the existence of groups of words where the attention distributions of the words in the same group are relatively similar. Interestingly, we have also discovered the fact that the groups of words we observed are fairly correlated with the constituents of the input sentence, as shown in Figure 1 (above) and Figure 3 (in Appendix A.1).
+
+Even though we have identified some patterns which match with the constituents of sentences, it is not enough to conclude that the pre-trained LMs are aware of syntactic phrases as found in phrasestructure grammars. To demonstrate the claim, we attempt to obtain constituency trees in a zero-shot learning fashion, relying only on the knowledge from the pre-trained LMs. To this end, we suggest the following, inspired from our finding: two words in a sentence are syntactically close to each other (i.e., the two words belong to the same constituent) if their attention distributions over words in the sentence are also close to each other. Note that this implicitly presumes that each word is more likely to attend more on the words in the same constituent to enrich its representation in the pre-trained LMs. Finally, we utilize the assumption to compute syntactic distances between each pair of adjacent words in a sentence, from which the corresponding constituency tree can be built.
+
+# 4 PROPOSED METHOD
+
+# 4.1 SYNTACTIC DISTANCE AND TREE CONSTRUCTION
+
+We leverage the concept of Syntactic Distance proposed by Shen et al. (2018a;b) to draw constituency trees from raw sentences in an intuitive way. Formally, given a sequence of words in a sentence, $w _ { 1 } , w _ { 2 } , \ldots , w _ { n }$ , we compute $\mathbf { d } = [ d _ { 1 } , d _ { 2 } , \dots , d _ { n - 1 } ]$ where $d _ { i }$ corresponds to the syntactic distance between $w _ { i }$ and $w _ { i + 1 }$ . Each $d _ { i }$ is defined as follows:
+
+$$
+d _ { i } = f ( g ( w _ { i } ) , g ( w _ { i + 1 } ) ) ,
+$$
+
+where $f ( \cdot , \cdot )$ and $g ( \cdot )$ are a distance measure function and representation extractor function, respectively. The function $g$ converts each word into the corresponding vector representation, while $f$ computes the syntactic distance between the two words given their representations. Once $\mathbf { d }$ is derived, it can be easily converted into the target constituency tree by a simple algorithm following Shen et al. (2018a).1 For details of the algorithm, we refer the reader to Appendix A.2.
+
+Although previous studies attempted to explicitly train the functions $f$ and $g$ with supervision (with access to gold-standard trees, Shen et al. (2018a)) or to obtain them as a by-product of training particular models that are carefully designed to recognize syntactic information (Shen et al., 2018b; 2019), in this work we stick to simple distance metric functions for $f$ and pre-trained LMs for $g$ , forgoing any training process. In other words, we focus on investigating the possibility of pre-trained LMs possessing constituency information in a form that can be readily extracted with straightforward computations. If the trees induced by the syntactic distances derived from the pre-trained LMs are similar enough to gold-standard syntax trees, we can reasonably claim that the LMs resemble phrase-structure.
+
+# 4.2 PRE-TRAINED LANGUAGE MODELS
+
+We consider four types of recently proposed language models. These are: BERT (Devlin et al., 2019), GPT-2 (Radford et al., 2019), RoBERTa (Liu et al., 2019b), and XLNet (Yang et al., 2019). They all have in common that they are based on the Transformer architecture and have been proven to be effective in natural language understanding (Wang et al., 2019) or generation. We handle two variants for each LM, varying in the number of layers, attention heads, and hidden dimensions, resulting in eight different cases in total. In particular, each LM has two variants. (1) base: consists of $l { = } 1 2$ layers, $a { = } 1 2$ attention heads, and $d { = } 7 6 8$ hidden dimensions, while (2) large: has $l { = } 2 4$ layers, $a { = } 1 6$ attention heads, and $d { = } 1 0 2 4$ hidden dimensions.2 We deal with a wide range of pre-trained LMs, unlike previous work which has mostly analyzed a specific model, particularly BERT. For details about each LM, we refer readers to the respective original papers.
+
+In terms of our formulation, each LM instance provides two categories of representation extractor functions, $G ^ { v }$ and $G ^ { d }$ . Specifically, $G ^ { v }$ refers to a set of functions $\{ g _ { j } ^ { v } | j = 1 , \ldots , l \}$ , each of which simply outputs the intermediate hidden representation of a given word on the $j$ th layer of the LM. Likewise, $\dot { G } ^ { d }$ is a set of functions $\{ g _ { ( j , k ) } ^ { d } | \bar { j } = 1 , \ldots , l , k = 1 , \ldots , a + 1 \}$ , each of which outputs the attention distribution of an input word by the $k$ th attention head on the $j$ th layer of the LM. Even though our main motivation comes from the self-attention mechanism, we also deal with the intermediate hidden representations present in the pre-trained LMs by introducing $G ^ { v }$ , considering that the hidden representations serve as storage of collective information taken from the processing of the pre-trained LMs. Note that $k$ ranges up to $a + 1$ , not $a$ , implying that we consider the average of all attention distributions on the same layer in addition to the individual ones. This averaging function can be regarded as an ensemble of other functions in the layer which are specialized for different aspects of information, and we expect that this technique will provide a better option in some cases as reported in previous work (Li et al., 2019).
+
+One remaining issue is that all the pre-trained LMs we use regard each input sentence as a sequence of subword tokens, while our formulation assumes words cannot be further divided into smaller tokens. To resolve this difference, we tested certain heuristics that guide how subword tokens for a complete word should be exploited to represent the word, and we have empirically found that the best result comes when each word is represented by an average of the representations of its subwords.3 Therefore, we adopt the above heuristic in this work for cases where a word is tokenized into more than two parts.
+
+# 4.3 DISTANCE MEASURE FUNCTIONS
+
+For the distance measure function $f$ , we prepare three options $( F ^ { v } )$ for $G ^ { v }$ and two options $( F ^ { d } )$ for $G ^ { d }$ . Formally, $f \in F ^ { v } \cup F ^ { d }$ , where $F ^ { v } = \{ \mathbf { C O S } , \mathbf { L 1 } , \mathbf { L 2 } \}$ , $F ^ { d } = \{ \mathrm { J S D } , \mathrm { H E L } \}$ . COS, L1, L2, JSD, and HEL correspond to Cosine, L1, and L2, Jensen-Shannon, and Hellinger distance respectively. The functions in $F ^ { v }$ are only compatible with the elements of $G ^ { v }$ , and the same holds for $F ^ { d }$ and $G ^ { d }$ . The exact definition of each function is listed in Appendix A.3.
+
+# 4.4 INJECTING BIAS INTO SYNTACTIC DISTANCES
+
+One of the main advantages we obtain by leveraging syntactic distances to derive parse trees is that we can easily inject inductive bias into our framework by simply modifying the values of the syntactic distances. Hence, we investigate whether the extracted trees from our method can be further refined with the aid of additional biases. To this end, we introduce a well-known bias for English constituency trees—the right-skewness bias—in a simple linear form.4 Namely, our intention is to influence the induced trees such that they are moderately right-skewed following the nature of gold-standard parse trees in English.
+
+Formally, we compute $\hat { d } _ { i }$ by appending the following linear bias term to every $d _ { i }$
+
+$$
+\hat { d } _ { i } = d _ { i } + \lambda \cdot \mathrm { A V G } ( \mathbf { d } ) \times ( 1 - 1 / ( m - 1 ) \times ( i - 1 ) ) ,
+$$
+
+where $\operatorname { A V G } ( \cdot )$ outputs an average of all elements in a vector, $\lambda$ is a hyperparameter, and $i$ ranges from 1 to $m = n - 1$ . We write $\hat { \mathbf { d } } = [ \hat { d } _ { 1 } , \hat { d } _ { 2 } , \dots , \hat { d } _ { m } ]$ in place of $\mathbf { d }$ to signify biased syntactic distances.
+
+The main purpose of introducing such a bias is examining what changes are made to the resulting tree structures rather than boosting quantitative performance per $s e$ , though it is of note that it serves this purpose as well. We believe that this additional consideration is necessary based on two points. First, English is what is known as a head-initial language. That is, given a selector and argument, the selector has a strong tendency to appear on the left, e.g., ‘eat food’, or ‘to Canada’. Head-initial languages therefore have an in-built preference for right-branching structures. By adjusting the bias injected into syntactic distances derived from pre-trained LMs, we can figure out whether the LMs are capable of inducing the right-branching bias, which is one of the main properties of English syntax; if injecting the bias does not influence the performance of the LMs on unsupervised parsing, we can conjecture they are inherently capturing the bias to some extent. Second, as mentioned before, we have witnessed some previous work (Shen et al., 2018b; 2019; Htut et al., 2018; Li et al., 2019; Shi et al., 2019) where the right-skewness bias is implicitly exploited, although it could be regarded as not ideal. What we intend to focus on is the question about which benefits the bias provides for such parsing models, leading to overall performance improvements. In other words, we look for what the exact contribution of the bias is when it is injected into grammar induction models, by explicitly controlling the bias using our framework.
+
+# 5 EXPERIMENTS
+
+# 5.1 GENERAL SETTINGS
+
+# 5.1.1 DATASETS
+
+In this section, we conduct unsupervised constituency parsing on two datasets. The first dataset is WSJ Penn Treebank (PTB, Marcus et al. (1993)), in which human-annotated gold-standard trees are available. We use the standard split of the dataset—2-21 for training, 22 for validation, and 23 for test. The second one is MNLI (Williams et al., 2018b), which is originally designed to test natural language inference but often utilized as a means of evaluating parsers. It contains constituency trees produced by an external parser (Klein & Manning, 2003). We leverage the union of two different versions of the MNLI development set as test data following convention (Htut et al., 2018; Drozdov et al., 2019), and we call it the MNLI test set in this paper. Moreover, we randomly sample 40K sentences from the training set of the MNLI to utilize them as a validation set. To preprocess the datasets, we follow the setting of Kim et al. (2019a) with the minor exceptions that words are not lower-cased and number characters are preserved instead of being substituted by a special character.
+
+# 5.1.2 IMPLEMENTATION DETAILS
+
+For implementation, to compare pre-trained LMs in an unified manner, we resort to an integrated PyTorch codebase that supports all the models we consider.5 For each LM, we tune the best combination of $f$ and $g$ functions using the validation set. Then, we derive a set of $\mathbf { d }$ for sentences in the test set using the chosen functions, followed by the resulting constituency trees converted from each d by the tree construction algorithm in Section 4.1. In addition to sentence-level F1 (S-F1) score, we report label recall scores for six main categories: SBAR, NP, VP, PP, ADJP, and ADVP. We also present the results of utilizing $\hat { \mathbf { d } }$ instead of $\mathbf { d }$ , empirically setting the bias hyperparameter $\lambda$ as 1.5. We do not fine-tune the LMs on domain-specific data, as we here focus on finding their universal characteristics.
+
+We take four na¨ıve baselines into account, random (averaged over 5 trials), balanced, left-branching, and right-branching binary trees. In addition, we present two more baselines which are identical to our models except that their $g$ functions are based on a randomly initialized XLNet-base rather than pre-trained ones. To be concrete, We provide ‘Random XLNet-base $( F ^ { v } ) ^ { : }$ ’ which applies the functions in $F ^ { v }$ on random hidden representations and ‘Random XLNet-base $( F ^ { d } )$ ’ that utilizes the functions in $F ^ { d }$ and random attention distributions, respectively. Considering the randomness of initialization and possible choices for $f$ , the final score for each of the baselines is calculated as an average over 5 trials of each possible $f$ , i.e., an average over $5 \times 3$ runs in case of $F ^ { v }$ and $5 \times 2$ runs for $F ^ { d }$ . These baselines enable us to estimate the exact advantage we obtain by pre-training LMs, effectively removing additional unexpected gains that may exist. Furthermore, we compare our parse trees against ones from existing grammar induction models.
+
+All scripts used in our experiments will be publicly available for reproduction and further analysis.6
+
+# 5.2 EXPERIMENTAL RESULTS ON PTB
+
+In Table 1, we report the results of the various models on the PTB test set. First of all, our method combined with pre-trained LMs shows competitive or comparable results in terms of S-F1 even without the right-skewness bias. This result implies that the extracted trees from our method can be regarded as a baseline for English grammar induction. Moreover, pre-trained LMs show substantial improvements over Random Transformers (XLNet-base), demonstrating that training language models on large corpora, in fact, enables the LMs to be more aware of syntactic information.
+
+When the right-skewness bias is applied to syntactic distances derived from pre-trained LMs, the S-F1 scores of the LMs increase by up to ten percentage points. This improvement indicates that the pre-trained LMs do not properly capture the largely right-branching nature of English syntax, at least when observed through the lens of our framework. By explicitly controlling the bias through our framework and observing the performance gap between our models with and without the bias, we confirm that the main contribution of the bias comes from its capability to capture subordinate clauses (SBAR) and verb phrases (VP). This observation provides a hint for what some previous work on unsupervised parsing desired to obtain by introducing the bias to their models. It is intriguing to see that all of the existing grammar induction models are inferior to the right-branching baseline in recognizing SBAR and VP (although some of them already utilized the right-skewness bias), implying that the same problem—models do not properly capture the right-branching nature— may also exist in current grammar induction models. One possible assumption is that the models do not need the bias to perform well in language modeling, although future work should provide a rigorous analysis about the phenomenon.
+
+Table 1: Results on the PTB test set. Bold numbers correspond to the top 3 results for each column. L: layer number, A: attention head number (AVG: the average of all attentions). $\dagger$ : Results reported by Kim et al. (2019a). $\ddagger$ : Approaches in which COO parser is utilized.   
+
+<table><tr><td>Model</td><td>f</td><td>L</td><td>A</td><td>S-F1</td><td>SBAR</td><td>NP</td><td>VP</td><td>PP</td><td>ADJP</td><td>ADVP</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random Trees</td><td></td><td>=</td><td></td><td>18.1</td><td>8%</td><td>23%</td><td>12%</td><td>18%</td><td>23%</td><td>28%</td></tr><tr><td>Balanced Trees</td><td></td><td>=</td><td>=</td><td>18.5</td><td>7%</td><td>27%</td><td>8%</td><td>18%</td><td>27%</td><td>25%</td></tr><tr><td>Left Branching Trees</td><td></td><td></td><td></td><td>8.7</td><td>5%</td><td>11%</td><td>0%</td><td>5%</td><td>2%</td><td>8%</td></tr><tr><td>Right Branching Trees</td><td></td><td></td><td></td><td>39.4</td><td>68%</td><td>24%</td><td>71%</td><td>42%</td><td>27%</td><td>38%</td></tr><tr><td>Random XLNet-base (F)</td><td></td><td></td><td></td><td>19.6</td><td>9%</td><td>26%</td><td>12%</td><td>20%</td><td>23%</td><td>24%</td></tr><tr><td>Random XLNet-base (Fd)</td><td></td><td></td><td></td><td>20.1</td><td>11%</td><td>25%</td><td>14%</td><td>19%</td><td>22%</td><td>26%</td></tr><tr><td>Pre-trainedLMs (w/o bias)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>JSD</td><td>9</td><td>AVG</td><td>32.4</td><td>28%</td><td>42%</td><td>28%</td><td>31%</td><td>35%</td><td>63%</td></tr><tr><td>BERT-large</td><td>HEL</td><td>17</td><td>AVG</td><td>34.2</td><td>34%</td><td>43%</td><td>27%</td><td>39%</td><td>37%</td><td>57%</td></tr><tr><td>GPT2</td><td>JSD</td><td>9</td><td>1</td><td>37.1</td><td>32%</td><td>47%</td><td>27%</td><td>55%</td><td>27%</td><td>36%</td></tr><tr><td>GPT2-medium</td><td>JSD</td><td>10</td><td>13</td><td>39.4</td><td>41%</td><td>51%</td><td>21%</td><td>67%</td><td>33%</td><td>44%</td></tr><tr><td>RoBERTa-base</td><td>JSD</td><td>9</td><td>4</td><td>33.8</td><td>40%</td><td>38%</td><td>33%</td><td>43%</td><td>42%</td><td>57%</td></tr><tr><td>RoBERTa-large</td><td>JSD</td><td>14</td><td>5</td><td>34.1</td><td>29%</td><td>46%</td><td>30%</td><td>37%</td><td>28%</td><td>40%</td></tr><tr><td>XLNet-base</td><td>HEL</td><td>9</td><td>AVG</td><td>40.1</td><td>35%</td><td>56%</td><td>26%</td><td>38%</td><td>47%</td><td>68%</td></tr><tr><td>XLNet-large</td><td>L2</td><td>11</td><td>-</td><td>38.1</td><td>36%</td><td>51%</td><td>26%</td><td>41%</td><td>45%</td><td>69%</td></tr><tr><td>Pre-trainedLMs (w/bias入=1.5)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>HEL</td><td>9</td><td>AVG</td><td>42.3</td><td>45%</td><td>46%</td><td>49%</td><td>43%</td><td>41%</td><td>65%</td></tr><tr><td>BERT-large</td><td>HEL</td><td>17</td><td>AVG</td><td>44.4</td><td>55%</td><td>48%</td><td>48%</td><td>52%</td><td>41%</td><td>62%</td></tr><tr><td>GPT2</td><td>JSD</td><td>9</td><td>1</td><td>41.3</td><td>43%</td><td>49%</td><td>38%</td><td>58%</td><td>27%</td><td>43%</td></tr><tr><td>GPT2-medium</td><td>HEL</td><td>2</td><td>1</td><td>42.3</td><td>54%</td><td>50%</td><td>39%</td><td>56%</td><td>24%</td><td>41%</td></tr><tr><td>RoBERTa-base</td><td>JSD</td><td>8</td><td>AVG</td><td>42.1</td><td>51%</td><td>44%</td><td>44%</td><td>55%</td><td>40%</td><td>66%</td></tr><tr><td>RoBERTa-large</td><td>JSD</td><td>12</td><td>AVG</td><td>42.3</td><td>40%</td><td>50%</td><td>43%</td><td>44%</td><td>48%</td><td>56%</td></tr><tr><td>XLNet-base</td><td>HEL</td><td>7</td><td>AVG</td><td>48.3</td><td>62%</td><td>53%</td><td>50%</td><td>58%</td><td>49%</td><td>74%</td></tr><tr><td>XLNet-large</td><td>HEL</td><td>11</td><td>AVG</td><td>46.7</td><td>57%</td><td>50%</td><td>54%</td><td>50%</td><td>57%</td><td>73%</td></tr><tr><td>Other models</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PRPN(tuned)† ‡</td><td></td><td></td><td>=</td><td>47.3</td><td>50%</td><td>59%</td><td>46%</td><td>57%</td><td>44%</td><td>32%</td></tr><tr><td>ON(tuned)† ‡</td><td></td><td></td><td></td><td>48.1</td><td>51%</td><td>64%</td><td>41%</td><td>54%</td><td>38%</td><td>31%</td></tr><tr><td>Neural PCFG†</td><td></td><td></td><td></td><td>50.8</td><td>52%</td><td>71%</td><td>33%</td><td>58%</td><td>32%</td><td>45%</td></tr><tr><td>Compound PCFG+</td><td>=</td><td>=</td><td>=</td><td>55.2</td><td>56%</td><td>74%</td><td>41%</td><td>68%</td><td>40%</td><td>52%</td></tr></table>
+
+On the other hand, the existing models show exceptionally high recall scores on noun phrases (NP), even though our pre-trained LMs also have success to some extent in capturing noun phrases compared to na¨ıve baselines. From this, we conjecture that neural models trained with a language modeling objective become largely equipped with the ability to understand the concept of NP. In contrast, the pre-trained LMs record the best recall scores on adjective and adverb phrases (ADJP and ADVP), suggesting that the LMs and existing models capture disparate aspects of English syntax to differing degrees. To further explain why some pre-trained LMs are good at capturing ADJPs and ADVPs, we manually investigated the attention heatmaps of the sentences that contain ADJPs or ADVPs. From the inspection, we empirically found that there are some keywords—including ‘two’, ‘ago’, ‘too’, and ‘far’—which have different patterns of attention distributions compared to those of their neighbors and that these keywords can be a clue for our framework to recognize the existence of ADJPs or ADJPs. It is also worth mentioning that ADJPs and ADVPs consist of a relatively smaller number of words than those of SBAR and VP, indicating that the LMs combined with our method have strength in correctly finding small chunks of words, i.e., low-level phrases.
+
+Meanwhile, in comparison with other LM models, GPT-2 and XLNet based models demonstrate their effectiveness and robustness in unsupervised parsing. Particularly, the XLNet-base model serves as a robust baseline achieving the top performance among LM candidates. One plausible explanation for this outcome is that the training objective of XLNet, which considers both autoencoding (AE) and autoregressive (AR) features, might encourage the model to be better aware of phrase structure than other LMs. Another possible hypothesis is that AR objective functions (e.g., typical language modeling) are more effective in training syntax-aware neural models than AE objectives (e.g., masked language modeling), as both GPT-2 and XLNet are pre-trained on AR variants. However, it is hard to conclude what factors contribute to their high performance at this stage.
+
+![](images/d982602134138e02110915208629d6ca404a31eb2399467fa65e34bb481667ef.jpg)  
+Figure 2: The best layer-wise S-F1 scores of each LM instance on the PTB test set. (Left) The performance of the X-‘base’ models. (Right) The performance of the X-‘large’ models.
+
+Interestingly, there is an obvious trend that the functions in $F ^ { d } .$ —the distance measure functions for attention distributions—lead most of the LM instances to the best parsing results, indicating that deriving parse trees from attention information can be more compact and efficient than extracting them from the LMs’ intermediate representations, which should contain linguistic knowledge beyond phrase structure. In addition, the results in Table 1 show that large parameterizations of the LMs generally increase their parsing performance, but this improvement is not always guaranteed. Meanwhile, as we expected in Section 4.2 and as seen in the $\mathbf { \^ A }$ (attention head number) column of Table 1, the average of attention distributions in the same layer often provides better results than individual attention distributions.
+
+# 5.3 EXPERIMENTAL RESULTS ON MNLI
+
+We present the results of various models on the MNLI test set in Table 3 of Appendix A.5. We observe trends in the results which mainly coincide with those of the PTB dataset. Particularly, (1) right-branching trees are strong baselines for the task, especially showing their strengths in capturing SBAR and VP clauses/phrases, (2) our method resorting to the LM instances is also comparable to the right-branching trees, demonstrating its superiority in recognizing different aspects of phrase categories including prepositional phrases (PP) and adverb phrases (ADVP), and (3) attention distributions seem more effective for distilling the phrase structures of sentences than intermediate representations.
+
+However, there are some issues worth mentioning. First, the right-branching baseline seems to be even stronger in the case of MNLI, recording a score of over 50 in sentence-level F1. We conjecture that this result comes principally from two reasons: (1) the average length of sentences in MNLI is much shorter than in PTB, giving a disproportionate advantage to na¨ıve baselines, and (2) our data preprocessing, which follows Kim et al. (2019a), removes all punctuation marks, unlike previous work (Htut et al., 2018; Drozdov et al., 2019), leading to an unexpected advantage for the rightbranching scheme. Moreover, it deserves to consider the fact that the gold-standard parse trees in MNLI are not human-annotated, rather automatically generated.
+
+Second, in terms of consistency in identifying the best choice of $f$ and $g$ for each LM, we observe that most of the best combinations of $f$ and $g$ tuned for PTB do not correspond well to the best ones for MNLI. Does this observation imply that a specific combination of these functions and the resulting performance do not generalize well across different data domains? To clarify, we manually investigated the performance of some combinations of $f$ and $g$ , which are tuned on PTB but tested on MNLI instead. As a result, we discover that particular combinations of $f$ and $g$ which are good at PTB are also competitive on MNLI, even though they fail to record the best scores on MNLI.
+
+Table 2: Results of training a pseudo-optimum $f _ { \mathrm { i d e a l } }$ with PTB and XLNet-base model.   
+
+<table><tr><td>Model</td><td>f</td><td>L</td><td>A</td><td>S-F1</td><td>SBAR</td><td>NP</td><td>VP</td><td>PP</td><td>ADJP</td><td>ADVP</td></tr><tr><td>Baselines (from Table1)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random XLNet-base (F)</td><td>1</td><td></td><td>-</td><td>19.6</td><td>9%</td><td>26%</td><td>12%</td><td>20%</td><td>23%</td><td>24%</td></tr><tr><td>Random XLNet-base (Fd)</td><td></td><td>1</td><td>-</td><td>20.1</td><td>11%</td><td>25%</td><td>14%</td><td>19%</td><td>22%</td><td>26%</td></tr><tr><td>XLNet-base (入=0)</td><td>JSD</td><td>9</td><td>AVG</td><td>40.1</td><td>35%</td><td>56%</td><td>26%</td><td>38%</td><td>47%</td><td>68%</td></tr><tr><td>XLNet-base (入=1.5)</td><td>HEL</td><td>7</td><td>AVG</td><td>48.3</td><td>62%</td><td>53%</td><td>50%</td><td>58%</td><td>49%</td><td>74%</td></tr><tr><td>Trained models(w/gold trees)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random XLNet-base</td><td>fideal</td><td>-</td><td>=</td><td>41.2</td><td>28%</td><td>58%</td><td>29%</td><td>50%</td><td>35%</td><td>41%</td></tr><tr><td>XLNet-base (worst case)</td><td>fideal</td><td>1</td><td>-</td><td>58.0</td><td>47%</td><td>75%</td><td>56%</td><td>71%</td><td>50%</td><td>61%</td></tr><tr><td>XLNet-base (best case)</td><td>fideal</td><td>7</td><td>=</td><td>65.1</td><td>61%</td><td>82%</td><td>67%</td><td>78%</td><td>55%</td><td>73%</td></tr></table>
+
+Concretely, the union of $f ^ { d } ( \mathrm { J S D } )$ and $g _ { ( 9 , 1 3 ) } ^ { d }$ —the best duo for the XLNet-base on PTB—achieves 39.2 in sentence-level F1 on MNLI, which is very close to the top performance (39.3) we can obtain when leveraging the XLNet-base. It is also worth noting that GPT-2 and XLNet are efficient in capturing PP and ADVP respectively, regardless of the data domain and the choice of $f$ and $g$ .
+
+# 6 FURTHER ANALYSIS
+
+# 6.1 PERFORMANCE COMPARISON BY LAYER
+
+To take a closer look at how different the layers of the pre-trained LMs are in terms of parsing performance, we retrieve the best sentence-level F1 scores from the lth layer of an LM from all combinations of $f$ and $g _ { l }$ , with regard to the PTB and MNLI respectively. Then we plot the scores as graphs in Figure 2 for the PTB and Figure 4 in Appendix A.4 for the MNLI. Each score is from the models to which the bias is not applied.
+
+From the graphs, we observe several patterns. First, XLNet-based models outperform other competitors across most of the layers. Second, the best outcomes are largely shown in the middle layers of the LMs akin to the observation from Shen et al. (2019), except for some cases where the first layers (especially in case of MNLI) record the best. Interestingly, GPT-2 shows a decreasing trend in its output values as the layer becomes high, while other models generally exhibit the opposite pattern. Moreover, we discover from raw statistics that regardless of the choice of $f$ and $g _ { l }$ , the parsing performance reported as S-F1 is moderately correlated with the layer number l. In other words, it seems that there are some particular layers in the LMs which are more sensitive to syntactic information.
+
+# 6.2 ESTIMATING THE UPPER LIMIT OF DISTANCE MEASURE FUNCTIONS
+
+Although we have introduced effective candidates for $f$ , we explore the potential of extracting more sophisticated trees from pre-trained LMs, supposing we are equipped with a pseudo-optimum $f$ , call it $f _ { \mathrm { i d e a l } }$ . To obtain $f _ { \mathrm { i d e a l } }$ , we train a simple linear layer on each layer of the pre-trained LMs with supervision from the gold-standard trees of the PTB training set, while $g$ remains unchanged—the pre-trained LMs are frozen during training. We choose the XLNet-base model as a representative for the pre-trained LMs. For more details about experimental settings, refer to Appendix A.6.
+
+In Table 2, we present three new results using $f _ { \mathrm { i d e a l } }$ . As a baseline, we report the performance of $f _ { \mathrm { i d e a l } }$ with a randomly initialized XLNet-base. Then, we list the worst and best result of $f _ { \mathrm { i d e a l } }$ according to $g$ , when it is combined with the pre-trained LM. We here mention some findings from the experiment. First, comparing the results with the pre-trained LM against one with the random LM, we reconfirm that pre-training an LM apparently enables the model to capture some aspects of grammar. Specifically, our method is comparable to the linear model trained on the gold-standard trees. Second, we find that there is a tendency for the performance of $f _ { \mathrm { i d e a l } }$ relying on different LM layers to follow one we already observed in Section 6.1—the best result comes from the middle layers of the LM while the worst from the first and last layer. Third, we identify that the LM has a potential to show improved performance on grammar induction by adopting a more sophisticated $f$ . However, we emphasize that our method equipped with a simple $f$ without gold-standard trees is remarkably reasonable in recognizing constituency grammar, being especially good at catching ADJP and ADVP.
+
+# 6.3 CONSTITUENCY TREE EXAMPLES
+
+We visualize several gold-standard trees from PTB and the corresponding tree predictions for comparison. For more details, we refer readers to Appendix A.7.
+
+# 7 CONCLUSIONS AND FUTURE WORK
+
+In this paper, we propose a simple but effective method of inducing constituency trees from pretrained language models in a zero-shot learning fashion. Furthermore, we report a set of intuitive findings observed from the extracted trees, demonstrating that the pre-trained LMs exhibit some properties similar to constituency grammar. In addition, we show that our method can serve as a strong baseline for English grammar induction when combined with (or even without) appropriate linguistic biases.
+
+On the other hand, there are still remaining issues that can be good starting points for future work. First, although we analyzed our method based on two popular datasets, we focused only on English grammar induction. As each language has its own properties (and correspondingly would need individualized biases), it is desirable to expand this work to other languages. Second, it would also be desirable to investigate whether further improvements can be achieved by directly grafting the pre-trained LMs onto existing grammar induction models. Lastly, by verifying the usefulness of the knowledge from the pre-trained LMs and linguistic biases for grammar induction, we want to point out that there is still much room for improvement in the existing grammar induction models.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Reinald Kim Amplayo and the anonymous reviewers for their thoughtful and valuable comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF2016M3C4A7952587).
+
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+
+# A APPENDIX
+
+![](images/4557215c033955f375eec54b05ca4b1a0d588cb45906a8bd42cdd74d3224d2f4.jpg)  
+A.1 ATTENTION HEATMAP EXAMPLES   
+Figure 3: Self-attention heatmaps for the average of all attention distributions from the 7th layer of the XLNet-base model, given a set of input sentences.
+
+# A.2 TREE CONSTRUCTION ALGORITHM WITH SYNTACTIC DISTANCES
+
+# Algorithm 1 Syntactic Distances to Binary Constituency Tree (originally from Shen et al. (2018a))
+
+1: $S = [ w _ { 1 } , w _ { 2 } , \ldots , w _ { n } ] ;$ a sequence of words in a sentence of length $_ n$ .   
+2: $\mathbf { d } = [ d _ { 1 } , d _ { 2 } , \dotsc , d _ { n - 1 } ]$ : a vector whose elements are the distances between every two adjacent words.   
+3: function TREE $( S , { \bf d } )$   
+4: if $\mathbf { d } = \left[ \begin{array} { l l l l l } \end{array} \right]$ then   
+5: node $ \mathrm { L e a f } ( S [ 0 ] )$   
+6: else   
+7: $i \gets \arg \operatorname* { m a x } _ { i } ( { \bf d } )$   
+8: $\mathrm { c h i l d } _ { l } \gets \mathrm { T R E E } ( S _ { \leq i } , { \bf d } _ { < i } )$   
+9: $\mathrm { c h i l d } _ { r } \gets \mathrm { T R E E } ( S _ { > i } , { \bf d } _ { > i } )$   
+10: node ← Node(childl, childr)   
+11: end if   
+12: return node   
+13: end function
+
+# A.3 DISTANCE MEASURE FUNCTIONS
+
+Table 3: The definitions of distance measure functions for computing syntactic distances between two adjacent words in a sentence. Note that $\mathbf { r } = g ^ { v } ( w _ { i } )$ , $\mathbf { s } = g ^ { v } ( w _ { i + 1 } )$ , $P = g ^ { d } ( w _ { i } )$ , and $Q =$ $g ^ { d } ( w _ { i + 1 } )$ , respectively. $d$ : hidden embedding size, $n$ : the number of words $( w )$ in a sentence $( S )$ .   
+A.4 PERFORMANCE COMPARISON BY LAYER ON MNLI   
+
+<table><tr><td>Function（f）Definition</td></tr><tr><td>Functions for intermediate representations (Fu)</td></tr><tr><td>(rTs/((∑(1r):(∑_1s2)2) +1)/2 Cos(r,s) L1(r,s)</td></tr><tr><td>L2(r,s) i-1(r -si）2） Functions for attention distributions (Fd)</td></tr><tr><td>JSD(PIIQ) ((DKL(P||M)+DKL(Q||M))/2) where M=(P+Q)/2</td></tr><tr><td>and DKL(A|lB) =∑ω∈sA(w)log(A(ω)/B(ω)) HEL(P,Q)</td></tr></table>
+
+![](images/57638ca8b998e629ceb0f21baa45992e8d097a5cba5affbe54347892a5bf5588.jpg)  
+Figure 4: The best layer-wise S-F1 scores of each LM instance on the MNLI test set. (Left) The performance of the X-‘base’ models. (Right) The performance of the X-‘large’ models.
+
+# A.5 EXPERIMENTAL RESULTS ON MNLI
+
+Table 4: Results on the MNLI test set. Bold numbers correspond to the top 3 results for each column. L: layer number, A: attention head number (AVG: the average of all attentions). $\dagger$ : Results reported by Htut et al. (2018) and Drozdov et al. (2019). $\ddagger$ : Approaches in which $\overline { { \mathbf { C O O } } }$ parser is utilized. $^ *$ : These results are not strictly comparable to ours, due to the difference in data preprocessing.   
+
+<table><tr><td>Model</td><td>f</td><td>L</td><td>A</td><td>S-F1</td><td>SBAR</td><td>NP</td><td>VP</td><td>PP</td><td>ADJP</td><td>ADVP</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random Trees</td><td></td><td></td><td></td><td>21.4</td><td>11%</td><td>25%</td><td>16%</td><td>22%</td><td>22%</td><td>27%</td></tr><tr><td>Balanced Trees</td><td></td><td></td><td></td><td>20.0</td><td>8%</td><td>29%</td><td>11%</td><td>20%</td><td>22%</td><td>32%</td></tr><tr><td>Left Branching Trees</td><td></td><td></td><td></td><td>8.4</td><td>6%</td><td>13%</td><td>1%</td><td>4%</td><td>1%</td><td>8%</td></tr><tr><td>Right Branching Trees</td><td></td><td></td><td></td><td>51.9</td><td>65%</td><td>28%</td><td>75%</td><td>47%</td><td>45%</td><td>30%</td></tr><tr><td>Random XLNet-base (F)</td><td></td><td></td><td></td><td>22.0</td><td>12%</td><td>26%</td><td>15%</td><td>22%</td><td>22%</td><td>25%</td></tr><tr><td>Random XLNet-base (Fd)</td><td>=</td><td></td><td>=</td><td>23.5</td><td>14%</td><td>26%</td><td>18%</td><td>22%</td><td>22%</td><td>25%</td></tr><tr><td>Pre-trained LMs (w/o bias)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>HEL</td><td>9</td><td>10</td><td>36.1</td><td>36%</td><td>37%</td><td>34%</td><td>45%</td><td>26%</td><td>42%</td></tr><tr><td>BERT-large</td><td>JSD</td><td>17</td><td>10</td><td>37.0</td><td>38%</td><td>32%</td><td>34%</td><td>50%</td><td>22%</td><td>39%</td></tr><tr><td>GPT2</td><td>JSD</td><td>1</td><td>10</td><td>44.0</td><td>43%</td><td>53%</td><td>31%</td><td>60%</td><td>24%</td><td>40%</td></tr><tr><td>GPT2-medium</td><td>JSD</td><td>3</td><td>12</td><td>49.1</td><td>57%</td><td>32%</td><td>61%</td><td>44%</td><td>35%</td><td>37%</td></tr><tr><td>RoBERTa-base</td><td>JSD</td><td>10</td><td>9</td><td>36.2</td><td>26%</td><td>35%</td><td>34%</td><td>50%</td><td>23%</td><td>44%</td></tr><tr><td>RoBERTa-large</td><td>JSD</td><td>3</td><td>6</td><td>39.8</td><td>20%</td><td>28%</td><td>35%</td><td>30%</td><td>28%</td><td>27%</td></tr><tr><td>XLNet-base</td><td>HEL</td><td>1</td><td>6</td><td>39.0</td><td>25%</td><td>39%</td><td>28%</td><td>59%</td><td>35%</td><td>44%</td></tr><tr><td>XLNet-large</td><td>HEL</td><td>1</td><td>15</td><td>42.2</td><td>32%</td><td>49%</td><td>27%</td><td>62%</td><td>32%</td><td>49%</td></tr><tr><td>Pre-trained LMs (w/bias入=1.5)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>HEL</td><td>2</td><td>12</td><td>52.7</td><td>64%</td><td>35%</td><td>70%</td><td>50%</td><td>46%</td><td>30%</td></tr><tr><td>BERT-large</td><td>HEL</td><td>4</td><td>4</td><td>51.7</td><td>63%</td><td>31%</td><td>71%</td><td>49%</td><td>46%</td><td>30%</td></tr><tr><td>GPT2</td><td>HEL</td><td>1</td><td>10</td><td>52.2</td><td>57%</td><td>53%</td><td>49%</td><td>62%</td><td>32%</td><td>42%</td></tr><tr><td>GPT2-medium</td><td>HEL</td><td>2</td><td>1</td><td>53.9</td><td>53%</td><td>57%</td><td>50%</td><td>62%</td><td>29%</td><td>44%</td></tr><tr><td>RoBERTa-base</td><td>HEL</td><td>2</td><td>3</td><td>52.0</td><td>64%</td><td>31%</td><td>72%</td><td>49%</td><td>47%</td><td>30%</td></tr><tr><td>RoBERTa-large</td><td>L1</td><td>23</td><td>=</td><td>52.7</td><td>55%</td><td>40%</td><td>65%</td><td>53%</td><td>43%</td><td>41%</td></tr><tr><td>XLNet-base</td><td>L2</td><td>8</td><td></td><td>54.9</td><td>57%</td><td>49%</td><td>61%</td><td>55%</td><td>44%</td><td>57%</td></tr><tr><td>XLNet-large</td><td>L2</td><td>12</td><td>=</td><td>53.5</td><td>54%</td><td>47%</td><td>59%</td><td>51%</td><td>48%</td><td>60%</td></tr><tr><td>Other models</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PRPN-UPt $</td><td></td><td></td><td></td><td>48.6*</td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>PRPN-LM†‡</td><td></td><td></td><td></td><td>50.4*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DIORA†</td><td></td><td></td><td></td><td>51.2*</td><td></td><td>=</td><td></td><td></td><td></td><td>=</td></tr><tr><td>DIORA(+PP)†</td><td></td><td></td><td></td><td>59.0*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# A.6 EXPERIMENTAL DETAILS FOR TRAINING IDEAL DISTANCE MEASURE FUNCTION
+
+In this part, we present the detailed specifications of the experiments introduced in Section 6.2. We assume $f _ { \mathrm { i d e a l } }$ is only compatible with the functions in $G ^ { v }$ , as the functions in $G ^ { d }$ are not suitable for training as the sizes of the representations provided by $G ^ { d }$ are variable according to the length of an input sentence. To train the pseudo-optimal function $f _ { \mathrm { i d e a l } }$ , we minimize a pair-wise learning-to-rank loss following previous work (Burges et al., 2005; Shen et al., 2018a):
+
+$$
+L _ { \mathrm { d i s t } } ^ { \mathrm { r a n k } } = \sum _ { i , j > i } [ 1 - \mathrm { s i g n } ( d _ { i } ^ { \mathrm { g o l d } } - d _ { j } ^ { \mathrm { g o l d } } ) ( d _ { i } ^ { \mathrm { p r e d } } - d _ { j } ^ { \mathrm { p r e d } } ) ] ^ { + } ,
+$$
+
+where $d ^ { \mathrm { g o l d } }$ and $d ^ { \mathrm { p r e d } }$ are computed from the gold tree and our predicted one, respectively. $[ x ] ^ { + }$ is defined as $m a x ( 0 , x )$ . We train the $f _ { \mathrm { i d e a l } }$ with the PTB training set for 5 epochs. Each batch of the training set contains 16 sentences. We use an ADAM optimizer (Kingma & Ba, 2014) with the learning rate 5e-4. We train the variations of $f _ { \mathrm { i d e a l } }$ differentiated by the choice of $g$ in $G ^ { v }$ and report the best result in the Table 2. Each $f _ { \mathrm { i d e a l } }$ is chosen based on its performance on the PTB validation set. Considering the randomness of training, every result for $f _ { \mathrm { i d e a l } }$ is averaged over 3 different trials.
+
+# A.7 CONSTITUENCY TREE EXAMPLES
+
+We randomly select six sentences from PTB and visualize their trees, where the resulting group of trees for each sentence consists of a gold constituency tree and two induced trees (one without the right-skewness bias and the other with the bias) from our best model—XLNet-base. The ‘T’ character in the induced trees indicates a dummy tag.
+
+![](images/2f5466c302db8af3968e112d6a6fc4110fb4ef20f050196c5a7a65dfa8733b06.jpg)  
+Figure 5: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘These include a child-care initiative and extensions of soon-to-expire tax breaks for low-income housing and research-and-development expenditures’.
+
+![](images/6138640de123c6c8c0d5b6307a743fd5691924d2c7b4d57ffee6d0c560e437e7.jpg)  
+Figure 6: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘But HOFI ‘s first offer would have given Ideal ’s other shareholders about $10 \%$ of the combined company’.
+
+![](images/7d0c9332a71f76b64f2aa2c57d9f2bc965adeb519c3ddb317aa2f119b1f2525e.jpg)  
+Figure 7: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘It was Friday the $1 3 \mathrm { t h }$ and the stock market plummeted nearly 200 points’.
+
+![](images/4a2a79c63bc955bde02bf004df0ecf68c2fd14a5b3664a006bf8bc3680db54a9.jpg)  
+Figure 8: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘Until recently national governments in Europe controlled most of the air time and allowed little or no advertising’.
+
+![](images/3317260942a98cdd94ddf89da58fac2f95eb78e8752f7828db2b29796cd36683.jpg)  
+Figure 9: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘Nevertheless Ms. Garzarelli said she was swamped with phone calls over the weekend from nervous shareholders’.
+
+![](images/60ab9c2749c684e51447319241b1e8da126a5f9ff2078536ebeed7661fa5fa58.jpg)  
+Figure 10: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘Analysts and competitors however doubt the numbers were that high’.

parse/train/H1xPR3NtPB/H1xPR3NtPB_content_list.json

@@ -0,0 +1,1717 @@
+[
+    {
+        "type": "text",
+        "text": "ARE PRE-TRAINED LANGUAGE MODELS AWARE OFPHRASES? SIMPLE BUT STRONG BASELINES FORGRAMMAR INDUCTION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Taeuk $\\mathbf { K i m ^ { 1 } }$ , Jihun Choi1, Daniel Edmiston2 & Sang-goo Lee1   \n1Dept. of Computer Science and Engineering, Seoul National University, Seoul, Korea   \n2Dept. of Linguistics, University of Chicago, Chicago, IL, USA   \n{taeuk,jhchoi,sglee}@europa.snu.ac.kr, danedmiston@uchicago.edu ",
+        "bbox": [
+            184,
+            194,
+            789,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            289,
+            544,
+            304
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "With the recent success and popularity of pre-trained language models (LMs) in natural language processing, there has been a rise in efforts to understand their inner workings. In line with such interest, we propose a novel method that assists us in investigating the extent to which pre-trained LMs capture the syntactic notion of constituency. Our method provides an effective way of extracting constituency trees from the pre-trained LMs without training. In addition, we report intriguing findings in the induced trees, including the fact that some pre-trained LMs outperform other approaches in correctly demarcating adverb phrases in sentences. ",
+        "bbox": [
+            233,
+            320,
+            764,
+            433
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Grammar induction, which is closely related to unsupervised parsing and latent tree learning, allows one to associate syntactic trees, i.e., constituency and dependency trees, with sentences. As grammar induction essentially assumes no supervision from gold-standard syntactic trees, the existing approaches for this task mainly rely on unsupervised objectives, such as language modeling (Shen et al., 2018b; 2019; Kim et al., 2019a;b) and cloze-style word prediction (Drozdov et al., 2019) to train their task-oriented models. On the other hand, there is a trend in the natural language processing (NLP) community of leveraging pre-trained language models (LMs), e.g., ELMo (Peters et al., 2018) and BERT (Devlin et al., 2019), as a means of acquiring contextualized word representations. These representations have proven to be surprisingly effective, playing key roles in recent improvements in various models for diverse NLP tasks. ",
+        "bbox": [
+            174,
+            492,
+            825,
+            631
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, inspired by the fact that the training objectives of both the approaches for grammar induction and for training LMs are identical, namely, (masked) language modeling, we investigate whether pre-trained LMs can also be utilized for grammar induction/unsupervised parsing, especially without training. Specifically, we focus on extracting constituency trees from pre-trained LMs without fine-tuning or introducing another task-specific module, at least one of which is usually required in other cases where representations from pre-trained LMs are employed. This restriction provides us with some advantages: (1) it enables us to derive strong baselines for grammar induction with reduced time and space complexity, offering a chance to reexamine the current status of existing grammar induction methods, (2) it facilitates an analysis on how much and what kind of syntactic information each pre-trained LM contains in its intermediate representations and attention distributions in terms of phrase-structure grammar, and (3) it allows us to easily inject biases into our framework, for instance, to encourage the right-skewness of the induced trees, resulting in performance gains in English unsupervised parsing. ",
+        "bbox": [
+            173,
+            638,
+            825,
+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "First, we briefly mention related work (§2). Then, we introduce the intuition behind our proposal in detail (§3), which is motivated by our observation that we can cluster words in a sentence according to the similarity of their attention distributions over words in the sentence. Based on this intuition, we define a straightforward yet effective method $( \\ S 4 )$ of drawing constituency trees directly from pretrained LMs with no fine-tuning or addition of task-specific parts, instead resorting to the concept of Syntactic Distance (Shen et al., 2018a;b). Then, we conduct experiments (§5) on the induced constituency trees, discovering some intriguing phenomena. Moreover, we analyze the pre-trained ",
+        "bbox": [
+            174,
+            827,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "LMs and constituency trees from various points of view, including looking into which layer(s) of the LMs is considered to be sensitive to phrase information (§6). ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To summarize, our contributions in this work are as follows: ",
+        "bbox": [
+            173,
+            138,
+            568,
+            154
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• By investigating the attention distributions from Transformer-based pre-trained LMs, we show that there is evidence to suggest that several attention heads of the LMs exhibit syntactic structure akin to constituency grammar. Inspired by the above observation, we propose a method that facilitates the derivation of constituency trees from pre-trained LMs without training. We also demonstrate that the induced trees can serve as a strong baseline for English grammar induction. We inspect, in view of our framework, what type of syntactic knowledge the pre-trained LMs capture, discovering interesting facts, e.g., that some pre-trained LMs are more aware of adverb phrases than other approaches. ",
+        "bbox": [
+            215,
+            166,
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+            306
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            327,
+            344,
+            344
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Grammar induction is a task whose goal is to infer from sequential data grammars which generalize, and are able to account for unseen data (Lari & Young (1990); Clark (2001); Klein & Manning (2002; 2004), to name a few). Traditionally, this was done by learning explicit grammar rules (e.g., context free rewrite rules), though more recent methods employ neural networks to learn such rules implicitly, focusing more on the induced grammars’ ability to generate or parse sequences. ",
+        "bbox": [
+            174,
+            361,
+            825,
+            431
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Specifically, Shen et al. (2018b) proposed Parsing-Reading-Predict Network (PRPN) where the concept of Syntactic Distance is first introduced. They devised a neural model for language modeling where the model is encouraged to recognize syntactic structure. The authors also probed the possibility of inducing constituency trees without access to gold-standard trees by adopting an algorithm that recursively splits a sequence of words into two parts, the split point being determined according to correlated syntactic distances; the point having the biggest distance becomes the first target of division. Shen et al. (2019) presented a model called Ordered Neurons (ON), which is a revised version of LSTM (Long Short-Term Memory, Hochreiter & Schmidhuber (1997)) which reflects the hierarchical biases of natural language and can be used to compute syntactic distances. Shen et al. (2018a) trained a supervised parser relying on the concept of syntactic distance. ",
+        "bbox": [
+            174,
+            438,
+            825,
+            577
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Other studies include Drozdov et al. (2019), who trained deep inside-outside recursive autoencoders (DIORA) to derive syntactic trees in an exhaustive way with the aid of the inside-outside algorithm, and Kim et al. (2019a) who proposed Compound Probabilistic Context-Free Grammars (compound PCFG), showing that neural PCFG models are capable of producing promising unsupervised parsing results. Li et al. (2019) proved that an ensemble of unsupervised parsing models can be beneficial, while Shi et al. (2019) utilized additional training signals from pictures related with input text. Dyer et al. (2016) proposed Recurrent Neural Network Grammars (RNNG) for both language modeling and parsing, and Kim et al. (2019b) suggested an unsupervised variant of the RNNG. There also exists another line of research on task-specific latent tree learning (Yogatama et al., 2017; Choi et al., 2018; Havrylov et al., 2019; Maillard et al., 2019). The goal here is not to construct linguistically plausible trees, but to induce trees fitted to improving target performance. Naturally, the induced performance-based trees need not resemble linguistically plausible trees, and some studies (Williams et al., 2018a; Nangia & Bowman, 2018) examined the apparent fact that performance-based and lingusitically plausible trees bear little resemblance to one another. ",
+        "bbox": [
+            173,
+            584,
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+            777
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Concerning pre-trained language models (Peters et al. (2018); Devlin et al. (2019); Radford et al. (2019); Yang et al. (2019); Liu et al. (2019b), inter alia)—particularly those employing a Transformer architecture (Vaswani et al., 2017)—these have proven to be helpful for diverse NLP downstream tasks. In spite of this, there is no vivid picture for explaining what particular factors contribute to performance gains, even though some recent work has attempted to shed light on this question. In detail, one group of studies (Raganato & Tiedemann (2018); Clark et al. (2019); Ethayarajh (2019); Hao et al. (2019); Voita et al. (2019), inter alia) has focused on dissecting the intermediate representations and attention distributions of the pre-trained LMs, while the another group of publications (Marecek & Rosa (2018); Goldberg (2019); Hewitt & Manning (2019); Liu et al. (2019a); Rosa & ˇ Marecek (2019), to name a few) delve into the question of the existence of syntactic knowledge in ˇ ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b59545393c8be32c7e4710ab90ef69ecbf77cabd130491c54d540a4d216ac52.jpg",
+        "image_caption": [
+            "Figure 1: Self-attention heatmaps from two different pre-trained LMs. (Left) A heatmap for the average of attention distributions from the 7th layer of the XLNet-base (Yang et al., 2019) model given the sample sentence. (Right) A heatmap for the average of attention distributions from the 9th layer of the BERT-base (Devlin et al., 2019) model given another sample sentence. We can easily spot the chunks of words on the two heatmaps that are correlated with the constituents of the input sentences, e.g., (Left) ‘the price of plastics’, ‘took off in 1987’, ‘Quantum Chemical Corp.’, (Right) ‘when price increases can be sustained’, and ‘he remarks’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            232,
+            99,
+            753,
+            263
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transformer-based models. Particularly, Marecek & Rosa (2019) proposed an algorithm for extract- ˇ ing constituency trees from Transformers trained for machine translation, which is similar to our approach. ",
+        "bbox": [
+            174,
+            424,
+            825,
+            467
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 MOTIVATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            505,
+            313,
+            521
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As pioneers in the literature have pointed out, the multi-head self-attention mechanism (Vaswani et al., 2017) is a key component in Transformer-based language models, and it seems this mechanism empowers the models to capture certain semantic and syntactic information existing in natural language. Among a diverse set of knowledge they may capture, in this work we concentrate on phrase-structure grammar by seeking to extract constituency trees directly from their attention information and intermediate weights. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            631
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In preliminary experiments, where we manually visualize and investigate the intermediate representations and attention distributions of several pre-trained LMs given input, we have found some evidence which suggests that the pre-trained LMs exhibit syntactic structure akin to constituency grammar in some degree. Specifically, we have noticed some patterns which are often displayed in self-attention heatmaps as explicit horizontal lines, or groups of rectangles of various sizes. As an attention distribution of a word in an input sentence corresponds to a row in a heatmap matrix, we can say that the appearance of these patterns indicates the existence of groups of words where the attention distributions of the words in the same group are relatively similar. Interestingly, we have also discovered the fact that the groups of words we observed are fairly correlated with the constituents of the input sentence, as shown in Figure 1 (above) and Figure 3 (in Appendix A.1). ",
+        "bbox": [
+            174,
+            638,
+            825,
+            777
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Even though we have identified some patterns which match with the constituents of sentences, it is not enough to conclude that the pre-trained LMs are aware of syntactic phrases as found in phrasestructure grammars. To demonstrate the claim, we attempt to obtain constituency trees in a zero-shot learning fashion, relying only on the knowledge from the pre-trained LMs. To this end, we suggest the following, inspired from our finding: two words in a sentence are syntactically close to each other (i.e., the two words belong to the same constituent) if their attention distributions over words in the sentence are also close to each other. Note that this implicitly presumes that each word is more likely to attend more on the words in the same constituent to enrich its representation in the pre-trained LMs. Finally, we utilize the assumption to compute syntactic distances between each pair of adjacent words in a sentence, from which the corresponding constituency tree can be built. ",
+        "bbox": [
+            174,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            377,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 SYNTACTIC DISTANCE AND TREE CONSTRUCTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            563,
+            148
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We leverage the concept of Syntactic Distance proposed by Shen et al. (2018a;b) to draw constituency trees from raw sentences in an intuitive way. Formally, given a sequence of words in a sentence, $w _ { 1 } , w _ { 2 } , \\ldots , w _ { n }$ , we compute $\\mathbf { d } = [ d _ { 1 } , d _ { 2 } , \\dots , d _ { n - 1 } ]$ where $d _ { i }$ corresponds to the syntactic distance between $w _ { i }$ and $w _ { i + 1 }$ . Each $d _ { i }$ is defined as follows: ",
+        "bbox": [
+            174,
+            160,
+            823,
+            215
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d4250423561bb4858740e95debd332a335e6c7ab11bbf3ee13a1d175948687c.jpg",
+        "text": "$$\nd _ { i } = f ( g ( w _ { i } ) , g ( w _ { i + 1 } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            223,
+            581,
+            241
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\cdot , \\cdot )$ and $g ( \\cdot )$ are a distance measure function and representation extractor function, respectively. The function $g$ converts each word into the corresponding vector representation, while $f$ computes the syntactic distance between the two words given their representations. Once $\\mathbf { d }$ is derived, it can be easily converted into the target constituency tree by a simple algorithm following Shen et al. (2018a).1 For details of the algorithm, we refer the reader to Appendix A.2. ",
+        "bbox": [
+            174,
+            247,
+            823,
+            318
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Although previous studies attempted to explicitly train the functions $f$ and $g$ with supervision (with access to gold-standard trees, Shen et al. (2018a)) or to obtain them as a by-product of training particular models that are carefully designed to recognize syntactic information (Shen et al., 2018b; 2019), in this work we stick to simple distance metric functions for $f$ and pre-trained LMs for $g$ , forgoing any training process. In other words, we focus on investigating the possibility of pre-trained LMs possessing constituency information in a form that can be readily extracted with straightforward computations. If the trees induced by the syntactic distances derived from the pre-trained LMs are similar enough to gold-standard syntax trees, we can reasonably claim that the LMs resemble phrase-structure. ",
+        "bbox": [
+            173,
+            324,
+            825,
+            449
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 PRE-TRAINED LANGUAGE MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            460,
+            481
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider four types of recently proposed language models. These are: BERT (Devlin et al., 2019), GPT-2 (Radford et al., 2019), RoBERTa (Liu et al., 2019b), and XLNet (Yang et al., 2019). They all have in common that they are based on the Transformer architecture and have been proven to be effective in natural language understanding (Wang et al., 2019) or generation. We handle two variants for each LM, varying in the number of layers, attention heads, and hidden dimensions, resulting in eight different cases in total. In particular, each LM has two variants. (1) base: consists of $l { = } 1 2$ layers, $a { = } 1 2$ attention heads, and $d { = } 7 6 8$ hidden dimensions, while (2) large: has $l { = } 2 4$ layers, $a { = } 1 6$ attention heads, and $d { = } 1 0 2 4$ hidden dimensions.2 We deal with a wide range of pre-trained LMs, unlike previous work which has mostly analyzed a specific model, particularly BERT. For details about each LM, we refer readers to the respective original papers. ",
+        "bbox": [
+            173,
+            492,
+            825,
+            632
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In terms of our formulation, each LM instance provides two categories of representation extractor functions, $G ^ { v }$ and $G ^ { d }$ . Specifically, $G ^ { v }$ refers to a set of functions $\\{ g _ { j } ^ { v } | j = 1 , \\ldots , l \\}$ , each of which simply outputs the intermediate hidden representation of a given word on the $j$ th layer of the LM. Likewise, $\\dot { G } ^ { d }$ is a set of functions $\\{ g _ { ( j , k ) } ^ { d } | \\bar { j } = 1 , \\ldots , l , k = 1 , \\ldots , a + 1 \\}$ , each of which outputs the attention distribution of an input word by the $k$ th attention head on the $j$ th layer of the LM. Even though our main motivation comes from the self-attention mechanism, we also deal with the intermediate hidden representations present in the pre-trained LMs by introducing $G ^ { v }$ , considering that the hidden representations serve as storage of collective information taken from the processing of the pre-trained LMs. Note that $k$ ranges up to $a + 1$ , not $a$ , implying that we consider the average of all attention distributions on the same layer in addition to the individual ones. This averaging function can be regarded as an ensemble of other functions in the layer which are specialized for different aspects of information, and we expect that this technique will provide a better option in some cases as reported in previous work (Li et al., 2019). ",
+        "bbox": [
+            173,
+            638,
+            825,
+            821
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One remaining issue is that all the pre-trained LMs we use regard each input sentence as a sequence of subword tokens, while our formulation assumes words cannot be further divided into smaller tokens. To resolve this difference, we tested certain heuristics that guide how subword tokens for a complete word should be exploited to represent the word, and we have empirically found that the best result comes when each word is represented by an average of the representations of its subwords.3 Therefore, we adopt the above heuristic in this work for cases where a word is tokenized into more than two parts. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 DISTANCE MEASURE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            232,
+            447,
+            246
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the distance measure function $f$ , we prepare three options $( F ^ { v } )$ for $G ^ { v }$ and two options $( F ^ { d } )$ for $G ^ { d }$ . Formally, $f \\in F ^ { v } \\cup F ^ { d }$ , where $F ^ { v } = \\{ \\mathbf { C O S } , \\mathbf { L 1 } , \\mathbf { L 2 } \\}$ , $F ^ { d } = \\{ \\mathrm { J S D } , \\mathrm { H E L } \\}$ . COS, L1, L2, JSD, and HEL correspond to Cosine, L1, and L2, Jensen-Shannon, and Hellinger distance respectively. The functions in $F ^ { v }$ are only compatible with the elements of $G ^ { v }$ , and the same holds for $F ^ { d }$ and $G ^ { d }$ . The exact definition of each function is listed in Appendix A.3. ",
+        "bbox": [
+            174,
+            263,
+            825,
+            333
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.4 INJECTING BIAS INTO SYNTACTIC DISTANCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            364,
+            535,
+            378
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One of the main advantages we obtain by leveraging syntactic distances to derive parse trees is that we can easily inject inductive bias into our framework by simply modifying the values of the syntactic distances. Hence, we investigate whether the extracted trees from our method can be further refined with the aid of additional biases. To this end, we introduce a well-known bias for English constituency trees—the right-skewness bias—in a simple linear form.4 Namely, our intention is to influence the induced trees such that they are moderately right-skewed following the nature of gold-standard parse trees in English. ",
+        "bbox": [
+            173,
+            395,
+            825,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Formally, we compute $\\hat { d } _ { i }$ by appending the following linear bias term to every $d _ { i }$ ",
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+    {
+        "type": "equation",
+        "img_path": "images/74474b452d0cb11b6e27012649daac141a7510dd7bbdee901a8d3907dd37d1aa.jpg",
+        "text": "$$\n\\hat { d } _ { i } = d _ { i } + \\lambda \\cdot \\mathrm { A V G } ( \\mathbf { d } ) \\times ( 1 - 1 / ( m - 1 ) \\times ( i - 1 ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\operatorname { A V G } ( \\cdot )$ outputs an average of all elements in a vector, $\\lambda$ is a hyperparameter, and $i$ ranges from 1 to $m = n - 1$ . We write $\\hat { \\mathbf { d } } = [ \\hat { d } _ { 1 } , \\hat { d } _ { 2 } , \\dots , \\hat { d } _ { m } ]$ in place of $\\mathbf { d }$ to signify biased syntactic distances. ",
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+        "text": "The main purpose of introducing such a bias is examining what changes are made to the resulting tree structures rather than boosting quantitative performance per $s e$ , though it is of note that it serves this purpose as well. We believe that this additional consideration is necessary based on two points. First, English is what is known as a head-initial language. That is, given a selector and argument, the selector has a strong tendency to appear on the left, e.g., ‘eat food’, or ‘to Canada’. Head-initial languages therefore have an in-built preference for right-branching structures. By adjusting the bias injected into syntactic distances derived from pre-trained LMs, we can figure out whether the LMs are capable of inducing the right-branching bias, which is one of the main properties of English syntax; if injecting the bias does not influence the performance of the LMs on unsupervised parsing, we can conjecture they are inherently capturing the bias to some extent. Second, as mentioned before, we have witnessed some previous work (Shen et al., 2018b; 2019; Htut et al., 2018; Li et al., 2019; Shi et al., 2019) where the right-skewness bias is implicitly exploited, although it could be regarded as not ideal. What we intend to focus on is the question about which benefits the bias provides for such parsing models, leading to overall performance improvements. In other words, we look for what the exact contribution of the bias is when it is injected into grammar induction models, by explicitly controlling the bias using our framework. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 GENERAL SETTINGS ",
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+        "type": "text",
+        "text": "5.1.1 DATASETS ",
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+        "type": "text",
+        "text": "In this section, we conduct unsupervised constituency parsing on two datasets. The first dataset is WSJ Penn Treebank (PTB, Marcus et al. (1993)), in which human-annotated gold-standard trees are available. We use the standard split of the dataset—2-21 for training, 22 for validation, and 23 for test. The second one is MNLI (Williams et al., 2018b), which is originally designed to test natural language inference but often utilized as a means of evaluating parsers. It contains constituency trees produced by an external parser (Klein & Manning, 2003). We leverage the union of two different versions of the MNLI development set as test data following convention (Htut et al., 2018; Drozdov et al., 2019), and we call it the MNLI test set in this paper. Moreover, we randomly sample 40K sentences from the training set of the MNLI to utilize them as a validation set. To preprocess the datasets, we follow the setting of Kim et al. (2019a) with the minor exceptions that words are not lower-cased and number characters are preserved instead of being substituted by a special character. ",
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+    {
+        "type": "text",
+        "text": "5.1.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For implementation, to compare pre-trained LMs in an unified manner, we resort to an integrated PyTorch codebase that supports all the models we consider.5 For each LM, we tune the best combination of $f$ and $g$ functions using the validation set. Then, we derive a set of $\\mathbf { d }$ for sentences in the test set using the chosen functions, followed by the resulting constituency trees converted from each d by the tree construction algorithm in Section 4.1. In addition to sentence-level F1 (S-F1) score, we report label recall scores for six main categories: SBAR, NP, VP, PP, ADJP, and ADVP. We also present the results of utilizing $\\hat { \\mathbf { d } }$ instead of $\\mathbf { d }$ , empirically setting the bias hyperparameter $\\lambda$ as 1.5. We do not fine-tune the LMs on domain-specific data, as we here focus on finding their universal characteristics. ",
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+        "type": "text",
+        "text": "We take four na¨ıve baselines into account, random (averaged over 5 trials), balanced, left-branching, and right-branching binary trees. In addition, we present two more baselines which are identical to our models except that their $g$ functions are based on a randomly initialized XLNet-base rather than pre-trained ones. To be concrete, We provide ‘Random XLNet-base $( F ^ { v } ) ^ { : }$ ’ which applies the functions in $F ^ { v }$ on random hidden representations and ‘Random XLNet-base $( F ^ { d } )$ ’ that utilizes the functions in $F ^ { d }$ and random attention distributions, respectively. Considering the randomness of initialization and possible choices for $f$ , the final score for each of the baselines is calculated as an average over 5 trials of each possible $f$ , i.e., an average over $5 \\times 3$ runs in case of $F ^ { v }$ and $5 \\times 2$ runs for $F ^ { d }$ . These baselines enable us to estimate the exact advantage we obtain by pre-training LMs, effectively removing additional unexpected gains that may exist. Furthermore, we compare our parse trees against ones from existing grammar induction models. ",
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+    {
+        "type": "text",
+        "text": "All scripts used in our experiments will be publicly available for reproduction and further analysis.6 ",
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+    {
+        "type": "text",
+        "text": "5.2 EXPERIMENTAL RESULTS ON PTB ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Table 1, we report the results of the various models on the PTB test set. First of all, our method combined with pre-trained LMs shows competitive or comparable results in terms of S-F1 even without the right-skewness bias. This result implies that the extracted trees from our method can be regarded as a baseline for English grammar induction. Moreover, pre-trained LMs show substantial improvements over Random Transformers (XLNet-base), demonstrating that training language models on large corpora, in fact, enables the LMs to be more aware of syntactic information. ",
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+        "type": "text",
+        "text": "When the right-skewness bias is applied to syntactic distances derived from pre-trained LMs, the S-F1 scores of the LMs increase by up to ten percentage points. This improvement indicates that the pre-trained LMs do not properly capture the largely right-branching nature of English syntax, at least when observed through the lens of our framework. By explicitly controlling the bias through our framework and observing the performance gap between our models with and without the bias, we confirm that the main contribution of the bias comes from its capability to capture subordinate clauses (SBAR) and verb phrases (VP). This observation provides a hint for what some previous work on unsupervised parsing desired to obtain by introducing the bias to their models. It is intriguing to see that all of the existing grammar induction models are inferior to the right-branching baseline in recognizing SBAR and VP (although some of them already utilized the right-skewness bias), implying that the same problem—models do not properly capture the right-branching nature— may also exist in current grammar induction models. One possible assumption is that the models do not need the bias to perform well in language modeling, although future work should provide a rigorous analysis about the phenomenon. ",
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+    {
+        "type": "table",
+        "img_path": "images/99029a5c4f5ac241dd71e58e413a4475da609effa3e42f899f7dc962553c93cc.jpg",
+        "table_caption": [
+            "Table 1: Results on the PTB test set. Bold numbers correspond to the top 3 results for each column. L: layer number, A: attention head number (AVG: the average of all attentions). $\\dagger$ : Results reported by Kim et al. (2019a). $\\ddagger$ : Approaches in which COO parser is utilized. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>f</td><td>L</td><td>A</td><td>S-F1</td><td>SBAR</td><td>NP</td><td>VP</td><td>PP</td><td>ADJP</td><td>ADVP</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random Trees</td><td></td><td>=</td><td></td><td>18.1</td><td>8%</td><td>23%</td><td>12%</td><td>18%</td><td>23%</td><td>28%</td></tr><tr><td>Balanced Trees</td><td></td><td>=</td><td>=</td><td>18.5</td><td>7%</td><td>27%</td><td>8%</td><td>18%</td><td>27%</td><td>25%</td></tr><tr><td>Left Branching Trees</td><td></td><td></td><td></td><td>8.7</td><td>5%</td><td>11%</td><td>0%</td><td>5%</td><td>2%</td><td>8%</td></tr><tr><td>Right Branching Trees</td><td></td><td></td><td></td><td>39.4</td><td>68%</td><td>24%</td><td>71%</td><td>42%</td><td>27%</td><td>38%</td></tr><tr><td>Random XLNet-base (F)</td><td></td><td></td><td></td><td>19.6</td><td>9%</td><td>26%</td><td>12%</td><td>20%</td><td>23%</td><td>24%</td></tr><tr><td>Random XLNet-base (Fd)</td><td></td><td></td><td></td><td>20.1</td><td>11%</td><td>25%</td><td>14%</td><td>19%</td><td>22%</td><td>26%</td></tr><tr><td>Pre-trainedLMs (w/o bias)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>JSD</td><td>9</td><td>AVG</td><td>32.4</td><td>28%</td><td>42%</td><td>28%</td><td>31%</td><td>35%</td><td>63%</td></tr><tr><td>BERT-large</td><td>HEL</td><td>17</td><td>AVG</td><td>34.2</td><td>34%</td><td>43%</td><td>27%</td><td>39%</td><td>37%</td><td>57%</td></tr><tr><td>GPT2</td><td>JSD</td><td>9</td><td>1</td><td>37.1</td><td>32%</td><td>47%</td><td>27%</td><td>55%</td><td>27%</td><td>36%</td></tr><tr><td>GPT2-medium</td><td>JSD</td><td>10</td><td>13</td><td>39.4</td><td>41%</td><td>51%</td><td>21%</td><td>67%</td><td>33%</td><td>44%</td></tr><tr><td>RoBERTa-base</td><td>JSD</td><td>9</td><td>4</td><td>33.8</td><td>40%</td><td>38%</td><td>33%</td><td>43%</td><td>42%</td><td>57%</td></tr><tr><td>RoBERTa-large</td><td>JSD</td><td>14</td><td>5</td><td>34.1</td><td>29%</td><td>46%</td><td>30%</td><td>37%</td><td>28%</td><td>40%</td></tr><tr><td>XLNet-base</td><td>HEL</td><td>9</td><td>AVG</td><td>40.1</td><td>35%</td><td>56%</td><td>26%</td><td>38%</td><td>47%</td><td>68%</td></tr><tr><td>XLNet-large</td><td>L2</td><td>11</td><td>-</td><td>38.1</td><td>36%</td><td>51%</td><td>26%</td><td>41%</td><td>45%</td><td>69%</td></tr><tr><td>Pre-trainedLMs (w/bias入=1.5)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>HEL</td><td>9</td><td>AVG</td><td>42.3</td><td>45%</td><td>46%</td><td>49%</td><td>43%</td><td>41%</td><td>65%</td></tr><tr><td>BERT-large</td><td>HEL</td><td>17</td><td>AVG</td><td>44.4</td><td>55%</td><td>48%</td><td>48%</td><td>52%</td><td>41%</td><td>62%</td></tr><tr><td>GPT2</td><td>JSD</td><td>9</td><td>1</td><td>41.3</td><td>43%</td><td>49%</td><td>38%</td><td>58%</td><td>27%</td><td>43%</td></tr><tr><td>GPT2-medium</td><td>HEL</td><td>2</td><td>1</td><td>42.3</td><td>54%</td><td>50%</td><td>39%</td><td>56%</td><td>24%</td><td>41%</td></tr><tr><td>RoBERTa-base</td><td>JSD</td><td>8</td><td>AVG</td><td>42.1</td><td>51%</td><td>44%</td><td>44%</td><td>55%</td><td>40%</td><td>66%</td></tr><tr><td>RoBERTa-large</td><td>JSD</td><td>12</td><td>AVG</td><td>42.3</td><td>40%</td><td>50%</td><td>43%</td><td>44%</td><td>48%</td><td>56%</td></tr><tr><td>XLNet-base</td><td>HEL</td><td>7</td><td>AVG</td><td>48.3</td><td>62%</td><td>53%</td><td>50%</td><td>58%</td><td>49%</td><td>74%</td></tr><tr><td>XLNet-large</td><td>HEL</td><td>11</td><td>AVG</td><td>46.7</td><td>57%</td><td>50%</td><td>54%</td><td>50%</td><td>57%</td><td>73%</td></tr><tr><td>Other models</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PRPN(tuned)† ‡</td><td></td><td></td><td>=</td><td>47.3</td><td>50%</td><td>59%</td><td>46%</td><td>57%</td><td>44%</td><td>32%</td></tr><tr><td>ON(tuned)† ‡</td><td></td><td></td><td></td><td>48.1</td><td>51%</td><td>64%</td><td>41%</td><td>54%</td><td>38%</td><td>31%</td></tr><tr><td>Neural PCFG†</td><td></td><td></td><td></td><td>50.8</td><td>52%</td><td>71%</td><td>33%</td><td>58%</td><td>32%</td><td>45%</td></tr><tr><td>Compound PCFG+</td><td>=</td><td>=</td><td>=</td><td>55.2</td><td>56%</td><td>74%</td><td>41%</td><td>68%</td><td>40%</td><td>52%</td></tr></table>",
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+        "text": "On the other hand, the existing models show exceptionally high recall scores on noun phrases (NP), even though our pre-trained LMs also have success to some extent in capturing noun phrases compared to na¨ıve baselines. From this, we conjecture that neural models trained with a language modeling objective become largely equipped with the ability to understand the concept of NP. In contrast, the pre-trained LMs record the best recall scores on adjective and adverb phrases (ADJP and ADVP), suggesting that the LMs and existing models capture disparate aspects of English syntax to differing degrees. To further explain why some pre-trained LMs are good at capturing ADJPs and ADVPs, we manually investigated the attention heatmaps of the sentences that contain ADJPs or ADVPs. From the inspection, we empirically found that there are some keywords—including ‘two’, ‘ago’, ‘too’, and ‘far’—which have different patterns of attention distributions compared to those of their neighbors and that these keywords can be a clue for our framework to recognize the existence of ADJPs or ADJPs. It is also worth mentioning that ADJPs and ADVPs consist of a relatively smaller number of words than those of SBAR and VP, indicating that the LMs combined with our method have strength in correctly finding small chunks of words, i.e., low-level phrases. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Meanwhile, in comparison with other LM models, GPT-2 and XLNet based models demonstrate their effectiveness and robustness in unsupervised parsing. Particularly, the XLNet-base model serves as a robust baseline achieving the top performance among LM candidates. One plausible explanation for this outcome is that the training objective of XLNet, which considers both autoencoding (AE) and autoregressive (AR) features, might encourage the model to be better aware of phrase structure than other LMs. Another possible hypothesis is that AR objective functions (e.g., typical language modeling) are more effective in training syntax-aware neural models than AE objectives (e.g., masked language modeling), as both GPT-2 and XLNet are pre-trained on AR variants. However, it is hard to conclude what factors contribute to their high performance at this stage. ",
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+    {
+        "type": "image",
+        "img_path": "images/d982602134138e02110915208629d6ca404a31eb2399467fa65e34bb481667ef.jpg",
+        "image_caption": [
+            "Figure 2: The best layer-wise S-F1 scores of each LM instance on the PTB test set. (Left) The performance of the X-‘base’ models. (Right) The performance of the X-‘large’ models. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+        "type": "text",
+        "text": "Interestingly, there is an obvious trend that the functions in $F ^ { d } .$ —the distance measure functions for attention distributions—lead most of the LM instances to the best parsing results, indicating that deriving parse trees from attention information can be more compact and efficient than extracting them from the LMs’ intermediate representations, which should contain linguistic knowledge beyond phrase structure. In addition, the results in Table 1 show that large parameterizations of the LMs generally increase their parsing performance, but this improvement is not always guaranteed. Meanwhile, as we expected in Section 4.2 and as seen in the $\\mathbf { \\^ A }$ (attention head number) column of Table 1, the average of attention distributions in the same layer often provides better results than individual attention distributions. ",
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+    {
+        "type": "text",
+        "text": "5.3 EXPERIMENTAL RESULTS ON MNLI ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present the results of various models on the MNLI test set in Table 3 of Appendix A.5. We observe trends in the results which mainly coincide with those of the PTB dataset. Particularly, (1) right-branching trees are strong baselines for the task, especially showing their strengths in capturing SBAR and VP clauses/phrases, (2) our method resorting to the LM instances is also comparable to the right-branching trees, demonstrating its superiority in recognizing different aspects of phrase categories including prepositional phrases (PP) and adverb phrases (ADVP), and (3) attention distributions seem more effective for distilling the phrase structures of sentences than intermediate representations. ",
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+    {
+        "type": "text",
+        "text": "However, there are some issues worth mentioning. First, the right-branching baseline seems to be even stronger in the case of MNLI, recording a score of over 50 in sentence-level F1. We conjecture that this result comes principally from two reasons: (1) the average length of sentences in MNLI is much shorter than in PTB, giving a disproportionate advantage to na¨ıve baselines, and (2) our data preprocessing, which follows Kim et al. (2019a), removes all punctuation marks, unlike previous work (Htut et al., 2018; Drozdov et al., 2019), leading to an unexpected advantage for the rightbranching scheme. Moreover, it deserves to consider the fact that the gold-standard parse trees in MNLI are not human-annotated, rather automatically generated. ",
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+    {
+        "type": "text",
+        "text": "Second, in terms of consistency in identifying the best choice of $f$ and $g$ for each LM, we observe that most of the best combinations of $f$ and $g$ tuned for PTB do not correspond well to the best ones for MNLI. Does this observation imply that a specific combination of these functions and the resulting performance do not generalize well across different data domains? To clarify, we manually investigated the performance of some combinations of $f$ and $g$ , which are tuned on PTB but tested on MNLI instead. As a result, we discover that particular combinations of $f$ and $g$ which are good at PTB are also competitive on MNLI, even though they fail to record the best scores on MNLI. ",
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+    {
+        "type": "table",
+        "img_path": "images/4a404207e4f6e394ff294f94f09fa00ce5e2b34bb1aeed1c5f58ff3a6d5c3378.jpg",
+        "table_caption": [
+            "Table 2: Results of training a pseudo-optimum $f _ { \\mathrm { i d e a l } }$ with PTB and XLNet-base model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>f</td><td>L</td><td>A</td><td>S-F1</td><td>SBAR</td><td>NP</td><td>VP</td><td>PP</td><td>ADJP</td><td>ADVP</td></tr><tr><td>Baselines (from Table1)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random XLNet-base (F)</td><td>1</td><td></td><td>-</td><td>19.6</td><td>9%</td><td>26%</td><td>12%</td><td>20%</td><td>23%</td><td>24%</td></tr><tr><td>Random XLNet-base (Fd)</td><td></td><td>1</td><td>-</td><td>20.1</td><td>11%</td><td>25%</td><td>14%</td><td>19%</td><td>22%</td><td>26%</td></tr><tr><td>XLNet-base (入=0)</td><td>JSD</td><td>9</td><td>AVG</td><td>40.1</td><td>35%</td><td>56%</td><td>26%</td><td>38%</td><td>47%</td><td>68%</td></tr><tr><td>XLNet-base (入=1.5)</td><td>HEL</td><td>7</td><td>AVG</td><td>48.3</td><td>62%</td><td>53%</td><td>50%</td><td>58%</td><td>49%</td><td>74%</td></tr><tr><td>Trained models(w/gold trees)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random XLNet-base</td><td>fideal</td><td>-</td><td>=</td><td>41.2</td><td>28%</td><td>58%</td><td>29%</td><td>50%</td><td>35%</td><td>41%</td></tr><tr><td>XLNet-base (worst case)</td><td>fideal</td><td>1</td><td>-</td><td>58.0</td><td>47%</td><td>75%</td><td>56%</td><td>71%</td><td>50%</td><td>61%</td></tr><tr><td>XLNet-base (best case)</td><td>fideal</td><td>7</td><td>=</td><td>65.1</td><td>61%</td><td>82%</td><td>67%</td><td>78%</td><td>55%</td><td>73%</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Concretely, the union of $f ^ { d } ( \\mathrm { J S D } )$ and $g _ { ( 9 , 1 3 ) } ^ { d }$ —the best duo for the XLNet-base on PTB—achieves 39.2 in sentence-level F1 on MNLI, which is very close to the top performance (39.3) we can obtain when leveraging the XLNet-base. It is also worth noting that GPT-2 and XLNet are efficient in capturing PP and ADVP respectively, regardless of the data domain and the choice of $f$ and $g$ . ",
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+    {
+        "type": "text",
+        "text": "6 FURTHER ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "6.1 PERFORMANCE COMPARISON BY LAYER ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To take a closer look at how different the layers of the pre-trained LMs are in terms of parsing performance, we retrieve the best sentence-level F1 scores from the lth layer of an LM from all combinations of $f$ and $g _ { l }$ , with regard to the PTB and MNLI respectively. Then we plot the scores as graphs in Figure 2 for the PTB and Figure 4 in Appendix A.4 for the MNLI. Each score is from the models to which the bias is not applied. ",
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+    {
+        "type": "text",
+        "text": "From the graphs, we observe several patterns. First, XLNet-based models outperform other competitors across most of the layers. Second, the best outcomes are largely shown in the middle layers of the LMs akin to the observation from Shen et al. (2019), except for some cases where the first layers (especially in case of MNLI) record the best. Interestingly, GPT-2 shows a decreasing trend in its output values as the layer becomes high, while other models generally exhibit the opposite pattern. Moreover, we discover from raw statistics that regardless of the choice of $f$ and $g _ { l }$ , the parsing performance reported as S-F1 is moderately correlated with the layer number l. In other words, it seems that there are some particular layers in the LMs which are more sensitive to syntactic information. ",
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+    {
+        "type": "text",
+        "text": "6.2 ESTIMATING THE UPPER LIMIT OF DISTANCE MEASURE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although we have introduced effective candidates for $f$ , we explore the potential of extracting more sophisticated trees from pre-trained LMs, supposing we are equipped with a pseudo-optimum $f$ , call it $f _ { \\mathrm { i d e a l } }$ . To obtain $f _ { \\mathrm { i d e a l } }$ , we train a simple linear layer on each layer of the pre-trained LMs with supervision from the gold-standard trees of the PTB training set, while $g$ remains unchanged—the pre-trained LMs are frozen during training. We choose the XLNet-base model as a representative for the pre-trained LMs. For more details about experimental settings, refer to Appendix A.6. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Table 2, we present three new results using $f _ { \\mathrm { i d e a l } }$ . As a baseline, we report the performance of $f _ { \\mathrm { i d e a l } }$ with a randomly initialized XLNet-base. Then, we list the worst and best result of $f _ { \\mathrm { i d e a l } }$ according to $g$ , when it is combined with the pre-trained LM. We here mention some findings from the experiment. First, comparing the results with the pre-trained LM against one with the random LM, we reconfirm that pre-training an LM apparently enables the model to capture some aspects of grammar. Specifically, our method is comparable to the linear model trained on the gold-standard trees. Second, we find that there is a tendency for the performance of $f _ { \\mathrm { i d e a l } }$ relying on different LM layers to follow one we already observed in Section 6.1—the best result comes from the middle layers of the LM while the worst from the first and last layer. Third, we identify that the LM has a potential to show improved performance on grammar induction by adopting a more sophisticated $f$ . However, we emphasize that our method equipped with a simple $f$ without gold-standard trees is remarkably reasonable in recognizing constituency grammar, being especially good at catching ADJP and ADVP. ",
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+    {
+        "type": "text",
+        "text": "6.3 CONSTITUENCY TREE EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We visualize several gold-standard trees from PTB and the corresponding tree predictions for comparison. For more details, we refer readers to Appendix A.7. ",
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we propose a simple but effective method of inducing constituency trees from pretrained language models in a zero-shot learning fashion. Furthermore, we report a set of intuitive findings observed from the extracted trees, demonstrating that the pre-trained LMs exhibit some properties similar to constituency grammar. In addition, we show that our method can serve as a strong baseline for English grammar induction when combined with (or even without) appropriate linguistic biases. ",
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+    {
+        "type": "text",
+        "text": "On the other hand, there are still remaining issues that can be good starting points for future work. First, although we analyzed our method based on two popular datasets, we focused only on English grammar induction. As each language has its own properties (and correspondingly would need individualized biases), it is desirable to expand this work to other languages. Second, it would also be desirable to investigate whether further improvements can be achieved by directly grafting the pre-trained LMs onto existing grammar induction models. Lastly, by verifying the usefulness of the knowledge from the pre-trained LMs and linguistic biases for grammar induction, we want to point out that there is still much room for improvement in the existing grammar induction models. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Reinald Kim Amplayo and the anonymous reviewers for their thoughtful and valuable comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF2016M3C4A7952587). ",
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+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017. ",
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+        "text": "Elena Voita, David Talbot, Fedor Moiseev, Rico Sennrich, and Ivan Titov. Analyzing multi-head self-attention: Specialized heads do the heavy lifting, the rest can be pruned. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 5797–5808, July 2019. ",
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+        "text": "Adina Williams, Andrew Drozdov, and Samuel R. Bowman. Do latent tree learning models identify meaningful structure in sentences? Transactions of the Association for Computational Linguistics, 6:253–267, 2018a. ",
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+        "text": "Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Russ R Salakhutdinov, and Quoc V Le. Xlnet: Generalized autoregressive pretraining for language understanding. Advances in Neural Information Processing Systems 32, pp. 5754–5764, 2019. ",
+        "bbox": [
+            176,
+            440,
+            823,
+            483
+        ],
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+    {
+        "type": "text",
+        "text": "Dani Yogatama, Phil Blunsom, Chris Dyer, Edward Grefenstette, and Wang Ling. Learning to compose words into sentences with reinforcement learning. In International Conference on Learning Representations, 2017. ",
+        "bbox": [
+            171,
+            492,
+            823,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/4557215c033955f375eec54b05ca4b1a0d588cb45906a8bd42cdd74d3224d2f4.jpg",
+        "image_caption": [
+            "A.1 ATTENTION HEATMAP EXAMPLES ",
+            "Figure 3: Self-attention heatmaps for the average of all attention distributions from the 7th layer of the XLNet-base model, given a set of input sentences. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            232,
+            171,
+            779,
+            868
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 TREE CONSTRUCTION ALGORITHM WITH SYNTACTIC DISTANCES",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            674,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Syntactic Distances to Binary Constituency Tree (originally from Shen et al. (2018a)) ",
+        "text_level": 1,
+        "bbox": [
+            168,
+            135,
+            820,
+            150
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1: $S = [ w _ { 1 } , w _ { 2 } , \\ldots , w _ { n } ] ;$ a sequence of words in a sentence of length $_ n$ .   \n2: $\\mathbf { d } = [ d _ { 1 } , d _ { 2 } , \\dotsc , d _ { n - 1 } ]$ : a vector whose elements are the distances between every two adjacent words.   \n3: function TREE $( S , { \\bf d } )$   \n4: if $\\mathbf { d } = \\left[ \\begin{array} { l l l l l } \\end{array} \\right]$ then   \n5: node $ \\mathrm { L e a f } ( S [ 0 ] )$   \n6: else   \n7: $i \\gets \\arg \\operatorname* { m a x } _ { i } ( { \\bf d } )$   \n8: $\\mathrm { c h i l d } _ { l } \\gets \\mathrm { T R E E } ( S _ { \\leq i } , { \\bf d } _ { < i } )$   \n9: $\\mathrm { c h i l d } _ { r } \\gets \\mathrm { T R E E } ( S _ { > i } , { \\bf d } _ { > i } )$   \n10: node ← Node(childl, childr)   \n11: end if   \n12: return node   \n13: end function ",
+        "bbox": [
+            176,
+            154,
+            821,
+            319
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.3 DISTANCE MEASURE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            347,
+            450,
+            361
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/d40ea323231b5354516b3efc3ab6d7da46e884c0f6f13343496af8876e419fcd.jpg",
+        "table_caption": [
+            "Table 3: The definitions of distance measure functions for computing syntactic distances between two adjacent words in a sentence. Note that $\\mathbf { r } = g ^ { v } ( w _ { i } )$ , $\\mathbf { s } = g ^ { v } ( w _ { i + 1 } )$ , $P = g ^ { d } ( w _ { i } )$ , and $Q =$ $g ^ { d } ( w _ { i + 1 } )$ , respectively. $d$ : hidden embedding size, $n$ : the number of words $( w )$ in a sentence $( S )$ . ",
+            "A.4 PERFORMANCE COMPARISON BY LAYER ON MNLI "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Function（f）Definition</td></tr><tr><td>Functions for intermediate representations (Fu)</td></tr><tr><td>(rTs/((∑(1r):(∑_1s2)2) +1)/2 Cos(r,s) L1(r,s)</td></tr><tr><td>L2(r,s) i-1(r -si）2） Functions for attention distributions (Fd)</td></tr><tr><td>JSD(PIIQ) ((DKL(P||M)+DKL(Q||M))/2) where M=(P+Q)/2</td></tr><tr><td>and DKL(A|lB) =∑ω∈sA(w)log(A(ω)/B(ω)) HEL(P,Q)</td></tr></table>",
+        "bbox": [
+            267,
+            434,
+            728,
+            602
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/57638ca8b998e629ceb0f21baa45992e8d097a5cba5affbe54347892a5bf5588.jpg",
+        "image_caption": [
+            "Figure 4: The best layer-wise S-F1 scores of each LM instance on the MNLI test set. (Left) The performance of the X-‘base’ models. (Right) The performance of the X-‘large’ models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            665,
+            812,
+            849
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5 EXPERIMENTAL RESULTS ON MNLI ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            468,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/d2e610d715e4f5c9829f017b6e3bde488a70b8d3bdbe2e708155e0df63d6dc09.jpg",
+        "table_caption": [
+            "Table 4: Results on the MNLI test set. Bold numbers correspond to the top 3 results for each column. L: layer number, A: attention head number (AVG: the average of all attentions). $\\dagger$ : Results reported by Htut et al. (2018) and Drozdov et al. (2019). $\\ddagger$ : Approaches in which $\\overline { { \\mathbf { C O O } } }$ parser is utilized. $^ *$ : These results are not strictly comparable to ours, due to the difference in data preprocessing. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>f</td><td>L</td><td>A</td><td>S-F1</td><td>SBAR</td><td>NP</td><td>VP</td><td>PP</td><td>ADJP</td><td>ADVP</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Random Trees</td><td></td><td></td><td></td><td>21.4</td><td>11%</td><td>25%</td><td>16%</td><td>22%</td><td>22%</td><td>27%</td></tr><tr><td>Balanced Trees</td><td></td><td></td><td></td><td>20.0</td><td>8%</td><td>29%</td><td>11%</td><td>20%</td><td>22%</td><td>32%</td></tr><tr><td>Left Branching Trees</td><td></td><td></td><td></td><td>8.4</td><td>6%</td><td>13%</td><td>1%</td><td>4%</td><td>1%</td><td>8%</td></tr><tr><td>Right Branching Trees</td><td></td><td></td><td></td><td>51.9</td><td>65%</td><td>28%</td><td>75%</td><td>47%</td><td>45%</td><td>30%</td></tr><tr><td>Random XLNet-base (F)</td><td></td><td></td><td></td><td>22.0</td><td>12%</td><td>26%</td><td>15%</td><td>22%</td><td>22%</td><td>25%</td></tr><tr><td>Random XLNet-base (Fd)</td><td>=</td><td></td><td>=</td><td>23.5</td><td>14%</td><td>26%</td><td>18%</td><td>22%</td><td>22%</td><td>25%</td></tr><tr><td>Pre-trained LMs (w/o bias)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>HEL</td><td>9</td><td>10</td><td>36.1</td><td>36%</td><td>37%</td><td>34%</td><td>45%</td><td>26%</td><td>42%</td></tr><tr><td>BERT-large</td><td>JSD</td><td>17</td><td>10</td><td>37.0</td><td>38%</td><td>32%</td><td>34%</td><td>50%</td><td>22%</td><td>39%</td></tr><tr><td>GPT2</td><td>JSD</td><td>1</td><td>10</td><td>44.0</td><td>43%</td><td>53%</td><td>31%</td><td>60%</td><td>24%</td><td>40%</td></tr><tr><td>GPT2-medium</td><td>JSD</td><td>3</td><td>12</td><td>49.1</td><td>57%</td><td>32%</td><td>61%</td><td>44%</td><td>35%</td><td>37%</td></tr><tr><td>RoBERTa-base</td><td>JSD</td><td>10</td><td>9</td><td>36.2</td><td>26%</td><td>35%</td><td>34%</td><td>50%</td><td>23%</td><td>44%</td></tr><tr><td>RoBERTa-large</td><td>JSD</td><td>3</td><td>6</td><td>39.8</td><td>20%</td><td>28%</td><td>35%</td><td>30%</td><td>28%</td><td>27%</td></tr><tr><td>XLNet-base</td><td>HEL</td><td>1</td><td>6</td><td>39.0</td><td>25%</td><td>39%</td><td>28%</td><td>59%</td><td>35%</td><td>44%</td></tr><tr><td>XLNet-large</td><td>HEL</td><td>1</td><td>15</td><td>42.2</td><td>32%</td><td>49%</td><td>27%</td><td>62%</td><td>32%</td><td>49%</td></tr><tr><td>Pre-trained LMs (w/bias入=1.5)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-base</td><td>HEL</td><td>2</td><td>12</td><td>52.7</td><td>64%</td><td>35%</td><td>70%</td><td>50%</td><td>46%</td><td>30%</td></tr><tr><td>BERT-large</td><td>HEL</td><td>4</td><td>4</td><td>51.7</td><td>63%</td><td>31%</td><td>71%</td><td>49%</td><td>46%</td><td>30%</td></tr><tr><td>GPT2</td><td>HEL</td><td>1</td><td>10</td><td>52.2</td><td>57%</td><td>53%</td><td>49%</td><td>62%</td><td>32%</td><td>42%</td></tr><tr><td>GPT2-medium</td><td>HEL</td><td>2</td><td>1</td><td>53.9</td><td>53%</td><td>57%</td><td>50%</td><td>62%</td><td>29%</td><td>44%</td></tr><tr><td>RoBERTa-base</td><td>HEL</td><td>2</td><td>3</td><td>52.0</td><td>64%</td><td>31%</td><td>72%</td><td>49%</td><td>47%</td><td>30%</td></tr><tr><td>RoBERTa-large</td><td>L1</td><td>23</td><td>=</td><td>52.7</td><td>55%</td><td>40%</td><td>65%</td><td>53%</td><td>43%</td><td>41%</td></tr><tr><td>XLNet-base</td><td>L2</td><td>8</td><td></td><td>54.9</td><td>57%</td><td>49%</td><td>61%</td><td>55%</td><td>44%</td><td>57%</td></tr><tr><td>XLNet-large</td><td>L2</td><td>12</td><td>=</td><td>53.5</td><td>54%</td><td>47%</td><td>59%</td><td>51%</td><td>48%</td><td>60%</td></tr><tr><td>Other models</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PRPN-UPt $</td><td></td><td></td><td></td><td>48.6*</td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>PRPN-LM†‡</td><td></td><td></td><td></td><td>50.4*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DIORA†</td><td></td><td></td><td></td><td>51.2*</td><td></td><td>=</td><td></td><td></td><td></td><td>=</td></tr><tr><td>DIORA(+PP)†</td><td></td><td></td><td></td><td>59.0*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            192,
+            204,
+            805,
+            529
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.6 EXPERIMENTAL DETAILS FOR TRAINING IDEAL DISTANCE MEASURE FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            563,
+            766,
+            577
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this part, we present the detailed specifications of the experiments introduced in Section 6.2. We assume $f _ { \\mathrm { i d e a l } }$ is only compatible with the functions in $G ^ { v }$ , as the functions in $G ^ { d }$ are not suitable for training as the sizes of the representations provided by $G ^ { d }$ are variable according to the length of an input sentence. To train the pseudo-optimal function $f _ { \\mathrm { i d e a l } }$ , we minimize a pair-wise learning-to-rank loss following previous work (Burges et al., 2005; Shen et al., 2018a): ",
+        "bbox": [
+            174,
+            587,
+            825,
+            657
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e27c12de29e9d89362a298f0feda60496cccd36773d8cfaebd549bd738f43405.jpg",
+        "text": "$$\nL _ { \\mathrm { d i s t } } ^ { \\mathrm { r a n k } } = \\sum _ { i , j > i } [ 1 - \\mathrm { s i g n } ( d _ { i } ^ { \\mathrm { g o l d } } - d _ { j } ^ { \\mathrm { g o l d } } ) ( d _ { i } ^ { \\mathrm { p r e d } } - d _ { j } ^ { \\mathrm { p r e d } } ) ] ^ { + } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            664,
+            673,
+            699
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where $d ^ { \\mathrm { g o l d } }$ and $d ^ { \\mathrm { p r e d } }$ are computed from the gold tree and our predicted one, respectively. $[ x ] ^ { + }$ is defined as $m a x ( 0 , x )$ . We train the $f _ { \\mathrm { i d e a l } }$ with the PTB training set for 5 epochs. Each batch of the training set contains 16 sentences. We use an ADAM optimizer (Kingma & Ba, 2014) with the learning rate 5e-4. We train the variations of $f _ { \\mathrm { i d e a l } }$ differentiated by the choice of $g$ in $G ^ { v }$ and report the best result in the Table 2. Each $f _ { \\mathrm { i d e a l } }$ is chosen based on its performance on the PTB validation set. Considering the randomness of training, every result for $f _ { \\mathrm { i d e a l } }$ is averaged over 3 different trials. ",
+        "bbox": [
+            174,
+            707,
+            825,
+            792
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.7 CONSTITUENCY TREE EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            452,
+            821
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We randomly select six sentences from PTB and visualize their trees, where the resulting group of trees for each sentence consists of a gold constituency tree and two induced trees (one without the right-skewness bias and the other with the bias) from our best model—XLNet-base. The ‘T’ character in the induced trees indicates a dummy tag. ",
+        "bbox": [
+            174,
+            833,
+            823,
+            890
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f5466c302db8af3968e112d6a6fc4110fb4ef20f050196c5a7a65dfa8733b06.jpg",
+        "image_caption": [
+            "Figure 5: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘These include a child-care initiative and extensions of soon-to-expire tax breaks for low-income housing and research-and-development expenditures’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            258,
+            799,
+            694
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/6138640de123c6c8c0d5b6307a743fd5691924d2c7b4d57ffee6d0c560e437e7.jpg",
+        "image_caption": [
+            "Figure 6: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘But HOFI ‘s first offer would have given Ideal ’s other shareholders about $10 \\%$ of the combined company’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            228,
+            799,
+            726
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d0c9332a71f76b64f2aa2c57d9f2bc965adeb519c3ddb317aa2f119b1f2525e.jpg",
+        "image_caption": [
+            "Figure 7: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘It was Friday the $1 3 \\mathrm { t h }$ and the stock market plummeted nearly 200 points’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            162,
+            751,
+            770
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/4a2a79c63bc955bde02bf004df0ecf68c2fd14a5b3664a006bf8bc3680db54a9.jpg",
+        "image_caption": [
+            "Figure 8: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘Until recently national governments in Europe controlled most of the air time and allowed little or no advertising’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            227,
+            797,
+            732
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/3317260942a98cdd94ddf89da58fac2f95eb78e8752f7828db2b29796cd36683.jpg",
+        "image_caption": [
+            "Figure 9: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘Nevertheless Ms. Garzarelli said she was swamped with phone calls over the weekend from nervous shareholders’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            199,
+            797,
+            758
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/60ab9c2749c684e51447319241b1e8da126a5f9ff2078536ebeed7661fa5fa58.jpg",
+        "image_caption": [
+            "Figure 10: Gold (top) and predicted trees (one without the bias in the middle, the other with the bias at the bottom) for the sentence ‘Analysts and competitors however doubt the numbers were that high’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            117,
+            753,
+            792
+        ],
+        "page_idx": 21
+    }
+]

parse/train/Syr8Qc1CW/Syr8Qc1CW.md

@@ -0,0 +1,318 @@
+# DNA-GAN: LEARNING DISENTANGLED REPRESEN-TATIONS FROM MULTI-ATTRIBUTE IMAGES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Disentangling factors of variation has always been a challenging problem in representation learning. Existing algorithms suffer from many limitations, such as unpredictable disentangling factors, bad quality of generated images from encodings, lack of identity information, etc. In this paper, we proposed a supervised algorithm called DNA-GAN trying to disentangle different attributes of images. The latent representations of images are DNA-like, in which each individual piece represents an independent factor of variation. By annihilating the recessive piece and swapping a certain piece of two latent representations, we obtain another two different representations which could be decoded into images. In order to obtain realistic images and also disentangled representations, we introduced the discriminator for adversarial training. Experiments on Multi-PIE and CelebA datasets demonstrate the effectiveness of our method and the advantage of overcoming limitations existing in other methods.
+
+# 1 INTRODUCTION
+
+The success of machine learning algorithms depends on data representation, because different representations can entangle different explanatory factors of variation behind the data. Although prior knowledge can help us design representations, the vast demand of AI algorithms in various domains cannot be met, since feature engineering is labor-intensive and needs domain expert knowledge. Therefore, algorithms that can automatically learn good representations of data will definitely make it easier for people to extract useful information when building classifiers or predictors.
+
+Of all criteria of learning good representations as discussed in Bengio et al. (2013), disentangling factors of variation is an important one that helps separate various explanatory factors. For example, given a human-face image, we can obtain various information about the person, including gender, hair style, facial expression, with/without eyeglasses and so on. All of these information are entangled in a single image, which renders the difficulty of training a single classifier to handle different facial attributes. If we could obtain a disentangled representation of the face image, we may build up only one classifier for multiple attributes.
+
+In this paper, we propose a supervised method called DNA-GAN to obtain disentangled representations of images. The idea of DNA-GAN is motivated by the DNA double helix structure, in which different kinds of traits are encoded in different DNA pieces. We make a similar assumption that different visual attributes in an image are controlled by different pieces of encodings in its latent representations. In DNA-GAN, an encoder is used to encode an image to the attribute-relevant part and the attribute-irrelevant part, where different pieces in the attribute-relevant part encode information of different attributes, and the attribute-irrelevant part encodes other information. For example, given a facial image, we are trying to obtain a latent representation that each individual part controls different attributes, such as hairstyles, genders, expressions and so on. Though annihilating recessive pieces and swapping certain pieces, we can obtain novel crossbreeds that can be decoded into new images. By the adversarial discriminator loss and the reconstruction loss, DNA-GAN can reconstruct the input images and generate new images with new attributes. Each attribute is disentangled from others gradually though iterative training. Finally, we are able to obtain disentangled representations in the latent representations.
+
+The summary of contributions of our work is as follows:
+
+1. We propose a supervised algorithm called DNA-GAN, that is able to disentangle multiple attributes as demonstrated by the experiments of interpolating multiple attributes on MultiPIE (Gross et al., 2010) and CelebA (Liu et al., 2015) datasets.   
+2. We introduce the annihilating operation that prevents from trivial solutions: the attributerelevant part encodes information of the whole image instead of a certain attribute.   
+3. We employ iterative training to address the problem of unbalanced multi-attribute image data, which was theoretically proved to be more efficient than random image pairs.
+
+# 2 RELATED WORK
+
+Traditional representation learning algorithms focus on (1) probabilistic graphical models, characterized by Restricted Boltzmann Machine (RBM) (Smolensky, 1986), Autoencoder (AE) and their variants; (2) manifold learning and geometrical approaches, such as Principal Components Analysis (PCA) (Pearson, 1901), Locally Linear Embedding (LLE) (Roweis & Saul, 2000), Local Coordinate Coding (LCC) (Yu et al., 2009), etc. However, recent research has actively focused on developing deep probabilistic models that learn to represent the distribution of data. Kingma & Welling (2013) employs an explicit model distribution and uses variational inference to learn its parameters. As the generative adversarial networks (GAN) (Goodfellow et al., 2014) has been invented, many implicit models are developed.
+
+In the semi-supervised setting, Siddharth et al. (2016) learns a disentangled representations by using an auxiliary variable. Bouchacourt et al. (2017) proposes the ML-VAE that can learn disentangled representations from a set of grouped observations. In the unsupervised setting, InfoGAN (Chen et al., 2016) tries to maximize mutual information between a small subset of latent variables and observations by introducing an auxiliary network to approximate the posterior. However, it relies much on the a-priori choice of distributions and suffered from unstable training. Another popular unsupervised method $\beta$ -VAE (Higgins et al., 2016), adapted from VAE, lays great stress on the KL distance between the approximate posterior and the prior. However, unsupervised approaches do not anchor a specific meaning into the disentanglement.
+
+More closely with our method, supervised methods take the advantage of labeled data and try to disentangle the factors as expected. DC-IGN (Kulkarni et al., 2015) asks the active attribute to explain certain factor of variation by feeding the other attributes by the average in a mini-batch. TD-GAN (Wang et al., 2017) uses a tag mapping net to boost the quality of disentangled representations, which are consistent with the representations extracted from images through the disentangling network. Besides, the quality of generated images is improved by implementing the adversarial training strategy. However, the identity information should be labeled so as to preserve the id information when swapping attributes, which renders the limitation of applying it into many other datasets without id labels. IcGAN (Perarnau et al., 2016) is a multi-stage training algorithm that first takes the advantage of cGAN (Mirza & Osindero, 2014) to learn a map from latent representations and conditional information to real images, and then learn its inverse map from images to the latent representations and conditions in a supervised manner. The overall effect depends on each training stage, therefore it is hard to obtain satisfying images. Unlike these models, our model requires neither explicit id information in labels nor multi-stage training.
+
+Many works have studied the image-to-image translation between unpaired image data using GANbased architectures, see Isola et al. (2016), Taigman et al. (2016), Zhu et al. (2017), Liu et al. (2017) and Zhou et al. (2017). Interestingly, these models require a form of 0/1 weak supervision that is similar to our setting. However, they are circumscribed in two image domains which are opposite to each other with respect to a single attribute. Our model differs from theirs as we generalize to the case of multi-attribute image data. Specifically, we employ the strategy of iterative training to overcome the difficulty of training on unbalanced multi-attribute image datasets.
+
+# 3 DNA-GAN APPROACH
+
+In this section, we formally outline our method. A set $\mathcal { X }$ of multi-labeled images and a set of labels $\mathcal { V }$ are considered in our setting. Let $\{ ( \mathbf { X } ^ { 1 } , \mathbf { Y } ^ { 1 } ) , \ldots , ( \mathbf { X } ^ { m } , \mathbf { Y } ^ { m } ) \}$ denote the whole training dataset, where $\mathbf { X } ^ { i } \in \mathcal X$ is the $i$ -th image with its label $\mathbf { Y } ^ { i } \in \mathcal { V }$ . The small letter $m$ denotes the number of samples in set $\mathcal { X }$ and $n$ denotes the number of attributes. The label $\mathbf { Y } ^ { i } = ( \mathbf { y } _ { 1 } ^ { i } , \ldots , \mathbf { y } _ { n } ^ { i } )$ is a $n$ -dimensional vector where each element represents whether $\mathbf { X } ^ { i }$ has certain attribute or not. For example, in the case of labels with three candidates [Bangs, Eyeglasses, Smiling], the facial image $\mathbf { X } ^ { i }$ whose label is $\mathbf { Y } ^ { i } = ( 1 , 0 , 1 )$ should depict a smiling face with bangs and no eyeglasses.
+
+# 3.1 MODEL
+
+As shown in Figure 1, DNA-GAN is mainly composed of three parts: an encoder $( { \mathrm { E n c } } )$ , a decoder (Dec) and a discriminator (D). The encoder maps the real-world images $A$ and $B$ into two latent disentangled representations
+
+$$
+\operatorname { E n c } ( A ) = [ a _ { 1 } , \dots , a _ { i } , \dots , a _ { n } , z _ { a } ] , \quad \operatorname { E n c } ( B ) = [ b _ { 1 } , \dots , b _ { i } , \dots , b _ { n } , z _ { b } ]
+$$
+
+where $[ a _ { 1 } , \ldots , a _ { i } , \ldots , a _ { n } ]$ is called the attribute-relevant part, and $z _ { a }$ is called the attribute-irrelevant part. $a _ { i }$ is supposed to be a DNA piece that controls $\mathbf { y } _ { i }$ , the $i$ -th attribute in the label, and $z _ { a }$ is for keeping other silent factors which do not appear in the attribute list as well as image identity information. The same thing applies for $\operatorname { E n c } ( B )$ .
+
+![](images/7d30a549e7003d1f960dc09538b3a978887025f279e99544c245587f3e976cda.jpg)  
+Figure 1: DNA-GAN architecture.
+
+e focus on one attribute each time in ourare required to have different labels, i.e. we a and $i$ $A$ and, re$B$ $( \mathbf { y } _ { 1 } ^ { A } , \ldots , 1 _ { i } ^ { A } , \ldots , \mathbf { y } _ { n } ^ { \bar { A } } )$ $( \mathbf { y } _ { 1 } ^ { B } , \ldots , 0 _ { i } ^ { B } , \ldots , \mathbf { y } _ { n } ^ { B } )$ spectively. In our convention, $A$ is always for the dominant pattern, while $B$ is for the recessive pattern. We copy $\operatorname { E n c } ( A )$ directly as the latent representation of $A _ { 1 }$ , and annihilate $b _ { i }$ in the copy of $\operatorname { E n c } ( B )$ as the latent representation of $B _ { 1 }$ . The annihilating operation means replacing all elements with zeros, and plays a key role in disentangling the attribute, which we will discuss in detail in Section 3.3. By swapping $a _ { i }$ and $0 _ { i }$ , we obtain two new latent representations $\left[ a _ { 1 } , \ldots , 0 _ { i } , \ldots , a _ { n } , z _ { a } \right]$ and $\left[ b _ { 1 } , \dots , a _ { i } , \dots , b _ { n } , z _ { b } \right]$ that are supposed to be decoded into $A _ { 2 }$ and $B _ { 2 }$ , respectively. Though a decoder Dec, we can get four newly generated images $A _ { 1 }$ , $B _ { 1 }$ , $A _ { 2 }$ and $B _ { 2 }$ .
+
+$$
+\begin{array} { r l } & { \mathrm { D e c } ( [ a _ { 1 } , \dots , a _ { i } , \dots , a _ { n } , z _ { a } ] ) = A _ { 1 } , \quad \mathrm { D e c } ( [ b _ { 1 } , \dots , 0 _ { i } , \dots , b _ { n } , z _ { b } ] ) = B _ { 1 } } \\ & { \mathrm { D e c } ( [ a _ { 1 } , \dots , 0 _ { i } , \dots , a _ { n } , z _ { a } ] ) = A _ { 2 } , \quad \mathrm { D e c } ( [ b _ { 1 } , \dots , a _ { i } , \dots , b _ { n } , z _ { b } ] ) = B _ { 2 } } \end{array}
+$$
+
+Out of these four children, $A _ { 1 }$ and $B _ { 1 }$ are reconstructions of $A$ and $B$ , while $A _ { 2 }$ and $B _ { 2 }$ are novel crossbreeds. The reconstruction losses between $A$ and $A _ { 1 }$ , $B$ and $B _ { 1 }$ ensure the quality of reconstructed samples. Besides, using an adversarial discriminator D that helps make generated samples $A _ { 2 }$ indistinguishable from $B$ , and $B _ { 2 }$ indistinguishable from $A$ , we can enforce attribute-related information to be encoded in $a _ { i }$ .
+
+# 3.2 LOSS FUNCTIONS
+
+Given two images $A$ and $B$ and their labels $\begin{array} { c c l } { \mathbf { Y } ^ { A } } & { = } & { ( \mathbf { y } _ { 1 } ^ { A } , \ldots , \mathbf { l } _ { i } ^ { A } , \ldots , \mathbf { y } _ { n } ^ { A } ) } \end{array}$ and $\begin{array} { r l } { \mathbf { Y } ^ { B } } & { { } = } \end{array}$ $( \mathbf { y } _ { 1 } ^ { B } , \ldots , 0 _ { i } ^ { B } , \ldots , \mathbf { y } _ { n } ^ { B } )$ which are different at the $i$ -th position, the data flow can be summarized by (1) and (2). We force the $i$ -th latent encoding of $B$ to be zero in order to prevent from trivial solutions as we will discuss in Section 3.3.
+
+The encoder and decoder receive two types of losses: (1) the reconstruction loss,
+
+$$
+L _ { r e c o n s t r u c t } = \| A - A _ { 1 } \| _ { 1 } + \| B - B _ { 1 } \| _ { 1 }
+$$
+
+which measures the reconstruction quality after a sequence of encoding and decoding; (2) the standard GAN loss,
+
+$$
+L _ { G A N } = - \mathbb { E } [ \log ( \mathrm { D } ( A _ { 2 } | \mathbf { y } _ { i } ^ { A } = 1 ) ) ] - \mathbb { E } [ \log ( \mathrm { D } ( B _ { 2 } | \mathbf { y } _ { i } ^ { B } = 0 ) ) ]
+$$
+
+which measures how realistic the generated images are. The discriminator takes the generated image and the $i$ -th element of its label as inputs, and outputs a number which indicates how realistic the input image is. The larger the number is, the more realistic the image is. Omitting the coefficient, the loss function for the encoder and decoder is
+
+$$
+L _ { G } = L _ { r e c o n s t r u c t } + L _ { G A N } .
+$$
+
+The discriminator D receives the standard GAN discriminator loss
+
+$$
+\begin{array} { r l } & { L _ { D _ { 1 } } = - \mathbb { E } [ \log ( \mathrm { D } ( A | \mathbf { y } _ { i } ^ { A } = 1 ) ) ] - \mathbb { E } [ \log ( 1 - \mathrm { D } ( B _ { 2 } | \mathbf { y } _ { i } ^ { A } = 1 ) ) ] } \\ & { L _ { D _ { 0 } } = - \mathbb { E } [ \log ( \mathrm { D } ( B | \mathbf { y } _ { i } ^ { B } = 0 ) ) ] - \mathbb { E } [ \log ( 1 - \mathrm { D } ( A _ { 2 } | \mathbf { y } _ { i } ^ { B } = 0 ) ) ] } \\ & { L _ { D } = L _ { D _ { 1 } } + L _ { D _ { 0 } } } \end{array}
+$$
+
+where $L _ { D _ { 1 } }$ drives $\mathrm { D }$ to tell $A$ from $B _ { 2 }$ , and $L _ { D _ { 0 } }$ drives $\mathrm { D }$ to tell $B$ from $A _ { 2 }$ .
+
+# 3.3 ANNIHILATING OPERATION PREVENTS FROM TRIVIAL SOLUTIONS
+
+Through experiments, we observe that there exist trivial solutions to our model without the annihilating operation. We just take the single-attribute case as an example. Suppose that $\operatorname { E n c } ( A ) = [ a , z _ { a } ]$ and $\bar { \mathrm { E n c } } ( B ) = [ b , \bar { z } _ { b } ]$ , we can get four children without annihilating operation
+
+$$
+A _ { 1 } = \mathrm { D e c } ( [ a , z _ { a } ] ) , \quad B _ { 1 } = \mathrm { D e c } ( [ b , z _ { b } ] ) , \quad A _ { 2 } = \mathrm { D e c } ( [ b , z _ { a } ] ) , \quad B _ { 2 } = \mathrm { D e c } ( [ a , z _ { b } ] )
+$$
+
+The reconstruction loss makes it invertible between the latent encoding space and image space. The adversarial discriminator D is supposed to disentangle the attribute from other information by telling whether $A _ { 2 }$ looks as real as $B$ and $B _ { 2 }$ looks as real as $A$ or not. As we know that the generative adversarial networks give the best solution when achieving the Nash equilibrium. But without the annihilating operation, information of the whole image could be encoded into the attribute-relevant part, which means
+
+$$
+\operatorname { E n c } ( A ) = [ a , 0 ] , \quad \operatorname { E n c } ( B ) = [ b , 0 ]
+$$
+
+Therefore, we obtain the following four children
+
+$$
+A _ { 1 } = \operatorname { D e c } ( [ a , 0 ] ) , \quad B _ { 1 } = \operatorname { D e c } ( [ b , 0 ] ) , \quad A _ { 2 } = \operatorname { D e c } ( [ b , 0 ] ) , \quad B _ { 2 } = \operatorname { D e c } ( [ a , 0 ] )
+$$
+
+In this situation, the discriminator $\mathrm { D }$ cannot discriminate $A _ { 2 }$ from $B$ , since they share the same latent encodings. By reconstruction loss, $A _ { 2 }$ and $B$ are exactly the same image, which is against our expectation that $A _ { 2 }$ should depict the person from $A$ with the attribute borrowed from $B$ . The same thing happens to $B _ { 2 }$ and $A$ as well.
+
+To prevent from learning trivial solutions, we adopt the annihilating operation by replacing the recessive pattern $b$ with a zero tensor of the same $\bar { \mathrm { s i z e } ^ { 1 } }$ . If information of the whole image were encoded into the attribute-relevant part, the four children in this case are
+
+$$
+A _ { 1 } = \operatorname { D e c } ( [ a , 0 ] ) , \quad B _ { 1 } = \operatorname { D e c } ( [ 0 , 0 ] ) , \quad A _ { 2 } = \operatorname { D e c } ( [ 0 , 0 ] ) , \quad B _ { 2 } = \operatorname { D e c } ( [ a , 0 ] )
+$$
+
+The encodings of $B _ { 1 }$ and $A _ { 2 }$ contain no information at all, thus neither the person in $B _ { 1 }$ nor $A _ { 2 }$ who is supposed to be the same as in $B$ can be reconstructed by Dec. This forces the attribute-irrelevant part to encode some information of images.
+
+# 3.4 ITERATIVE TRAINING
+
+To reduce the difficulty of disentangling multiple attributes, we take the strategy of iterative training: we update our model using a pair of images with opposite labels at a certain position each time. Suppose that we are at the $i$ -th position, the label of image $A$ is $( \mathbf { y } _ { 1 } ^ { A } , \ldots , 1 _ { i } ^ { A } , \cdot \cdot . , \mathbf { y } _ { n } ^ { A } )$ , while the label of image $B$ is $( \mathbf { y } _ { 1 } ^ { B } , \ldots , 0 _ { i } ^ { B } , \ldots , \mathbf { y } _ { n } ^ { B } )$ . During each iteration, as $i$ goes through from 1 to $n$ repeatedly, our model fed with such a pair of images can disentangle multiple attributes one-by-one.
+
+Compared with training with random pairs of images, iterative training is proved to be more effective. Random pairs of images means randomly selecting pairs of images each time without label constraints. A pair of images with different labels is called a useful pair.
+
+We theoretically show that our iterative training is much more efficient than random image pairs especially when the dataset is unbalanced. All proofs can be found in the Appendix.
+
+Theorem 1. Let $\mathcal { X } = \{ ( \mathbf { X } ^ { 1 } , \mathbf { Y } ^ { 1 } ) , \dots , ( \mathbf { X } ^ { m } , \mathbf { Y } ^ { m } ) \}$ denote the whole multi-attribute image dataset,   
+where $\mathbf { X } ^ { i }$ is a multi-attribute image and its label $\mathbf { \dot { Y } } ^ { i } = ( \mathbf { y } _ { 1 } ^ { i } , \dots , \mathbf { y } _ { n } ^ { i } )$ is an $n$ -dimensional vector.   
+Therelabel iterat reis ns ally , anded $2 ^ { n }$ ls, denoted by . To select all lecting pairs an $\mathcal { L } = \{ l _ { 1 } , \ldots , l _ { 2 ^ { n } } \}$ . The number of ionce, the expecteding are denoted by ges wmberand $l _ { i }$ $m _ { i }$ $\textstyle \sum _ { i = 1 } ^ { 2 ^ { n } } m _ { i } = m$ $\mathrm { E _ { 1 } }$ $\mathrm { E _ { 2 } }$ respectively. Then,
+
+$$
+\begin{array} { l } { \mathrm { E } _ { 1 } = m ^ { 2 } \left( 1 + \displaystyle \frac { 1 } { 2 } + \cdot \cdot \cdot + \frac { 1 } { m ^ { 2 } - \sum _ { i = 1 } ^ { 2 n } m _ { i } ^ { 2 } } \right) } \\ { \mathrm { E } _ { 2 } \leq 2 n \cdot \displaystyle \operatorname* { m a x } _ { s = 1 , \ldots , n } \displaystyle \sum _ { i \in I _ { s } , j \in J _ { s } } m _ { i } m _ { j } \left( 1 + \frac { 1 } { 2 } + \cdot \cdot \cdot + \frac { 1 } { m ^ { 2 } - \sum _ { k _ { 1 } = 1 } ^ { 2 n - 1 } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } } \right) } \end{array}
+$$
+
+where $I _ { s }$ represents the indices of labels where the s-th element is $1$ , and $J _ { s }$ represents the indices of labels where the $s$ -th element is 0.
+
+Definition 1. (Balancedness) Define the balancedness of a dataset $\mathcal { X }$ described above with respect to the $s$ -th attribute as follows:
+
+$$
+\rho _ { s } = \frac { \sum _ { i \in I _ { s } } m _ { i } } { \sum _ { j \in J _ { s } } m _ { j } }
+$$
+
+where $I _ { s }$ represents the indices of labels where the $s$ -th element is 1, and $J _ { s }$ represents the indices of labels where the $s$ -th element is 0.
+
+Theorem 2. We have $\mathrm { E } _ { 2 } \leq \mathrm { E } _ { 1 }$ , when
+
+$$
+n \leq \operatorname* { m i n } _ { s } \frac { ( \rho _ { s } + 1 ) ^ { 2 } } { 2 \rho _ { s } } .
+$$
+
+Specifically, $\mathrm { E } _ { 2 } \leq \mathrm { E } _ { 1 }$ holds true for all $n \leq 2$
+
+The property of the function $( \rho + 1 ) ^ { 2 } / ( 2 \rho )$ suits well with the definition of balancedness, because it attains the same value for $\rho$ and $1 / \rho$ , which is invariant to different labeling methods. Its value gets larger as the dataset becomes more unbalanced. The minimum is obtained at $\rho = 1$ , which is the case of a balanced dataset.
+
+Theorem 2 demonstrates that the iterative training mechanism is always more efficient than random pairs of images when the number of attributes met the criterion (16). As the dataset becomes more unbalanced, $\dot { ( \rho _ { s } + 1 ) } ^ { 2 } / ( 2 \rho _ { s } )$ goes larger, which means (16) can be more easily satisfied. More importantly, iterative training helps stabilize the training process on unbalanced datasets. For example, given a two-attribute dataset, the number of data of each kind is as follows:
+
+Table 1: The example of an unbalanced two-attribute dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Label</td><td rowspan=1 colspan=1>(0,0)</td><td rowspan=1 colspan=1>(0,1)</td><td rowspan=1 colspan=1>(1,0)</td><td rowspan=1 colspan=1>(1,1)</td></tr><tr><td rowspan=1 colspan=1>Number of data</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>m</td></tr></table>
+
+If $m \gg 1$ is a very large number, then it is highly likely that we will select a pair of images whose labels are $( 1 , 0 )$ and $( 1 , 1 )$ each time by randomly selecting pairs. We ignore the pair of images whose labels are $( 1 , 0 )$ and $( 1 , 0 )$ or $( 1 , 1 )$ and $( 1 , 1 )$ , though these two cases have equal probabilities of being chosen. Because they are not useful pairs, thus do not participated in training. In this case, most of the time the model is trained with respect to the second attribute, which will cause the final learnt model less effective to the first attribute. However, iterative training can prevent this from happening, since we update our model evenly with respect to two attributes.
+
+![](images/42278b7c4d2b1d49753c4445c4a11a8a8c9c804e77fceacb234d8a1b49c813d2.jpg)  
+Figure 2: Manipulating illumination factors on the Multi-PIE dataset. From left to right, the six images in a row are: original images $A$ with light illumination and $B$ with the dark illumination, newly generated images $A _ { 2 }$ and $B _ { 2 }$ by swapping the illumination-relevant piece in disentangled representations, and reconstructed images $A _ { 1 }$ and $B _ { 1 }$ .
+
+# 4 EXPERIMENTS
+
+In this section, we perform different kinds of experiments on two real-world datasets to validate the effectiveness of our methods. We use the RMSProp (Sutskever et al., 2013) optimization method initialized by a learning rate of 5e-5 and momentum 0. All neural networks are equipped with Batch Normalization (Ioffe & Szegedy, 2015) after convolutions or deconvolutions. We used Leaky Relu (Maas et al., 2013) as the activation function in the encoder. Besides, we adopt strategies mentioned in Wasserstein GAN (Arjovsky et al., 2017) for stable training. More details will be available online. We divide all images into training images and test images according to the ratio of 9:1. All of the following results are from test images without cherry-picking.
+
+# 4.1 MULTI-PIE DATABASE
+
+The Multi-PIE (Gross et al., 2010) face database contains over 750,000 images of 337 subjects captured under 15 view points and 19 illumination conditions. We collecte all front faces images of different illuminations and align them based on 5-point landmarks on eyes, nose and mouth. All aligned images are resized into $1 2 8 \times 1 2 8$ as inputs in our experiments. We label the light illumination face images by 1 and the dark illumination face images by 0. As shown in Figure 2, the illumination on one face is successfully transferred into the other face without modifying any other information in the images. This demonstrates that DNA-GAN can effectively disentangle the illumination factor from other factors in the latent space.
+
+# 4.2 CELEBA DATASET
+
+CelebA (Liu et al., 2015) is a dataset composed of 202599 face images and 40 attribute binary vectors and 5 landmark locations. We use the aligned and cropped version and scaled all images down to $6 4 \times 6 4$ . To better demonstrate the advantage of our method, we choose TD-GAN (Wang et al., 2017) and IcGAN (Perarnau et al., 2016) for comparisons.
+
+As we mentioned before, TD-GAN requires the explicit id information in the label, thus cannot be applied to the CelebA dataset directly. To overcome this limitation, we use some channels to encode the id information in its latent representations. In our experiments, the id information is preserved when swapping the attribute information in the latent encodings. We also compared the experimental results of IcGAN with ours in the celebA dataset. The following results are obtained using the the official code and pre-trained celebA model provided by the author2.
+
+![](images/2ecc4949d6e212e1bf23ef9de4b47d50e299cd2832f3f41476204d10ce9d8fab.jpg)  
+Figure 3: The experimental results of TD-GAN and IcGAN on CelebA dataset. Three rows indicates the swapping attributes of Bangs, Eyeglasses and Smiling. For each model, the four images in a row are: two original images, and two newly generated images by swapping the attributes. The third image is generated by adding the attribute to the first one, and the fourth image is generated by removing the attribute from the second one.
+
+As displayed in Figure 3a, modified TD-GAN encounters the problem of trivial solutions. Without id information explicitly contained in the label, TD-GAN encodes the information of the whole image into the attribute-related part in the latent representations. As a result, two faces are swapped directly. Whereas in Figure 3b, the quality of images generated by IcGAN are very bad, which is probably due to the multi-stage training process of IcGAN. Since the overall effect of the model relies much on the each stage.
+
+DNA-GAN is able to disentangle multiple attributes in the latent representations as shown in Figure 4. Since different attributes are encoded in different DNA pieces in our latent representations, we are able to interpolate the attribute subspaces by linear combination of disentangled encodings. Figure 4a, 4b and 4c present disentangled attribute subspaces spanned by any two attributes of Bangs, Eyeglasses and Smiling. They demonstrate that our model is effective in learning disentangled representations. Figure 4d shows the hairstyle transfer process among different Bangs styles. It is worth mentioning that the top-left image in Figure 4d is outside the CelebA dataset, which further validate the generalization potential of our model on unseen data. Please refer to Figure 5 in the Appendix for more results.
+
+![](images/aa58c88e87268845652e308670f5efc003ac56f4242bd70d79371ee40a8ddf34.jpg)  
+Figure 4: The interpolation results of DNA-GAN. Figure 4a, 4b and 4c display the disentangled attribute subspaces spanned by any two attributes of Bangs, Eyeglasses and Smiling. Figure 4d shows the attribute subspaces spanned by several Bangs feature vectors. Besides, the top-left image in Figure 4d is outside the CelebA dataset.
+
+# 5 CONCLUSION
+
+In this paper, we propose a supervised algorithm called DNA-GAN that can learn disentangled representations from multi-attribute images. The latent representations of images are DNA-like, consisting of attribute-relevant and attribute-irrelevant parts. By the annihilating operation and attribute hybridization, we are able to create new latent representations which could be decoded into novel images with designed attributes. The iterative training strategy effectively overcomes the difficulty of training on unbalanced datasets and helps disentangle multiple attributes in the latent space.
+
+The experimental results not only demonstrate that DNA-GAN is effective in learning disentangled representations and image editing, but also point out its potential in interpretable deep learning, image understanding and transfer learning.
+
+There also exist some limitations of our model. Without strong guidance on the attribute-irrelevant parts, some background information is encoded into the attribute-relevant part. As we can see in Figure 4, the background color gets changed when swapping attributes. Besides, our model may fail when several attributes are highly correlated with each other. For example, Male and Mustache are statistically dependent, which are hard to disentangle in the latent representations. These are left as our future work.
+
+# REFERENCES
+
+Mart´ın Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein GAN. ´ CoRR, abs/1701.07875, 2017.
+
+Yoshua Bengio, Aaron C. Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell., 35(8):1798–1828, 2013.
+
+Diane Bouchacourt, Ryota Tomioka, and Sebastian Nowozin. Multi-level variational autoencoder: Learning disentangled representations from grouped observations. CoRR, abs/1705.08841, 2017.
+
+Xi Chen, Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in Neural Information Processing Systems 29, pp. 2172–2180, 2016.
+
+Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron C. Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems 27, pp. 2672–2680, 2014.
+
+Ralph Gross, Iain A. Matthews, Jeffrey F. Cohn, Takeo Kanade, and Simon Baker. Multi-pie. Image Vision Comput., 28(5):807–813, 2010.
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+Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. 2016.
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+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, pp. 448–456, 2015.
+
+Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A. Efros. Image-to-image translation with conditional adversarial networks. CoRR, abs/1611.07004, 2016.
+
+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. CoRR, abs/1312.6114, 2013.
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+Tejas D. Kulkarni, William F. Whitney, Pushmeet Kohli, and Joshua B. Tenenbaum. Deep convolutional inverse graphics network. In Advances in Neural Information Processing Systems 28, pp. 2539–2547, 2015.
+
+Ming-Yu Liu, Thomas Breuel, and Jan Kautz. Unsupervised image-to-image translation networks. CoRR, abs/1703.00848, 2017.
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+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In 2015 IEEE International Conference on Computer Vision, ICCV 2015, pp. 3730–3738, 2015.
+
+Andrew L Maas, Awni Y Hannun, and Andrew Y Ng. Rectifier nonlinearities improve neural network acoustic models. In Proc. ICML, volume 30, 2013.
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+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. CoRR, abs/1411.1784, 2014.
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+Karl Pearson. Liii. on lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11):559–572, 1901.
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+Guim Perarnau, Joost van de Weijer, Bogdan Raducanu, and Jose M. Alvarez. Invertible conditional ´ gans for image editing. CoRR, abs/1611.06355, 2016.
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+Sam T Roweis and Lawrence K Saul. Nonlinear dimensionality reduction by locally linear embedding. science, 290(5500):2323–2326, 2000.
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+N Siddharth, Brooks Paige, Alban Desmaison, Jan-Willem van de Meent, Frank Wood, Noah D Goodman, Pushmeet Kohli, and Philip HS Torr. Learning disentangled representations in deep generative models. 2016.
+
+Paul Smolensky. Information processing in dynamical systems: Foundations of harmony theory. Technical report, COLORADO UNIV AT BOULDER DEPT OF COMPUTER SCIENCE, 1986.
+
+Ilya Sutskever, James Martens, George E. Dahl, and Geoffrey E. Hinton. On the importance of initialization and momentum in deep learning. In Proceedings of the 30th International Conference on Machine Learning, ICML 2013, pp. 1139–1147, 2013.
+
+Yaniv Taigman, Adam Polyak, and Lior Wolf. Unsupervised cross-domain image generation. CoRR, abs/1611.02200, 2016.
+
+Chaoyue Wang, Chaohui Wang, Chang Xu, and Dacheng Tao. Tag disentangled generative adversarial network for object image re-rendering. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, pp. 2901–2907, 2017.
+
+Kai Yu, Tong Zhang, and Yihong Gong. Nonlinear learning using local coordinate coding. In Advances in Neural Information Processing Systems 22, pp. 2223–2231, 2009.
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+Shuchang Zhou, Taihong Xiao, Yi Yang, Dieqiao Feng, Qinyao He, and Weiran He. Genegan: Learning object transfiguration and attribute subspace from unpaired data. CoRR, abs/1705.04932, 2017.
+
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. CoRR, abs/1703.10593, 2017.
+
+# APPENDIX
+
+To prove Theorem 1, we need the following lemma.
+
+Lemma 1. $A$ set $S = \{ s _ { 1 } , \ldots , s _ { m } \}$ has m different elements, from which elements are being selected equally likely with replacement. The expected number of trials needed to collect a subset $R = \{ s _ { 1 } , \ldots , s _ { n } \}$ of $n ( 1 \leq n \leq m )$ elements is
+
+$$
+m \cdot \left( { \frac { 1 } { 1 } } + { \frac { 1 } { 2 } } + \cdots + { \frac { 1 } { n } } \right) .
+$$
+
+Proof. Let $T$ be the time to collect all $n$ elements in the subset $R$ , and let $t _ { i }$ be the time to collect the $i$ -th new elements after $i - 1$ elements in $R$ have been collected. Observe that the probability of collecting a new element is $p _ { i } = ( n - ( i - 1 ) ) / m$ . Therefore, $t _ { i }$ is a geometrically distributed random variable with expectation $1 / p _ { i }$ . By the linearity of expectations, we have:
+
+$$
+{ \begin{array} { r l } & { \mathbb { E } ( T ) = \mathbb { E } ( t _ { 1 } ) + \mathbb { E } ( t _ { 2 } ) + \dots + \mathbb { E } ( t _ { n } ) } \\ & { \qquad = { \frac { 1 } { p _ { 1 } } } + { \frac { 1 } { p _ { 2 } } } + \dots + { \frac { 1 } { p _ { n } } } } \\ & { \qquad = { \frac { m } { n } } + { \frac { m } { n - 1 } } + \dots + { \frac { m } { 1 } } } \\ & { \qquad = m \cdot \left( { \frac { 1 } { 1 } } + { \frac { 1 } { 2 } } + \dots + { \frac { 1 } { n } } \right) . } \end{array} }
+$$
+
+Proof. (of Theorem 1)
+
+We first consider the case of randomly selecting pairs. All possible image pairs are actually in the product space $\mathcal X \times \mathcal X$ , whose cardinality is $m ^ { 2 }$ . If we take the order of two images in a pair into consideration, the number of possible pairs is $m ^ { 2 }$ . Recall that the useful pair denotes a pair of image of different labels. Therefore, the number of all useful pairs is $\textstyle \sum _ { i \neq j } m _ { i } m _ { j }$ . By Lemma 1, the expected number of iterations for randomly selecting pairs to select all useful pairs at least once is
+
+$$
+\begin{array} { r l } & { \mathrm { E } _ { 1 } = m ^ { 2 } \left( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { \sum _ { i \neq j } m _ { i } m _ { j } } \right) } \\ & { \quad = m ^ { 2 } \left( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { \sum _ { i = 1 } ^ { 2 n } ( m _ { i } \sum _ { j \neq i } m _ { j } ) } \right) } \\ & { \quad = m ^ { 2 } \left( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { \sum _ { i = 1 } ^ { 2 n } m _ { i } ( m - m _ { i } ) } \right) } \\ & { \quad = m ^ { 2 } \left( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { m ^ { 2 } - \sum _ { i = 1 } ^ { 2 n } m _ { i } ^ { 2 } } \right) . } \end{array}
+$$
+
+Now we consider the case of iterative training. We always select a pair of images of different labels each time. Suppose we are selecting images with opposite labels at the $s$ -th position. Let $I _ { s }$ denote the indices of all labels with the $s$ -th element 1, and $J _ { s }$ denote the indices of all labels with the $s$ -th element 0, where $| I _ { s } | = | J _ { s } | = 2 ^ { n - 1 }$ . Then we consider the subproblem by neglecting the first position in data labels, the number of all possible pairs is $2 \sum _ { i \in I _ { s } , j \in J _ { s } } m _ { i } m _ { j }$ (regarding of order),
+
+and the number of useful pairs is
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { k _ { 1 } \neq k _ { 2 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ( m _ { i _ { k _ { 2 } } } + m _ { j _ { k _ { 2 } } } ) } \\ & { = \displaystyle \sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } \sum _ { k _ { 2 } \neq k _ { 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ( m _ { i _ { k _ { 2 } } } + m _ { j _ { k _ { 2 } } } ) } \\ & { = \displaystyle \sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ( m - m _ { i _ { k _ { 1 } } } - m _ { j _ { k _ { 1 } } } ) } \\ & { = m ^ { 2 } - \displaystyle \sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } . } \end{array}
+$$
+
+Therefore, the expectation to select all useful pairs at least once regardless of the $s$ -th element in the label is
+
+$$
+\mathrm { E } _ { \backslash s } = 2 \sum _ { i \in I _ { s } , j \in J _ { s } } m _ { i } m _ { j } \left( 1 + { \frac { 1 } { 2 } } + \cdots + { \frac { 1 } { m ^ { 2 } - \sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } } } \right)
+$$
+
+Since we rotate the subscript $s$ from 1 to $n$ , the expected number of iterations for iterative training to select all useful pairs at least once is
+
+$$
+\begin{array} { r l r } {  { \mathrm { E } _ { 2 } \le n \cdot \operatorname* { m a x } _ { s = 1 , \ldots , n } \mathrm { E } _ { \mathsf { V } ^ { s } } } } \\ & { } & { = 2 n \cdot \operatorname* { m a x } _ { s = 1 , \ldots , n } \sum _ { i \in I _ { s } , j \in J _ { s } } m _ { i } m _ { j } ( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { m ^ { 2 } - \sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } } ) . } \end{array}
+$$
+
+Proof. (of Theorem 2) We firstly show that
+
+$$
+\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } \geq \sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } ^ { 2 } + m _ { j _ { k _ { 1 } } } ^ { 2 } ) = \sum _ { i = 1 } ^ { 2 ^ { n } } m _ { i } ^ { 2 }
+$$
+
+According to the result of Theorem 1 and the Definition 1 of balancedness, we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { 2 } = 2 n \cdot \operatorname* { m a x } \displaystyle \sum _ { s \in L _ { s } \times \xi _ { s } } m _ { s t } ( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { m ^ { 2 } - \sum _ { k = 1 } ^ { 2 n } ( m _ { k _ { 1 } } + m _ { \delta _ { 1 } } ) ^ { 2 } } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { = 2 n \cdot \operatorname* { m a x } \displaystyle \sum _ { s \in L _ { s } \times \xi _ { s } } m _ { s t } ( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { m ^ { 2 } - \sum _ { s = 1 } ^ { 2 n } m _ { s t } ^ { 2 } } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { = 2 n \cdot \operatorname* { m a x } \displaystyle \sum _ { s \in L _ { s } \times \xi _ { s } } m _ { s t } ( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { m ^ { 2 } - \sum _ { s = 1 } ^ { 2 n } m _ { s t } ^ { 2 } } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { = 2 n \cdot \operatorname* { m a x } \displaystyle \frac { 2 n \rho _ { 3 } } { \rho _ { 3 } + 1 } \cdots m ^ { 2 } ( 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { m ^ { 2 } - \sum _ { s = 1 } ^ { 2 n } m _ { s t } ^ { 2 } } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  = \operatorname* { m a x } \displaystyle \frac { 2 n \rho _ { 3 } } { s } \cdots m ^ { 2 } ( 1 + \frac { 1 } { 2 } + \cdots + \ \end{array}
+$$
+
+Specifically, if $n \leq 2$ ,
+
+$$
+\frac { 2 n \rho _ { s } } { ( \rho _ { s } + 1 ) ^ { 2 } } \leq \frac { 4 \rho _ { s } } { ( \rho _ { s } + 1 ) ^ { 2 } } \leq 1 .
+$$
+
+The inequality holds true forever.
+
+![](images/01582e6778fd2c6a1e43ba8985b6f2f757a12e7f81e62d5cd93e936a006ec71e.jpg)
+
+![](images/e207376bc660c66977376b6de530ed48d15a212cb7ebb3b54a7a0215adaba738.jpg)  
+Figure 5: More experimental results of DNA-GAN.

parse/train/Syr8Qc1CW/Syr8Qc1CW_content_list.json

@@ -0,0 +1,1573 @@
+[
+    {
+        "type": "text",
+        "text": "DNA-GAN: LEARNING DISENTANGLED REPRESEN-TATIONS FROM MULTI-ATTRIBUTE IMAGES",
+        "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": "Disentangling factors of variation has always been a challenging problem in representation learning. Existing algorithms suffer from many limitations, such as unpredictable disentangling factors, bad quality of generated images from encodings, lack of identity information, etc. In this paper, we proposed a supervised algorithm called DNA-GAN trying to disentangle different attributes of images. The latent representations of images are DNA-like, in which each individual piece represents an independent factor of variation. By annihilating the recessive piece and swapping a certain piece of two latent representations, we obtain another two different representations which could be decoded into images. In order to obtain realistic images and also disentangled representations, we introduced the discriminator for adversarial training. Experiments on Multi-PIE and CelebA datasets demonstrate the effectiveness of our method and the advantage of overcoming limitations existing in other methods. ",
+        "bbox": [
+            233,
+            268,
+            764,
+            449
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            336,
+            496
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The success of machine learning algorithms depends on data representation, because different representations can entangle different explanatory factors of variation behind the data. Although prior knowledge can help us design representations, the vast demand of AI algorithms in various domains cannot be met, since feature engineering is labor-intensive and needs domain expert knowledge. Therefore, algorithms that can automatically learn good representations of data will definitely make it easier for people to extract useful information when building classifiers or predictors. ",
+        "bbox": [
+            174,
+            513,
+            823,
+            597
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Of all criteria of learning good representations as discussed in Bengio et al. (2013), disentangling factors of variation is an important one that helps separate various explanatory factors. For example, given a human-face image, we can obtain various information about the person, including gender, hair style, facial expression, with/without eyeglasses and so on. All of these information are entangled in a single image, which renders the difficulty of training a single classifier to handle different facial attributes. If we could obtain a disentangled representation of the face image, we may build up only one classifier for multiple attributes. ",
+        "bbox": [
+            174,
+            603,
+            825,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a supervised method called DNA-GAN to obtain disentangled representations of images. The idea of DNA-GAN is motivated by the DNA double helix structure, in which different kinds of traits are encoded in different DNA pieces. We make a similar assumption that different visual attributes in an image are controlled by different pieces of encodings in its latent representations. In DNA-GAN, an encoder is used to encode an image to the attribute-relevant part and the attribute-irrelevant part, where different pieces in the attribute-relevant part encode information of different attributes, and the attribute-irrelevant part encodes other information. For example, given a facial image, we are trying to obtain a latent representation that each individual part controls different attributes, such as hairstyles, genders, expressions and so on. Though annihilating recessive pieces and swapping certain pieces, we can obtain novel crossbreeds that can be decoded into new images. By the adversarial discriminator loss and the reconstruction loss, DNA-GAN can reconstruct the input images and generate new images with new attributes. Each attribute is disentangled from others gradually though iterative training. Finally, we are able to obtain disentangled representations in the latent representations. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The summary of contributions of our work is as follows: ",
+        "bbox": [
+            174,
+            909,
+            544,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. We propose a supervised algorithm called DNA-GAN, that is able to disentangle multiple attributes as demonstrated by the experiments of interpolating multiple attributes on MultiPIE (Gross et al., 2010) and CelebA (Liu et al., 2015) datasets.   \n2. We introduce the annihilating operation that prevents from trivial solutions: the attributerelevant part encodes information of the whole image instead of a certain attribute.   \n3. We employ iterative training to address the problem of unbalanced multi-attribute image data, which was theoretically proved to be more efficient than random image pairs. ",
+        "bbox": [
+            209,
+            103,
+            825,
+            213
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            232,
+            344,
+            248
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Traditional representation learning algorithms focus on (1) probabilistic graphical models, characterized by Restricted Boltzmann Machine (RBM) (Smolensky, 1986), Autoencoder (AE) and their variants; (2) manifold learning and geometrical approaches, such as Principal Components Analysis (PCA) (Pearson, 1901), Locally Linear Embedding (LLE) (Roweis & Saul, 2000), Local Coordinate Coding (LCC) (Yu et al., 2009), etc. However, recent research has actively focused on developing deep probabilistic models that learn to represent the distribution of data. Kingma & Welling (2013) employs an explicit model distribution and uses variational inference to learn its parameters. As the generative adversarial networks (GAN) (Goodfellow et al., 2014) has been invented, many implicit models are developed. ",
+        "bbox": [
+            174,
+            263,
+            825,
+            390
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the semi-supervised setting, Siddharth et al. (2016) learns a disentangled representations by using an auxiliary variable. Bouchacourt et al. (2017) proposes the ML-VAE that can learn disentangled representations from a set of grouped observations. In the unsupervised setting, InfoGAN (Chen et al., 2016) tries to maximize mutual information between a small subset of latent variables and observations by introducing an auxiliary network to approximate the posterior. However, it relies much on the a-priori choice of distributions and suffered from unstable training. Another popular unsupervised method $\\beta$ -VAE (Higgins et al., 2016), adapted from VAE, lays great stress on the KL distance between the approximate posterior and the prior. However, unsupervised approaches do not anchor a specific meaning into the disentanglement. ",
+        "bbox": [
+            173,
+            396,
+            825,
+            522
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "More closely with our method, supervised methods take the advantage of labeled data and try to disentangle the factors as expected. DC-IGN (Kulkarni et al., 2015) asks the active attribute to explain certain factor of variation by feeding the other attributes by the average in a mini-batch. TD-GAN (Wang et al., 2017) uses a tag mapping net to boost the quality of disentangled representations, which are consistent with the representations extracted from images through the disentangling network. Besides, the quality of generated images is improved by implementing the adversarial training strategy. However, the identity information should be labeled so as to preserve the id information when swapping attributes, which renders the limitation of applying it into many other datasets without id labels. IcGAN (Perarnau et al., 2016) is a multi-stage training algorithm that first takes the advantage of cGAN (Mirza & Osindero, 2014) to learn a map from latent representations and conditional information to real images, and then learn its inverse map from images to the latent representations and conditions in a supervised manner. The overall effect depends on each training stage, therefore it is hard to obtain satisfying images. Unlike these models, our model requires neither explicit id information in labels nor multi-stage training. ",
+        "bbox": [
+            174,
+            529,
+            825,
+            723
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Many works have studied the image-to-image translation between unpaired image data using GANbased architectures, see Isola et al. (2016), Taigman et al. (2016), Zhu et al. (2017), Liu et al. (2017) and Zhou et al. (2017). Interestingly, these models require a form of 0/1 weak supervision that is similar to our setting. However, they are circumscribed in two image domains which are opposite to each other with respect to a single attribute. Our model differs from theirs as we generalize to the case of multi-attribute image data. Specifically, we employ the strategy of iterative training to overcome the difficulty of training on unbalanced multi-attribute image datasets. ",
+        "bbox": [
+            174,
+            729,
+            825,
+            828
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 DNA-GAN APPROACH",
+        "text_level": 1,
+        "bbox": [
+            176,
+            848,
+            400,
+            864
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we formally outline our method. A set $\\mathcal { X }$ of multi-labeled images and a set of labels $\\mathcal { V }$ are considered in our setting. Let $\\{ ( \\mathbf { X } ^ { 1 } , \\mathbf { Y } ^ { 1 } ) , \\ldots , ( \\mathbf { X } ^ { m } , \\mathbf { Y } ^ { m } ) \\}$ denote the whole training dataset, where $\\mathbf { X } ^ { i } \\in \\mathcal X$ is the $i$ -th image with its label $\\mathbf { Y } ^ { i } \\in \\mathcal { V }$ . The small letter $m$ denotes the number of samples in set $\\mathcal { X }$ and $n$ denotes the number of attributes. The label $\\mathbf { Y } ^ { i } = ( \\mathbf { y } _ { 1 } ^ { i } , \\ldots , \\mathbf { y } _ { n } ^ { i } )$ is a $n$ -dimensional vector where each element represents whether $\\mathbf { X } ^ { i }$ has certain attribute or not. For example, in the case of labels with three candidates [Bangs, Eyeglasses, Smiling], the facial image $\\mathbf { X } ^ { i }$ whose label is $\\mathbf { Y } ^ { i } = ( 1 , 0 , 1 )$ should depict a smiling face with bangs and no eyeglasses. ",
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+    {
+        "type": "text",
+        "text": "3.1 MODEL ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1, DNA-GAN is mainly composed of three parts: an encoder $( { \\mathrm { E n c } } )$ , a decoder (Dec) and a discriminator (D). The encoder maps the real-world images $A$ and $B$ into two latent disentangled representations ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/862a9018ec059cea213406b52f5f9af48338a1889a5203e19fd5a5048289b32d.jpg",
+        "text": "$$\n\\operatorname { E n c } ( A ) = [ a _ { 1 } , \\dots , a _ { i } , \\dots , a _ { n } , z _ { a } ] , \\quad \\operatorname { E n c } ( B ) = [ b _ { 1 } , \\dots , b _ { i } , \\dots , b _ { n } , z _ { b } ]\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $[ a _ { 1 } , \\ldots , a _ { i } , \\ldots , a _ { n } ]$ is called the attribute-relevant part, and $z _ { a }$ is called the attribute-irrelevant part. $a _ { i }$ is supposed to be a DNA piece that controls $\\mathbf { y } _ { i }$ , the $i$ -th attribute in the label, and $z _ { a }$ is for keeping other silent factors which do not appear in the attribute list as well as image identity information. The same thing applies for $\\operatorname { E n c } ( B )$ . ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7d30a549e7003d1f960dc09538b3a978887025f279e99544c245587f3e976cda.jpg",
+        "image_caption": [
+            "Figure 1: DNA-GAN architecture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "e focus on one attribute each time in ourare required to have different labels, i.e. we a and $i$ $A$ and, re$B$ $( \\mathbf { y } _ { 1 } ^ { A } , \\ldots , 1 _ { i } ^ { A } , \\ldots , \\mathbf { y } _ { n } ^ { \\bar { A } } )$ $( \\mathbf { y } _ { 1 } ^ { B } , \\ldots , 0 _ { i } ^ { B } , \\ldots , \\mathbf { y } _ { n } ^ { B } )$ spectively. In our convention, $A$ is always for the dominant pattern, while $B$ is for the recessive pattern. We copy $\\operatorname { E n c } ( A )$ directly as the latent representation of $A _ { 1 }$ , and annihilate $b _ { i }$ in the copy of $\\operatorname { E n c } ( B )$ as the latent representation of $B _ { 1 }$ . The annihilating operation means replacing all elements with zeros, and plays a key role in disentangling the attribute, which we will discuss in detail in Section 3.3. By swapping $a _ { i }$ and $0 _ { i }$ , we obtain two new latent representations $\\left[ a _ { 1 } , \\ldots , 0 _ { i } , \\ldots , a _ { n } , z _ { a } \\right]$ and $\\left[ b _ { 1 } , \\dots , a _ { i } , \\dots , b _ { n } , z _ { b } \\right]$ that are supposed to be decoded into $A _ { 2 }$ and $B _ { 2 }$ , respectively. Though a decoder Dec, we can get four newly generated images $A _ { 1 }$ , $B _ { 1 }$ , $A _ { 2 }$ and $B _ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58992e71ad3df820585aff5d0db75eb6d561ba270b05faacae435a8d7bc17efb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { D e c } ( [ a _ { 1 } , \\dots , a _ { i } , \\dots , a _ { n } , z _ { a } ] ) = A _ { 1 } , \\quad \\mathrm { D e c } ( [ b _ { 1 } , \\dots , 0 _ { i } , \\dots , b _ { n } , z _ { b } ] ) = B _ { 1 } } \\\\ & { \\mathrm { D e c } ( [ a _ { 1 } , \\dots , 0 _ { i } , \\dots , a _ { n } , z _ { a } ] ) = A _ { 2 } , \\quad \\mathrm { D e c } ( [ b _ { 1 } , \\dots , a _ { i } , \\dots , b _ { n } , z _ { b } ] ) = B _ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Out of these four children, $A _ { 1 }$ and $B _ { 1 }$ are reconstructions of $A$ and $B$ , while $A _ { 2 }$ and $B _ { 2 }$ are novel crossbreeds. The reconstruction losses between $A$ and $A _ { 1 }$ , $B$ and $B _ { 1 }$ ensure the quality of reconstructed samples. Besides, using an adversarial discriminator D that helps make generated samples $A _ { 2 }$ indistinguishable from $B$ , and $B _ { 2 }$ indistinguishable from $A$ , we can enforce attribute-related information to be encoded in $a _ { i }$ . ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 LOSS FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given two images $A$ and $B$ and their labels $\\begin{array} { c c l } { \\mathbf { Y } ^ { A } } & { = } & { ( \\mathbf { y } _ { 1 } ^ { A } , \\ldots , \\mathbf { l } _ { i } ^ { A } , \\ldots , \\mathbf { y } _ { n } ^ { A } ) } \\end{array}$ and $\\begin{array} { r l } { \\mathbf { Y } ^ { B } } & { { } = } \\end{array}$ $( \\mathbf { y } _ { 1 } ^ { B } , \\ldots , 0 _ { i } ^ { B } , \\ldots , \\mathbf { y } _ { n } ^ { B } )$ which are different at the $i$ -th position, the data flow can be summarized by (1) and (2). We force the $i$ -th latent encoding of $B$ to be zero in order to prevent from trivial solutions as we will discuss in Section 3.3. ",
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+    {
+        "type": "text",
+        "text": "The encoder and decoder receive two types of losses: (1) the reconstruction loss, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07466c7a4826e26c64201dca70be516401caae58df9ace79a04a5086feff7e53.jpg",
+        "text": "$$\nL _ { r e c o n s t r u c t } = \\| A - A _ { 1 } \\| _ { 1 } + \\| B - B _ { 1 } \\| _ { 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which measures the reconstruction quality after a sequence of encoding and decoding; (2) the standard GAN loss, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ee58a3cfc37580a17c2d2480f2686c3ad149a54cfca6a1ca1e2319f4492e2dd.jpg",
+        "text": "$$\nL _ { G A N } = - \\mathbb { E } [ \\log ( \\mathrm { D } ( A _ { 2 } | \\mathbf { y } _ { i } ^ { A } = 1 ) ) ] - \\mathbb { E } [ \\log ( \\mathrm { D } ( B _ { 2 } | \\mathbf { y } _ { i } ^ { B } = 0 ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which measures how realistic the generated images are. The discriminator takes the generated image and the $i$ -th element of its label as inputs, and outputs a number which indicates how realistic the input image is. The larger the number is, the more realistic the image is. Omitting the coefficient, the loss function for the encoder and decoder is ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e6d6f0431289d21191bb1499c4b159a5b48786ba608a0f94d4593bd884ba9a6.jpg",
+        "text": "$$\nL _ { G } = L _ { r e c o n s t r u c t } + L _ { G A N } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The discriminator D receives the standard GAN discriminator loss ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a82d22fc3d8d8b66457300e6ed2afe790d0114be92a9d0131745aea4cca54c6.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { D _ { 1 } } = - \\mathbb { E } [ \\log ( \\mathrm { D } ( A | \\mathbf { y } _ { i } ^ { A } = 1 ) ) ] - \\mathbb { E } [ \\log ( 1 - \\mathrm { D } ( B _ { 2 } | \\mathbf { y } _ { i } ^ { A } = 1 ) ) ] } \\\\ & { L _ { D _ { 0 } } = - \\mathbb { E } [ \\log ( \\mathrm { D } ( B | \\mathbf { y } _ { i } ^ { B } = 0 ) ) ] - \\mathbb { E } [ \\log ( 1 - \\mathrm { D } ( A _ { 2 } | \\mathbf { y } _ { i } ^ { B } = 0 ) ) ] } \\\\ & { L _ { D } = L _ { D _ { 1 } } + L _ { D _ { 0 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $L _ { D _ { 1 } }$ drives $\\mathrm { D }$ to tell $A$ from $B _ { 2 }$ , and $L _ { D _ { 0 } }$ drives $\\mathrm { D }$ to tell $B$ from $A _ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 ANNIHILATING OPERATION PREVENTS FROM TRIVIAL SOLUTIONS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Through experiments, we observe that there exist trivial solutions to our model without the annihilating operation. We just take the single-attribute case as an example. Suppose that $\\operatorname { E n c } ( A ) = [ a , z _ { a } ]$ and $\\bar { \\mathrm { E n c } } ( B ) = [ b , \\bar { z } _ { b } ]$ , we can get four children without annihilating operation ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b78ef2d4ce7f2fdce9d5e27f928f8ee478428a7411397fed7be3cfae5048b1e.jpg",
+        "text": "$$\nA _ { 1 } = \\mathrm { D e c } ( [ a , z _ { a } ] ) , \\quad B _ { 1 } = \\mathrm { D e c } ( [ b , z _ { b } ] ) , \\quad A _ { 2 } = \\mathrm { D e c } ( [ b , z _ { a } ] ) , \\quad B _ { 2 } = \\mathrm { D e c } ( [ a , z _ { b } ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The reconstruction loss makes it invertible between the latent encoding space and image space. The adversarial discriminator D is supposed to disentangle the attribute from other information by telling whether $A _ { 2 }$ looks as real as $B$ and $B _ { 2 }$ looks as real as $A$ or not. As we know that the generative adversarial networks give the best solution when achieving the Nash equilibrium. But without the annihilating operation, information of the whole image could be encoded into the attribute-relevant part, which means ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b54ec0ca354a878b2aded056d8b7b38590700449dd9574b6c60a8499d20c51f.jpg",
+        "text": "$$\n\\operatorname { E n c } ( A ) = [ a , 0 ] , \\quad \\operatorname { E n c } ( B ) = [ b , 0 ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we obtain the following four children ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/50277d639067aa79ee24f2d1963c9ff94e2f25f50ebcfa5107338b719ab0f46f.jpg",
+        "text": "$$\nA _ { 1 } = \\operatorname { D e c } ( [ a , 0 ] ) , \\quad B _ { 1 } = \\operatorname { D e c } ( [ b , 0 ] ) , \\quad A _ { 2 } = \\operatorname { D e c } ( [ b , 0 ] ) , \\quad B _ { 2 } = \\operatorname { D e c } ( [ a , 0 ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this situation, the discriminator $\\mathrm { D }$ cannot discriminate $A _ { 2 }$ from $B$ , since they share the same latent encodings. By reconstruction loss, $A _ { 2 }$ and $B$ are exactly the same image, which is against our expectation that $A _ { 2 }$ should depict the person from $A$ with the attribute borrowed from $B$ . The same thing happens to $B _ { 2 }$ and $A$ as well. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To prevent from learning trivial solutions, we adopt the annihilating operation by replacing the recessive pattern $b$ with a zero tensor of the same $\\bar { \\mathrm { s i z e } ^ { 1 } }$ . If information of the whole image were encoded into the attribute-relevant part, the four children in this case are ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0bfb7f393066e2f9187ab9da3970d59533691430390260d14b27a9afc3f6d116.jpg",
+        "text": "$$\nA _ { 1 } = \\operatorname { D e c } ( [ a , 0 ] ) , \\quad B _ { 1 } = \\operatorname { D e c } ( [ 0 , 0 ] ) , \\quad A _ { 2 } = \\operatorname { D e c } ( [ 0 , 0 ] ) , \\quad B _ { 2 } = \\operatorname { D e c } ( [ a , 0 ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The encodings of $B _ { 1 }$ and $A _ { 2 }$ contain no information at all, thus neither the person in $B _ { 1 }$ nor $A _ { 2 }$ who is supposed to be the same as in $B$ can be reconstructed by Dec. This forces the attribute-irrelevant part to encode some information of images. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4 ITERATIVE TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To reduce the difficulty of disentangling multiple attributes, we take the strategy of iterative training: we update our model using a pair of images with opposite labels at a certain position each time. Suppose that we are at the $i$ -th position, the label of image $A$ is $( \\mathbf { y } _ { 1 } ^ { A } , \\ldots , 1 _ { i } ^ { A } , \\cdot \\cdot . , \\mathbf { y } _ { n } ^ { A } )$ , while the label of image $B$ is $( \\mathbf { y } _ { 1 } ^ { B } , \\ldots , 0 _ { i } ^ { B } , \\ldots , \\mathbf { y } _ { n } ^ { B } )$ . During each iteration, as $i$ goes through from 1 to $n$ repeatedly, our model fed with such a pair of images can disentangle multiple attributes one-by-one. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Compared with training with random pairs of images, iterative training is proved to be more effective. Random pairs of images means randomly selecting pairs of images each time without label constraints. A pair of images with different labels is called a useful pair. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We theoretically show that our iterative training is much more efficient than random image pairs especially when the dataset is unbalanced. All proofs can be found in the Appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $\\mathcal { X } = \\{ ( \\mathbf { X } ^ { 1 } , \\mathbf { Y } ^ { 1 } ) , \\dots , ( \\mathbf { X } ^ { m } , \\mathbf { Y } ^ { m } ) \\}$ denote the whole multi-attribute image dataset,   \nwhere $\\mathbf { X } ^ { i }$ is a multi-attribute image and its label $\\mathbf { \\dot { Y } } ^ { i } = ( \\mathbf { y } _ { 1 } ^ { i } , \\dots , \\mathbf { y } _ { n } ^ { i } )$ is an $n$ -dimensional vector.   \nTherelabel iterat reis ns ally , anded $2 ^ { n }$ ls, denoted by . To select all lecting pairs an $\\mathcal { L } = \\{ l _ { 1 } , \\ldots , l _ { 2 ^ { n } } \\}$ . The number of ionce, the expecteding are denoted by ges wmberand $l _ { i }$ $m _ { i }$ $\\textstyle \\sum _ { i = 1 } ^ { 2 ^ { n } } m _ { i } = m$ $\\mathrm { E _ { 1 } }$ $\\mathrm { E _ { 2 } }$ respectively. Then, ",
+        "bbox": [
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+            287,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19d786c7c69fdbbe50531bf2dde0caf9c798ff67eaa267dd0f79dc6152a27c25.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathrm { E } _ { 1 } = m ^ { 2 } \\left( 1 + \\displaystyle \\frac { 1 } { 2 } + \\cdot \\cdot \\cdot + \\frac { 1 } { m ^ { 2 } - \\sum _ { i = 1 } ^ { 2 n } m _ { i } ^ { 2 } } \\right) } \\\\ { \\mathrm { E } _ { 2 } \\leq 2 n \\cdot \\displaystyle \\operatorname* { m a x } _ { s = 1 , \\ldots , n } \\displaystyle \\sum _ { i \\in I _ { s } , j \\in J _ { s } } m _ { i } m _ { j } \\left( 1 + \\frac { 1 } { 2 } + \\cdot \\cdot \\cdot + \\frac { 1 } { m ^ { 2 } - \\sum _ { k _ { 1 } = 1 } ^ { 2 n - 1 } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            378,
+            763,
+            468
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $I _ { s }$ represents the indices of labels where the s-th element is $1$ , and $J _ { s }$ represents the indices of labels where the $s$ -th element is 0. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. (Balancedness) Define the balancedness of a dataset $\\mathcal { X }$ described above with respect to the $s$ -th attribute as follows: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1d4656bff670afa9322c085cd8d8f692c14c1c6aa00ddbc7c13f5e41fd5a746.jpg",
+        "text": "$$\n\\rho _ { s } = \\frac { \\sum _ { i \\in I _ { s } } m _ { i } } { \\sum _ { j \\in J _ { s } } m _ { j } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            525,
+            557,
+            564
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $I _ { s }$ represents the indices of labels where the $s$ -th element is 1, and $J _ { s }$ represents the indices of labels where the $s$ -th element is 0. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. We have $\\mathrm { E } _ { 2 } \\leq \\mathrm { E } _ { 1 }$ , when ",
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d88f84453b0f379304dc3eb22966c93fbe04475d6a40d88ff7fd2fcb71cac786.jpg",
+        "text": "$$\nn \\leq \\operatorname* { m i n } _ { s } \\frac { ( \\rho _ { s } + 1 ) ^ { 2 } } { 2 \\rho _ { s } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Specifically, $\\mathrm { E } _ { 2 } \\leq \\mathrm { E } _ { 1 }$ holds true for all $n \\leq 2$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The property of the function $( \\rho + 1 ) ^ { 2 } / ( 2 \\rho )$ suits well with the definition of balancedness, because it attains the same value for $\\rho$ and $1 / \\rho$ , which is invariant to different labeling methods. Its value gets larger as the dataset becomes more unbalanced. The minimum is obtained at $\\rho = 1$ , which is the case of a balanced dataset. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 demonstrates that the iterative training mechanism is always more efficient than random pairs of images when the number of attributes met the criterion (16). As the dataset becomes more unbalanced, $\\dot { ( \\rho _ { s } + 1 ) } ^ { 2 } / ( 2 \\rho _ { s } )$ goes larger, which means (16) can be more easily satisfied. More importantly, iterative training helps stabilize the training process on unbalanced datasets. For example, given a two-attribute dataset, the number of data of each kind is as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/96a0a3f1b0864738700a0fd2e730f849e909de8fbfcac2512d130bd49fb9d92e.jpg",
+        "table_caption": [
+            "Table 1: The example of an unbalanced two-attribute dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Label</td><td rowspan=1 colspan=1>(0,0)</td><td rowspan=1 colspan=1>(0,1)</td><td rowspan=1 colspan=1>(1,0)</td><td rowspan=1 colspan=1>(1,1)</td></tr><tr><td rowspan=1 colspan=1>Number of data</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>m</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If $m \\gg 1$ is a very large number, then it is highly likely that we will select a pair of images whose labels are $( 1 , 0 )$ and $( 1 , 1 )$ each time by randomly selecting pairs. We ignore the pair of images whose labels are $( 1 , 0 )$ and $( 1 , 0 )$ or $( 1 , 1 )$ and $( 1 , 1 )$ , though these two cases have equal probabilities of being chosen. Because they are not useful pairs, thus do not participated in training. In this case, most of the time the model is trained with respect to the second attribute, which will cause the final learnt model less effective to the first attribute. However, iterative training can prevent this from happening, since we update our model evenly with respect to two attributes. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/42278b7c4d2b1d49753c4445c4a11a8a8c9c804e77fceacb234d8a1b49c813d2.jpg",
+        "image_caption": [
+            "Figure 2: Manipulating illumination factors on the Multi-PIE dataset. From left to right, the six images in a row are: original images $A$ with light illumination and $B$ with the dark illumination, newly generated images $A _ { 2 }$ and $B _ { 2 }$ by swapping the illumination-relevant piece in disentangled representations, and reconstructed images $A _ { 1 }$ and $B _ { 1 }$ . "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we perform different kinds of experiments on two real-world datasets to validate the effectiveness of our methods. We use the RMSProp (Sutskever et al., 2013) optimization method initialized by a learning rate of 5e-5 and momentum 0. All neural networks are equipped with Batch Normalization (Ioffe & Szegedy, 2015) after convolutions or deconvolutions. We used Leaky Relu (Maas et al., 2013) as the activation function in the encoder. Besides, we adopt strategies mentioned in Wasserstein GAN (Arjovsky et al., 2017) for stable training. More details will be available online. We divide all images into training images and test images according to the ratio of 9:1. All of the following results are from test images without cherry-picking. ",
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+    {
+        "type": "text",
+        "text": "4.1 MULTI-PIE DATABASE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Multi-PIE (Gross et al., 2010) face database contains over 750,000 images of 337 subjects captured under 15 view points and 19 illumination conditions. We collecte all front faces images of different illuminations and align them based on 5-point landmarks on eyes, nose and mouth. All aligned images are resized into $1 2 8 \\times 1 2 8$ as inputs in our experiments. We label the light illumination face images by 1 and the dark illumination face images by 0. As shown in Figure 2, the illumination on one face is successfully transferred into the other face without modifying any other information in the images. This demonstrates that DNA-GAN can effectively disentangle the illumination factor from other factors in the latent space. ",
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+        "type": "text",
+        "text": "",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 CELEBA DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "CelebA (Liu et al., 2015) is a dataset composed of 202599 face images and 40 attribute binary vectors and 5 landmark locations. We use the aligned and cropped version and scaled all images down to $6 4 \\times 6 4$ . To better demonstrate the advantage of our method, we choose TD-GAN (Wang et al., 2017) and IcGAN (Perarnau et al., 2016) for comparisons. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As we mentioned before, TD-GAN requires the explicit id information in the label, thus cannot be applied to the CelebA dataset directly. To overcome this limitation, we use some channels to encode the id information in its latent representations. In our experiments, the id information is preserved when swapping the attribute information in the latent encodings. We also compared the experimental results of IcGAN with ours in the celebA dataset. The following results are obtained using the the official code and pre-trained celebA model provided by the author2. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2ecc4949d6e212e1bf23ef9de4b47d50e299cd2832f3f41476204d10ce9d8fab.jpg",
+        "image_caption": [
+            "Figure 3: The experimental results of TD-GAN and IcGAN on CelebA dataset. Three rows indicates the swapping attributes of Bangs, Eyeglasses and Smiling. For each model, the four images in a row are: two original images, and two newly generated images by swapping the attributes. The third image is generated by adding the attribute to the first one, and the fourth image is generated by removing the attribute from the second one. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            377,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As displayed in Figure 3a, modified TD-GAN encounters the problem of trivial solutions. Without id information explicitly contained in the label, TD-GAN encodes the information of the whole image into the attribute-related part in the latent representations. As a result, two faces are swapped directly. Whereas in Figure 3b, the quality of images generated by IcGAN are very bad, which is probably due to the multi-stage training process of IcGAN. Since the overall effect of the model relies much on the each stage. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DNA-GAN is able to disentangle multiple attributes in the latent representations as shown in Figure 4. Since different attributes are encoded in different DNA pieces in our latent representations, we are able to interpolate the attribute subspaces by linear combination of disentangled encodings. Figure 4a, 4b and 4c present disentangled attribute subspaces spanned by any two attributes of Bangs, Eyeglasses and Smiling. They demonstrate that our model is effective in learning disentangled representations. Figure 4d shows the hairstyle transfer process among different Bangs styles. It is worth mentioning that the top-left image in Figure 4d is outside the CelebA dataset, which further validate the generalization potential of our model on unseen data. Please refer to Figure 5 in the Appendix for more results. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/aa58c88e87268845652e308670f5efc003ac56f4242bd70d79371ee40a8ddf34.jpg",
+        "image_caption": [
+            "Figure 4: The interpolation results of DNA-GAN. Figure 4a, 4b and 4c display the disentangled attribute subspaces spanned by any two attributes of Bangs, Eyeglasses and Smiling. Figure 4d shows the attribute subspaces spanned by several Bangs feature vectors. Besides, the top-left image in Figure 4d is outside the CelebA dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a supervised algorithm called DNA-GAN that can learn disentangled representations from multi-attribute images. The latent representations of images are DNA-like, consisting of attribute-relevant and attribute-irrelevant parts. By the annihilating operation and attribute hybridization, we are able to create new latent representations which could be decoded into novel images with designed attributes. The iterative training strategy effectively overcomes the difficulty of training on unbalanced datasets and helps disentangle multiple attributes in the latent space. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The experimental results not only demonstrate that DNA-GAN is effective in learning disentangled representations and image editing, but also point out its potential in interpretable deep learning, image understanding and transfer learning. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "There also exist some limitations of our model. Without strong guidance on the attribute-irrelevant parts, some background information is encoded into the attribute-relevant part. As we can see in Figure 4, the background color gets changed when swapping attributes. Besides, our model may fail when several attributes are highly correlated with each other. For example, Male and Mustache are statistically dependent, which are hard to disentangle in the latent representations. These are left as our future work. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "type": "text",
+        "text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. CoRR, abs/1703.10593, 2017. ",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            264,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To prove Theorem 1, we need the following lemma. ",
+        "bbox": [
+            173,
+            138,
+            514,
+            154
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. $A$ set $S = \\{ s _ { 1 } , \\ldots , s _ { m } \\}$ has m different elements, from which elements are being selected equally likely with replacement. The expected number of trials needed to collect a subset $R = \\{ s _ { 1 } , \\ldots , s _ { n } \\}$ of $n ( 1 \\leq n \\leq m )$ elements is ",
+        "bbox": [
+            173,
+            161,
+            826,
+            205
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b66a6e79384835f1c90ce6e07cf7361d4f54595af05d88b8b33f58c93fe99950.jpg",
+        "text": "$$\nm \\cdot \\left( { \\frac { 1 } { 1 } } + { \\frac { 1 } { 2 } } + \\cdots + { \\frac { 1 } { n } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            219,
+            588,
+            255
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $T$ be the time to collect all $n$ elements in the subset $R$ , and let $t _ { i }$ be the time to collect the $i$ -th new elements after $i - 1$ elements in $R$ have been collected. Observe that the probability of collecting a new element is $p _ { i } = ( n - ( i - 1 ) ) / m$ . Therefore, $t _ { i }$ is a geometrically distributed random variable with expectation $1 / p _ { i }$ . By the linearity of expectations, we have: ",
+        "bbox": [
+            173,
+            295,
+            825,
+            352
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2081993145fc79439626001eb13bd9e7041c8f72aac4a3f3696e8d923bb187b.jpg",
+        "text": "$$\n{ \\begin{array} { r l } & { \\mathbb { E } ( T ) = \\mathbb { E } ( t _ { 1 } ) + \\mathbb { E } ( t _ { 2 } ) + \\dots + \\mathbb { E } ( t _ { n } ) } \\\\ & { \\qquad = { \\frac { 1 } { p _ { 1 } } } + { \\frac { 1 } { p _ { 2 } } } + \\dots + { \\frac { 1 } { p _ { n } } } } \\\\ & { \\qquad = { \\frac { m } { n } } + { \\frac { m } { n - 1 } } + \\dots + { \\frac { m } { 1 } } } \\\\ & { \\qquad = m \\cdot \\left( { \\frac { 1 } { 1 } } + { \\frac { 1 } { 2 } } + \\dots + { \\frac { 1 } { n } } \\right) . } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            366,
+            624,
+            483
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. (of Theorem 1) ",
+        "bbox": [
+            174,
+            553,
+            325,
+            568
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We first consider the case of randomly selecting pairs. All possible image pairs are actually in the product space $\\mathcal X \\times \\mathcal X$ , whose cardinality is $m ^ { 2 }$ . If we take the order of two images in a pair into consideration, the number of possible pairs is $m ^ { 2 }$ . Recall that the useful pair denotes a pair of image of different labels. Therefore, the number of all useful pairs is $\\textstyle \\sum _ { i \\neq j } m _ { i } m _ { j }$ . By Lemma 1, the expected number of iterations for randomly selecting pairs to select all useful pairs at least once is ",
+        "bbox": [
+            173,
+            574,
+            826,
+            647
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f43a617a0c143d83d982b51f6229cfbd05fa3caa670184b983a78f21c7e02fe.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { E } _ { 1 } = m ^ { 2 } \\left( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { \\sum _ { i \\neq j } m _ { i } m _ { j } } \\right) } \\\\ & { \\quad = m ^ { 2 } \\left( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { \\sum _ { i = 1 } ^ { 2 n } ( m _ { i } \\sum _ { j \\neq i } m _ { j } ) } \\right) } \\\\ & { \\quad = m ^ { 2 } \\left( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { \\sum _ { i = 1 } ^ { 2 n } m _ { i } ( m - m _ { i } ) } \\right) } \\\\ & { \\quad = m ^ { 2 } \\left( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { m ^ { 2 } - \\sum _ { i = 1 } ^ { 2 n } m _ { i } ^ { 2 } } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            661,
+            666,
+            834
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Now we consider the case of iterative training. We always select a pair of images of different labels each time. Suppose we are selecting images with opposite labels at the $s$ -th position. Let $I _ { s }$ denote the indices of all labels with the $s$ -th element 1, and $J _ { s }$ denote the indices of all labels with the $s$ -th element 0, where $| I _ { s } | = | J _ { s } | = 2 ^ { n - 1 }$ . Then we consider the subproblem by neglecting the first position in data labels, the number of all possible pairs is $2 \\sum _ { i \\in I _ { s } , j \\in J _ { s } } m _ { i } m _ { j }$ (regarding of order), ",
+        "bbox": [
+            173,
+            853,
+            825,
+            928
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "and the number of useful pairs is ",
+        "bbox": [
+            173,
+            103,
+            392,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/62f9d9dd209ee36047a4396ef53a4d632aa865865c3b837509cb8e8dc42d7262.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { k _ { 1 } \\neq k _ { 2 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ( m _ { i _ { k _ { 2 } } } + m _ { j _ { k _ { 2 } } } ) } \\\\ & { = \\displaystyle \\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } \\sum _ { k _ { 2 } \\neq k _ { 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ( m _ { i _ { k _ { 2 } } } + m _ { j _ { k _ { 2 } } } ) } \\\\ & { = \\displaystyle \\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ( m - m _ { i _ { k _ { 1 } } } - m _ { j _ { k _ { 1 } } } ) } \\\\ & { = m ^ { 2 } - \\displaystyle \\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            121,
+            645,
+            297
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the expectation to select all useful pairs at least once regardless of the $s$ -th element in the label is ",
+        "bbox": [
+            173,
+            297,
+            826,
+            324
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1ab780bff59ff6d8d4adca0032b2a9f5bf711971f7196a93a01bbb66f1929c8.jpg",
+        "text": "$$\n\\mathrm { E } _ { \\backslash s } = 2 \\sum _ { i \\in I _ { s } , j \\in J _ { s } } m _ { i } m _ { j } \\left( 1 + { \\frac { 1 } { 2 } } + \\cdots + { \\frac { 1 } { m ^ { 2 } - \\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            323,
+            743,
+            368
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Since we rotate the subscript $s$ from 1 to $n$ , the expected number of iterations for iterative training to select all useful pairs at least once is ",
+        "bbox": [
+            173,
+            368,
+            823,
+            396
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9991e42608c20e5a8f6c92201723e1d7ad4649ed2021eaa47181793e365c5fd1.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\mathrm { E } _ { 2 } \\le n \\cdot \\operatorname* { m a x } _ { s = 1 , \\ldots , n } \\mathrm { E } _ { \\mathsf { V } ^ { s } } } } \\\\ & { } & { = 2 n \\cdot \\operatorname* { m a x } _ { s = 1 , \\ldots , n } \\sum _ { i \\in I _ { s } , j \\in J _ { s } } m _ { i } m _ { j } ( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { m ^ { 2 } - \\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            397,
+            767,
+            468
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. (of Theorem 2) We firstly show that ",
+        "bbox": [
+            174,
+            503,
+            460,
+            518
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a48e578e69dc8825accd38dde7f6f74831c5ba094feff2f94b25a5ae7d29cc58.jpg",
+        "text": "$$\n\\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } + m _ { j _ { k _ { 1 } } } ) ^ { 2 } \\geq \\sum _ { k _ { 1 } = 1 } ^ { 2 ^ { n - 1 } } ( m _ { i _ { k _ { 1 } } } ^ { 2 } + m _ { j _ { k _ { 1 } } } ^ { 2 } ) = \\sum _ { i = 1 } ^ { 2 ^ { n } } m _ { i } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            518,
+            679,
+            565
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "According to the result of Theorem 1 and the Definition 1 of balancedness, we have ",
+        "bbox": [
+            174,
+            573,
+            723,
+            588
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a901205c457c39f1963c07bc0f2c90a57ce856779d01e591c538e747bef1aeb0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { 2 } = 2 n \\cdot \\operatorname* { m a x } \\displaystyle \\sum _ { s \\in L _ { s } \\times \\xi _ { s } } m _ { s t } ( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { m ^ { 2 } - \\sum _ { k = 1 } ^ { 2 n } ( m _ { k _ { 1 } } + m _ { \\delta _ { 1 } } ) ^ { 2 } } ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { = 2 n \\cdot \\operatorname* { m a x } \\displaystyle \\sum _ { s \\in L _ { s } \\times \\xi _ { s } } m _ { s t } ( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { m ^ { 2 } - \\sum _ { s = 1 } ^ { 2 n } m _ { s t } ^ { 2 } } ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { = 2 n \\cdot \\operatorname* { m a x } \\displaystyle \\sum _ { s \\in L _ { s } \\times \\xi _ { s } } m _ { s t } ( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { m ^ { 2 } - \\sum _ { s = 1 } ^ { 2 n } m _ { s t } ^ { 2 } } ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { = 2 n \\cdot \\operatorname* { m a x } \\displaystyle \\frac { 2 n \\rho _ { 3 } } { \\rho _ { 3 } + 1 } \\cdots m ^ { 2 } ( 1 + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { m ^ { 2 } - \\sum _ { s = 1 } ^ { 2 n } m _ { s t } ^ { 2 } } ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ &  = \\operatorname* { m a x } \\displaystyle \\frac { 2 n \\rho _ { 3 } } { s } \\cdots m ^ { 2 } ( 1 + \\frac { 1 } { 2 } + \\cdots + \\ \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            589,
+            766,
+            820
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Specifically, if $n \\leq 2$ , ",
+        "bbox": [
+            173,
+            832,
+            318,
+            847
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4426c40508d4bc4887515586da1e58169a88080c8109608e068056c69e02d0a6.jpg",
+        "text": "$$\n\\frac { 2 n \\rho _ { s } } { ( \\rho _ { s } + 1 ) ^ { 2 } } \\leq \\frac { 4 \\rho _ { s } } { ( \\rho _ { s } + 1 ) ^ { 2 } } \\leq 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            847,
+            596,
+            880
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The inequality holds true forever. ",
+        "bbox": [
+            174,
+            880,
+            393,
+            895
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/01582e6778fd2c6a1e43ba8985b6f2f757a12e7f81e62d5cd93e936a006ec71e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            77,
+            818,
+            929
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/e207376bc660c66977376b6de530ed48d15a212cb7ebb3b54a7a0215adaba738.jpg",
+        "image_caption": [
+            "Figure 5: More experimental results of DNA-GAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            71,
+            816,
+            900
+        ],
+        "page_idx": 13
+    }
+]

parse/train/wHoIjrT6MMb/wHoIjrT6MMb.md

@@ -0,0 +1,376 @@
+# Robust Compressed Sensing MRI with Deep Generative Priors
+
+Ajil Jalal∗ ECE, UT Austin ajiljalal@utexas.edu
+
+Marius Arvinte\* ECE, UT Austin arvinte@utexas.edu
+
+Giannis Daras CS, UT Austin giannisdaras@utexas.edu
+
+Eric Price CS, UT Austin ecprice@cs.utexas.edu
+
+Alexandros G. Dimakis ECE, UT Austin dimakis@austin.utexas.edu
+
+Jonathan I. Tamir ECE, UT Austin jtamir@utexas.edu
+
+# Abstract
+
+The CSGM framework (Bora-Jalal-Price-Dimakis’17) has shown that deep generative priors can be powerful tools for solving inverse problems. However, to date this framework has been empirically successful only on certain datasets (for example, human faces and MNIST digits), and it is known to perform poorly on out-of-distribution samples. In this paper, we present the first successful application of the CSGM framework on clinical MRI data. We train a generative prior on brain scans from the fastMRI dataset, and show that posterior sampling via Langevin dynamics achieves high quality reconstructions. Furthermore, our experiments and theory show that posterior sampling is robust to changes in the ground-truth distribution and measurement process. Our code and models are available at: https://github.com/utcsilab/csgm-mri-langevin.
+
+# 1 Introduction
+
+Compressed sensing [23, 15] has enabled reductions to the number of measurements needed for successful reconstruction in a variety of imaging inverse problems. In particular, it has led to shorter scan times for magnetic resonance imaging (MRI) [62, 90], and most MRI vendors have released products leveraging this framework to accelerate clinical workflows. Despite their successes, sparsitybased methods are limited by the achievable acceleration rates, as the sparsity assumptions are either hand-crafted or are limited to simple learned sparse codes [72, 73].
+
+More recently, deep learning techniques have been used as powerful data-driven reconstruction methods for inverse problems [49, 68]. There are two broad families of deep learning inversion techniques [68]: end-to-end supervised and distribution-learning approaches. End-to-end supervised techniques use a training set of measured images and deploy convolutional neural networks (CNNs) and other architectures to learn the inverse mapping from measurements to image. Network architectures that include both CNN blocks and the imaging forward model have grown in popularity, as they combine deep learning with the compressed sensing optimization framework, see e.g. [32, 3, 64]. End-to-end methods are trained for specific imaging anatomy and measurement models and show excellent performance in these tasks. However, reconstruction quality is known to suffer when applied out of distribution, and recently has been shown to severely degrade [4, 19] under certain types of natural measurement and anatomy perturbations.
+
+In this paper we study deep learning inversion techniques based on distribution learning. These models are trained without reference to measurements, and so easily adapt to changes in the measurement process. The most common family of such techniques, known also as Compressed Sensing with Generative Models (CSGM) [13] uses pre-trained generative models as priors. Generative models are extremely powerful at representing image statistics and CSGM has been successfully applied to numerous inverse problems [13, 34] including non-linear phase retrieval [35], and improved with invertible models [6], sparsity based deviations [21], image adaptivity [42], and posterior sampling [79, 45]. These methods have only recently been applied to MRI and have not yet been shown to be competitive with supervised end-to-end methods. The very recent work [53] trains a StyleGAN for magnitude-only DICOM images but requires the presence of side-information and studies Gaussian, real-valued measurements for reconstruction. The deviation from the true MRI measurement model and the use of magnitude images are known to be problematic when evaluating performance [77]. Another work [54] trained an Invertible Neural Network on complex-valued single-coil MR images and showed very good performance in comparison to sparsity and GAN priors. Untrained and unamortized generators [37] have also been recently explored [19], showing promising results in some cases. Further, [17] studies the harder problem of learning a generative model for a class of images using only partial observations, as first proposed in AmbientGAN [14].
+
+In this paper we train the first score-based generative model [80] for MR images. We show that we can faithfully represent MR images without any assumptions on the measurement system. As a consequence, we are able to reconstruct retrospectively under-sampled MRI data under a variety of realistic sampling schemes. We show that our reconstruction algorithm is competitive with end-to-end supervised training when the test-data are matched to the training data and that it is robust to various out-of-distribution shifts, while in some cases end-to-end methods significantly degrade.
+
+# 1.1 Contributions
+
+• We successfully train a score-based deep generative model for complex-valued, T2-weighted brain MR images without any assumptions on the measurement scheme. When applied to multi-coil MRI reconstruction under the CSGM framework, we achieve competitive performance compared to end-to-end deep learning methods when the test-time data are sampled within distribution.
+
+• We give evidence that posterior sampling should give high-quality reconstructions. First, we show that for any measurements (including the Fourier measurements in MRI) that posterior sampling with the correct prior is within constant factors of the optimal recovery method; second, even if the prior is wrong but gives $\alpha$ mass to the true distribution, we show that posterior sampling for Gaussian measurements is nearly optimal with just an additive ${ \cal O } ( \log ( 1 / \alpha ) )$ loss.
+
+• We empirically show that our approach is robust to test-time distribution shifts including different sampling patterns and imaging anatomy. The former is unsurprising given that our model was trained without knowledge of the measurement scheme. As a consequence, our approach provides a degree of flexibility in choosing scan parameters – a common situation in routine clinical imaging. Perhaps surprisingly, the latter indicates that a specialized training set may offer sufficient regularization for a larger class of images. In contrast, we empirically show that end-to-end methods do not always enjoy the same robustness guarantees, in some cases leading to severe degradation in reconstruction quality when applied out-of-distribution.
+
+• Our method can be used to obtain multiple samples from the posterior by running Langevin dynamics with different random initializations. This allows us to get multiple reconstructions which can be used to obtain confidence intervals for each reconstructed voxel and visualize our reconstruction uncertainty on a voxel-by-voxel resolution. Uncertainty quantification can be incorporated into end-to-end methods, e.g., using variational auto-encoders [24], but this requires changes to the architecture. Our method does not require any modification and multiple reconstruction samplers can be run in parallel.
+
+Our main results are succinctly summarized in Figure 1: we achieve equivalent reconstruction performance using a reduced training set when evaluated in-distribution and are robust when evaluated out-of-distribution.
+
+# 1.2 Related Work
+
+Generative priors have shown great utility to improving compressed sensing and other inverse problems, starting with [13], who generalized the theoretical framework of compressed sensing and restricted eigenvalue conditions [85, 23, 12, 15, 40, 11, 10, 25] for signals lying on the range of a deep generative model [29, 55, 81]. Lower bounds in [51, 61, 48] established that the sample complexities in [13] are order optimal. The approach in [13] has been generalized to tackle different inverse problems [47, 35, 7, 71, 60, 63, 74, 9], and different reconstruction algorithms [21, 50, 69, 27, 26, 64, 37, 38, 18]. The complexity of optimization algorithms using generative models have been analyzed in [28, 39, 58, 36]. Our prior work shows that posterior sampling is instance-optimal for compressed sensing [45], and satisfies certain fairness guarantees without explicit information about protected sensitive groups [46].
+
+![](images/2271a0cb7f02f9550b6dba08cb9ce9313a3b43e69cf78b71f113305da794db8e.jpg)  
+Figure 1: Comparison of reconstruction methods for in-distribution, sampling-shift, and anatomy-shift images. All methods and hyperparameters were optimized on T2-weighted brain scans with a vertical sampling mask, and tested at higher accelerations, horizontal masks, and on knee & abdomen scans. Our reconstructions are competitive with state-of-the-art methods, and introduce fewer artifacts out of distribution. All measurements are multicoil k-space from the NYU fastMRI dataset and the supervised baselines are trained from scratch on MVUE targets for a fair comparison.
+
+Using compressed sensing for multi-coil MRI reconstruction has led to a rich body of work in the past two decades [62, 20, 87, 75]. See [22] and the recent special issue [44] for an overview of these methods. Classical approaches impose sparsity in a well-chosen basis, such as the wavelet domain [62], or apply shallow learning that leverages low-level redundancy in the images [72, 73, 93]. Recent research has demonstrated the superior performance of deep neural networks for MR image reconstruction [76, 32, 3, 82, 83]. A broad class of approaches is represented by end-to-end unrolled methods, which use deep networks as learned data priors in the image [3, 32, 82] or k-space domain [84]. Recent work has also investigated the performance of untrained methods [89, 38] for MR reconstruction and has shown competitive results. A much less explored line of research is MR image reconstruction with generative priors. The work in [67] proposes a CSGM-like algorithm that finetunes an entire pre-trained generator that requires a carefully tuned optimization algorithm during inference.
+
+![](images/aa8365114dade2d73b532b23e5dd1bac6e3a03cbc1f245453109db8cadcc4644.jpg)  
+Figure 2: Average test PSNR in various scenarios, across a range of acceleration factors $R$ . Higher $R$ indicates a smaller number of acquired measurements. All methods and hyperparameters were optimized on brains with an equispaced vertical mask. Our approach mostly shows the best performance and lowest reconstruction variance both in- and out-of-distribution at test-time. Shaded regions indicate $9 5 \%$ confidence intervals. Note that we trained baselines on MVUE images and hence these numerical values should not be compared with those in literature trained on RSS images (see Appendix A.1 for a more detailed discussion).
+
+# 2 System Model and Algorithm
+
+# 2.1 Multi-coil Magnetic Resonance Imaging
+
+MRI is a medical imaging modality that makes measurements using an array of radio-frequency coils placed around the body. Each coil is spatially sensitive to a local region, and measurements are acquired directly in the spatial frequency, or $k$ -space, domain. To decrease scan time, reduce operating costs, and improve patient comfort, a reduced number of $\mathbf { k }$ -space measurements are acquired in clinical use and reconstructed by incorporating explicit or implicit knowledge of the spatial sensitivity maps [78, 70, 30]. Formally, the vector of measurements $\dot { y } _ { i } \in \mathbb { C } ^ { L }$ acquired by the $\mathrm { i ^ { t h } }$ coil can be characterized by the forward model [70]:
+
+$$
+y _ { i } = P F S _ { i } x ^ { * } + w _ { i } , \quad i = 1 , . . . , N _ { c } ,
+$$
+
+where $\boldsymbol { x } ^ { * } \in \mathbb { C } ^ { N }$ is the image containing $N$ pixels, $S _ { i }$ is an operator representing the point-wise multiplication of the $\mathrm { i ^ { t h } }$ coil sensitivity map, $F$ is the spatial Fourier transform operator, $P$ represents the $\mathbf { k }$ -space sampling operator, and we assume $w _ { i } \sim \dot { \mathcal { N } } _ { c } \left( 0 , \sigma ^ { 2 } I \right)$ for simplicity. Importantly, note that the same under-sampling operator is applied to all $N _ { c }$ coils.
+
+The acceleration factor $R$ denotes the degree of under-sampling in the $k$ -space domain, i.e., $R = N / L$ Due to the multiple coils, the measurements may not be compressive for small $R$ . However, due to redundancy between the coils, the measurements are compressive for moderate values of $R$ (even if $N _ { c } \cdot L > N )$ [41]. Also note that we use the true acceleration factor $R$ , and this does not match the values in fastMRI [56] 2 on certain sampling patterns.
+
+Given multi-coil measurements $y$ , sensitivity maps represented by $S$ and the sampling operator $P$ , the goal of MR image reconstruction is to estimate the underlying image variable $x ^ { * }$ . Prior work formulates this as a regularized optimization problem:
+
+$$
+\underset { x } { \arg \operatorname* { m i n } } \| y - A x \| _ { 2 } ^ { 2 } + \lambda Q ( x ) ,
+$$
+
+where we use the operator $A \in \mathbb { C } ^ { M \times N }$ ( with $M = N _ { c } \cdot L )$ ) to subsume the discrete approximation to all linear effects, and $Q$ is a suitably chosen functional prior for the image variable $x$ . For example, to enforce a sparsity prior, one can penalize the $\ell _ { 1 }$ norm in the wavelet representation of $x$ [62]. More recent approaches involve learned regularization terms parameterized by deep neural networks [76, 32, 3]. These models are typically trained end-to-end using a fixed training set and certain assumptions about the sampling operator. In the sequel, we present how score-based generative models can be combined with the posterior sampling [45] mechanism to reformulate (2) and achieve good quality reconstructions without any a priori assumptions about the sampling scheme.
+
+When k-space is fully sampled at the Nyquist rate and no regularization is applied, the solution to (2) corresponds to the minimum-variance unbiased estimator (MVUE) of $x ^ { * }$ , denoted by $\hat { x } _ { \mathrm { M V U E } }$ [70]. Given fully sampled $\mathbf { k }$ -space data, this estimate can act as a reference image for evaluating reconstruction error as well as for end-to-end training. Alternatively, a reference image called the root-sum-of-squares (RSS) estimate can be formed by taking the inverse Fourier transform of each coil and subsequently applying the $\ell _ { 2 }$ norm for each pixel across the coil dimension, i.e. $\begin{array} { r } { \hat { x } _ { \mathrm { R S S } } = \sqrt { \sum _ { i = 1 } ^ { N _ { c } } \left| \left( F ^ { H } y _ { i } \right) \right| ^ { 2 } } } \end{array}$ , where $F ^ { H }$ is the Hermitian transpose of $F$ (here the inverse DFT). Although the RSS estimate is a biased estimator, it is often used as it does not make any assumptions about the sensitivity maps, which are not explicitly measured by the MRI system. However, even if solving (2) results in perfect recovery of $x ^ { * }$ , there will be a bias when comparing the result to $\scriptstyle { \hat { x } } _ { \mathrm { R S S } }$ and thus the RSS and MVUE cannot be directly compared numerically.
+
+# 2.2 Posterior Sampling
+
+The algorithm we consider is posterior sampling [45]. That is, given an observation of the form $y = A x ^ { * } + w$ , where $y \in \mathbb { C } ^ { M }$ , $A \in \mathbb { C } ^ { M \times N }$ , $w \stackrel { - } { \sim } \mathcal { N } _ { c } ( 0 , \sigma ^ { 2 } I )$ , and $x ^ { * } \sim \mu$ , the posterior sampling recovery algorithm outputs $\widehat { x }$ according to the posterior distribution $\mu ( \cdot | y )$ .
+
+In order to sample from the posterior, we use Langevin Dynamics [8]. Assuming we have access to $\nabla _ { x } \log \mu ( x | y )$ , we can sample from $\mu ( x | y )$ by running noisy gradient ascent:
+
+$$
+x _ { t + 1 } \gets x _ { t } + \eta _ { t } \nabla _ { x _ { t } } \log \mu ( x _ { t } | y ) + \sqrt { 2 \eta _ { t } } \ \zeta _ { t } , \quad \zeta _ { t } \sim \mathcal { N } ( 0 , 1 ) .
+$$
+
+Prior work [8] has shown that as $t \to \infty$ and $\eta _ { t }  0$ , Langevin dynamics will correctly sample from $\mu ( x | y )$ . In practice, vanilla Langevin Dynamics are slow to converge. Hence, the work in [79] proposes annealed Langevin Dynamics, where the marginal distribution of $x$ at iteration $t$ is modelled as $\mu _ { t } = \mu * \mathcal { N } ( 0 , \beta _ { t } ^ { 2 } )$ and the generative model is trained to estimate the score function $f ( x _ { t } ; \beta _ { t } ) : = \nabla _ { x _ { t } } \log ( ( \mu * \mathcal { N } ( 0 , \beta _ { t } ^ { 2 } ) ( x _ { t } ) )$ .
+
+Since the distribution of $y | x ^ { * }$ is Gaussian in Eqn (2), we obtain $\begin{array} { r } { \nabla _ { x _ { t } } \log \mu ( y | x _ { t } ) = \frac { A ^ { H } ( y - A x _ { t } ) } { \sigma ^ { 2 } } } \end{array}$ . We find that it is also helpful to anneal this term, and we set it to AH (y−Axt)σ2+γ2 , where γt → 0 is a decreasing sequence. An application of Bayes’ rule gives: $\begin{array} { r } { \nabla _ { x _ { t } } \log \mu ( x _ { t } | y ) = f ( x _ { t } ; \beta _ { t } ) + \frac { A ^ { H } ( y - A x _ { t } ) } { \sigma ^ { 2 } + \gamma _ { t } ^ { 2 } } , } \end{array}$ .
+
+Putting everything together, our final algorithm is: for $x _ { 0 } \sim \mathcal { N } _ { c } ( 0 , I )$ and for all $t = 0 , \cdots , T - 1$ ,
+
+$$
+x _ { t + 1 }  x _ { t } + \eta _ { t } ( f ( x _ { t } ; \beta _ { t } ) + \frac { A ^ { H } ( y - A x _ { t } ) } { \gamma _ { t } ^ { 2 } + \sigma ^ { 2 } } ) + \sqrt { 2 \eta _ { t } } \ \zeta _ { t } , \quad \zeta _ { t } \sim \mathcal { N } ( 0 ; I ) .
+$$
+
+Note that the parameters $T , \{ \beta _ { t } \} _ { t = 0 } ^ { T - 1 }$ were fixed during training of the generative model, and hence the only hyperparameters during inference are $\{ \eta _ { t } \} _ { t = 0 } ^ { T - 1 } , \sigma$ and $\{ \gamma _ { t } \} _ { t = 0 } ^ { T - 1 }$ . Scripts in our codebase describe hyperparameter values used in our experiments.
+
+# 3 Theoretical Results
+
+Background and Notation. We first introduce background and notation required for our theoretical results. $\| \cdot \|$ refers to the $\ell _ { 2 }$ norm. In this section alone, for simplicity of exposition, we will assume that all matrices and vectors are real valued.
+
+For two probability distributions $\mu , \nu$ on some normed space $\Omega$ , and for any $q \geq 1$ , the Wasserstein$q$ [91, 5] and Wasserstein- $\infty$ [16] distances are defined as:
+
+$$
+\mathcal { W } _ { q } ( \mu , \nu ) : = \operatorname* { i n f } _ { \gamma \in \Pi ( \mu , \nu ) } \left( \bigoplus _ { ( u , v ) \sim \gamma } [ \| u - v \| ^ { q } ] \right) ^ { 1 / q } , \quad \mathcal { W } _ { \infty } ( \mu , \nu ) : = \operatorname* { i n f } _ { \gamma \in \Pi ( \mu , \nu ) } \left( \gamma _ { ( u , v ) \in \Omega ^ { 2 } } ^ { - \mathrm { e s s } } \| u - v \| \right) .
+$$
+
+where $\Pi ( \mu , \nu )$ denotes the set of joint distributions whose marginals are $\mu , \nu$ . The above definition says that if $\mathcal { W } _ { \infty } ( \mu , \nu ) \leq \varepsilon$ , and $( u , v ) \sim \gamma$ , then $\| u - v \| \leq \varepsilon$ almost surely.
+
+The $( \varepsilon , \delta ) \cdot$ −approximate covering number [45], is defined as the smallest number of $\varepsilon$ -radius balls required to cover $1 - \delta$ mass under a distribution.
+
+Definition 3.1 $( \varepsilon , \delta )$ -approximate covering number). Let $\mu$ be a distribution on $\mathbb { R } ^ { N }$ . For some parameters $\varepsilon > 0 , \delta \in [ 0 , 1 ]$ , the $( \varepsilon , \delta )$ -approximate covering number of $\mu$ is defined as
+
+$$
+\mathrm { C o v } _ { \varepsilon , \delta } ( \mu ) : = \operatorname* { m i n } \left\{ k : \mu \left[ \cup _ { i = 1 } ^ { k } B ( x _ { i } , \varepsilon ) \right] \geq 1 - \delta , x _ { i } \in \mathbb { R } ^ { N } \right\} ,
+$$
+
+where $B ( x , \varepsilon )$ is the $\ell _ { 2 }$ ball of radius $\varepsilon$ centered at $x$ .
+
+Distributional robustness under Gaussian measurements. First, we consider mismatch between the ground-truth distribution, denoted by $\mu$ , and the generator distribution, denoted by $\nu$ . Prior work [45] has shown that if (i) $\mathcal { W } _ { q } ( \mu , \nu ) \leq \varepsilon$ for some $q \geq 1$ and (ii) we are given $M \geq O ( \log \mathrm { C o v } _ { \varepsilon , \delta } ( \mu ) )$ Gaussian measurements, then posterior sampling with respect to $\nu$ will recover $x ^ { * } \sim \mu$ up to an error of $\varepsilon / \delta ^ { 1 / q }$ with probability $1 - \delta$ . Closeness in Wasserstein distance is a reasonable assumption in certain examples, such as when $\mu$ is the distribution of celebrity faces and $\nu$ is the distribution of a generator trained on FlickrFaces [52]. However, this assumption is unsatisfactory when we consider distributions of abdominal and brain MR scans, for example, since images of these anatomies look entirely different.
+
+We define the following weaker notion of divergence between distributions. Informally, this new definition tells us that $\nu$ and $\mu$ are “close” if they can each be split into components which are close in $\mathcal { W } _ { \infty }$ distance, such that the close components contain a sufficiently large fraction under $\nu$ and $\mu$ . Formally, this is defined as:
+
+Definition 3.2 $( ( \delta , \alpha ) – \mathcal { W } _ { \infty }$ divergence). For two probability distributions $\nu$ and $\mu ,$ , and parameters $\delta , \alpha \in [ 0 , 1 ]$ , the $( \delta , \alpha ) \ – \mathcal { W } _ { \infty }$ divergence is defined as
+
+$$
+\begin{array} { r l } & { ( \delta , \alpha ) {  { \mathcal W } } _ { \infty } ( \mu , \nu ) : = \operatorname* { i n f } \{ \varepsilon \geq 0 : } \\ & { \exists \mu ^ { \prime } , \mu ^ { \prime \prime } , \nu ^ { \prime } , \nu ^ { \prime \prime } \in {  { \mathcal W } } ( {  { \mathbb R } } ^ { N } ) \ s . t . \ \mu = ( 1 - \delta ) \mu ^ { \prime } + \delta \mu ^ { \prime \prime } , \nu = ( 1 - \alpha ) \nu ^ { \prime } + \alpha \nu ^ { \prime \prime } , \mathcal W _ { \infty } ( \mu ^ { \prime } , \nu ^ { \prime } ) = \varepsilon . \} } \end{array}
+$$
+
+Lemma B.1 highlights that this is a strict generalization of Wasserstein distances, in the sense that closeness in Wasserstein distance implies closeness in this new divergence.
+
+Since the $( \delta , \alpha ) \ – \mathcal { W } _ { \infty }$ divergence is a generalization of Wasserstein distances, it is not clear that the main Theorem in [45] holds for distributions that are close in this new divergence. The following result shows a rather surprising fact: if $( \delta , \alpha ) \ – \mathcal { W } _ { \infty } ( \mu , \nu ) \leq \varepsilon$ then posterior sampling with $M =$ $\begin{array} { r } { O \left( \log \left( \frac { 1 } { 1 - \alpha } \right) + \log \mathrm { C o v } _ { \varepsilon , \delta } ( \mu ) \right) } \end{array}$ measurements will still succeed with probability $\ge 1 - O ( \delta )$ .
+
+Theorem 3.3. Let $\delta , \alpha \in [ 0 , 1 ]$ , and $\varepsilon > 0$ be parameters. Let $\mu , \nu$ be arbitrary distributions over $\mathbb { R } ^ { N }$ satisfying $( \delta , \alpha ) \ – \mathcal { W } _ { \infty } ( \mu , \nu ) \leq \varepsilon$ . Let $x ^ { * } \sim \mu$ and suppose $y = A x ^ { * } + w$ , where $A \in \mathbb { R } ^ { M \times N }$ and $w \in \dot { \mathbb { R } } ^ { M }$ are i.i.d. Gaussian normalized such that $A _ { i j } \sim \mathcal { N } ( 0 , 1 / M )$ and $w _ { i } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } / M )$ , with $\sigma \gtrsim \varepsilon$ . Given $y$ and the fixed matrix $A$ , let $\widehat { x }$ be the output of posterior sampling with respect to $\nu$ .
+
+Then for $\begin{array} { r } { M \geq O \left( \log \left( \frac { 1 } { 1 - \alpha } \right) + \operatorname* { m i n } ( \log \mathrm { C o v } _ { \sigma , \delta } ( \mu ) , \log \mathrm { C o v } _ { \sigma , \delta } ( \nu ) ) \right) } \end{array}$ , there exists a universal constant $c > 0$ such that with probability at least $1 - e ^ { - \Omega ( M ) }$ over $A , w$ ,
+
+$$
+\operatorname* { P r } _ { x ^ { * } \sim \mu , \widehat { x } \sim \nu ( \cdot | y ) } \left[ \| x ^ { * } - \widehat { x } \| \geq c ( \varepsilon + \sigma ) \right] \leq \delta + e ^ { - \Omega ( M ) } .
+$$
+
+![](images/90078d3b474935e152b912aec68c52df9e50dcb9e83b5141f2339a17391f80a8.jpg)  
+Figure 3: Comparative reconstructions of a 2D abdominal scan with uniform random under-sampling in the horizontal direction at $R = 4$ . None of the methods were trained to reconstruct abdomen MRI. Our method uses a score-based generative model trained on brain images (as explained) and obtains good reconstructions. The red arrows indicate missing details or artifacts in the kidney structure.
+
+For our running example of $\nu$ being a generator trained on brain scans, and $\mu$ the distribution of abdominal scans, we can set $\nu ^ { \prime }$ to be the distribution of our generator restricted to abdominal scans, and we can let $\mu ^ { \prime }$ be the distribution restricted to “inliers” in $\mu$ . This shows that even if our generator places an exponentially small probability mass(i.e., $1 - \alpha \ll 1 ,$ ) on the set of abdominal scans, we can still recover abdominal scans with a polynomial additive increase in the number of measurements (i.e., $\log ( 1 / ( 1 - \alpha ) )$ ).
+
+Near-optimality under arbitrary measurement processes. The previous result required Gaussian matrices to handle the distribution shift. Our next result shows that for an arbitrary measurement process, and assuming that there is no distribution shift between the generator and the ground truth distribution, posterior sampling is almost the best algorithm for this fixed measurement process. This result also shows that posterior sampling is good with respect to any metric.
+
+Theorem 3.4. Let $d ( \cdot , \cdot )$ be an arbitrary metric over $\mathbb { R } ^ { N } \times \mathbb { R } ^ { N }$ . Let $x ^ { * } \sim \mu$ and let $y = \mathcal { A } ( x ^ { * } )$ be measurements generated from $x ^ { * }$ for some arbitrary forward operator $\mathcal { A } : \mathbb { R } ^ { N } \to \mathbb { R } ^ { M }$ . Then if there exists an algorithm that uses y as inputs and outputs $x ^ { \prime }$ such that
+
+$$
+d ( x ^ { * } , x ^ { \prime } ) \leq \varepsilon { \mathit { w i t h } } p r o b a b i l i t y \ 1 - \delta ,
+$$
+
+then posterior sampling $\widehat { x } \sim \mu ( \cdot | y )$ will satisfy
+
+$$
+d ( x ^ { * } , \widehat { x } ) \leq 2 \varepsilon w i t h p r o b a b i l i t y \ \geq 1 - 2 \delta .
+$$
+
+Remark on combining these results. Our theoretical results above show that posterior sampling is (1) highly robust to distribution shift under Gaussian measurements, and (2) accurate with arbitrary measurements without distribution shift. A natural hope would be to combine these two results and show that it is robust to distribution shift under Fourier measurements. Unfortunately, this is not true for general distributions: for example, if $\mu$ and $\nu$ are both random distributions over Fourier-sparse signals, then Fourier measurements will usually give zero information about the signal, so cannot convince the sampler to sample near $\mu$ rather than $\nu$ .
+
+# 4 Experimental Results
+
+We perform retrospective under-sampling in all experiments, i.e., given fully-sampled $\mathbf { k }$ -space measurements from the NYU fastMRI [56, 94] and Stanford MRI [1] datasets, we apply sampling masks and evaluate the performance of all considered algorithms on the reconstructed data. Depending on scan parameters (e.g., 3D scans for the Stanford knee data in Appendix F), we appropriately slice and sample the data in the proper dimension so as to not commit any inverse crime [31, 77].
+
+We first highlight that an advantage of the proposed approach is the invariance to the sampling scheme during training. In contrast, this is a design choice that must be made for supervised end-to-end methods, which here were trained on equispaced, vertical sampling masks, following the fastMRI 2020 challenge guidelines [94, 66]. As our results show, this affords us a significant degree of robustness across a wide distribution of sampling masks during inference.
+
+We train a score-based model, NCSNv2 [80], on a small subset of scans from the NYU fastMRI brain dataset. Specifically, we train using T2-weighted images at a field strength of 3 Tesla for a total of 14,539 2D training slices. We calculate the MVUE from the fully sampled data and use the ESPIRiT algorithm [87, 43] applied to the fully-sampled central portion of $\mathbf { k }$ -space to estimate the sensitivity maps. The backbone network for our model is a RefineNet [59]. Since the generator’s output is expected to be complex-valued, we treat the real and imaginary parts as separate image channels. Details about the architectures are given in Appendix G.
+
+We use an $\ell _ { 1 }$ -Wavelet regularized reconstruction algorithm [62] as a parallel imaging and compressed sensing baseline. This aims to solve the optimization problem given in (2) with $\mathbf { \bar { Q } } ( x ) = | | \mathbf { \bar { W } } x | | _ { 1 }$ where $W$ is a 2D Wavelet transform. We use the publicly available implementation from the BART toolbox [88, 86] and optimize the regularization hyper-parameter using the same subset of samples from the brain dataset that was used to train our method. We find that $\lambda = 0 . 0 1$ performs the best on the training data and use this value for all experiments. We consider three different deep learning baselines: MoDL [3], E2E-VarNet [82], and the ConvDecoder architecture [19].
+
+We train the MoDL and E2E-VarNet baselines from scratch on the same training dataset as our method, at acceleration factors $R = \{ 3 , 6 \}$ and equispaced under-sampling, with a supervised SSIM loss on the magnitude MVUE image, for 40 and 15 epochs, respectively, using a batch size of 1. For the ConvDecoder baseline, we use the architecture for brain data in [19] that outputs a complex image estimate and optimize the number of fitting iterations on a subset of samples from the training data. We find that 10000 iterations are sufficient to reach a stable average performance at $R = 3$ . Put together, all of our baselines are tailored to estimate the complex image $x$ , thus all comparisons are fair. We evaluate reconstruction performance using the complex MVUE of the fully sampled data as a reference image and measure the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) [92] between the absolute values of the reconstruction and ground-truth MVUE images.
+
+# 4.1 In-Distribution Performance
+
+In this experiment, we test all models using the same forward model that matches the training conditions for the baselines: vertical, equispaced sampling patterns. Examples of various sampling patterns are shown in Appendix C.
+
+Figure 1 (top three rows) shows qualitative results and Figures 2a & 5a respectively show PSNR & SSIM values, for the case where there is no mismatch between the training and inference sampling patterns. As the baselines were trained to maximize SSIM at $R = 3 ~ \& ~ 6$ , we see that they achieve better SSIM scores than us at these accelerations, although there is clear aliasing in the baselines at $R = 6$ . We achieve better PSNR values at these accelerations, which supports the claim that our method does not overfit to a particular metric (Theorem 3.4). This also highlights the importance of qualitative evaluations in medical image reconstruction and the limitations of existing image quality metrics [65]. From the third row of Figure 1, and Figures 2a & 5a, we notice that our method surpasses baselines at higher accelerations.
+
+We find that $\ell _ { 1 }$ -Wavelet suffers both qualitatively and quantitatively at high acceleration factors, while the ConvDecoder is also a competitive architecture, but incurs a large computational cost. When benchmarked on an NVIDIA RTX 2080Ti GPU, our method takes 16 minutes and $0 . 9 5 \mathrm { G B }$ of memory to reconstruct a high-resolution brain scan, whereas the ConvDecoder takes longer than 80 minutes and $6 . 6 \mathrm { G B }$ of memory. While our method is limited by the inference time and is not in the range of end-to-end models (where reconstruction takes at most on the order of seconds and $3 . 5 \mathrm { G B }$ of memory), multiple scans can be reconstructed in parallel due to the reduced memory footprint.
+
+# 4.2 Out-of-Distribution Performance
+
+Test-time sampling pattern shifts. Here we consider shifts in the forward sampling operator at test-time, while still evaluating on the same anatomy as the training conditions. We measure robustness by evaluating the average incurred performance loss when the sampling pattern changes. Recall that our proposed approach does not use any explicit information about the sampling pattern $P$ during training, hence we anticipate the highest degree of robustness.
+
+![](images/b018ec33241411896dc3fc3986cc59d405d15f8a24b43f8fcf25f05147bb0981.jpg)  
+Figure 4: Our method successfully recovers fine details and can provide an estimate of the reconstruction error. The left column shows a knee from the fastMRI dataset, along with an annotated meniscus tear (indicated by red arrow in zoomed inset). Given measurements at an acceleration factor of $R = 4$ , we obtain 48 independent reconstructions via posterior sampling. The second column shows the pixel-wise average of reconstructions, the third column shows the pixel-wise standard deviation, and the fourth column shows the magnitude of the error between the ground truth and the mean reconstruction. Note that our generative prior has never seen such pathology, as it was trained on T2-weighted brain scans.
+
+Figure 1 (fourth row) shows qualitative reconstructions when the measurements are obtained from an equispaced, horizontal sampling mask, with an acceleration factor $R = 3$ . It can be observed that the reconstructions output by E2E-VarNet show aliasing artifacts. Based on the statistical results in Figure 2b & 5b, our method retains its performance.
+
+Furthermore, this experiment reveals that MoDL is more robust to this type of mask shift when compared to E2E-VarNet, even though it uses a smaller network. This is explained by the fact that E2E-VarNet does not use external sensitivity map estimates, but uses a deep neural network for endto-end map estimation. While this improves performance on in-distribution samples, the performance drop is strong evidence that accurate sensitivity map estimation is vital for robust generalization, and both our proposed approach and MoDL benefit from the external ESPIRiT algorithm, which is compatible with different sampling patterns.
+
+We do note that retrospectively flipping the horizontal and vertical sampling direction is not necessarily representative of prospective sampling in the horizontal direction due to the discrete nature of the phase encoding direction in MRI, and this may contribute to the higher scores compared to the vertical mask experiments.
+
+Test-time anatomy shifts. We now consider the more difficult problem of reconstructing different anatomies than the ones seen during. This was previously investigated in [19], which concluded that all methods suffer a drastic shift due to the various changes in scan parameters between body parts. In contrast to prior work, our main finding is that the proposed score-based model retains a significant degree of robustness under these shifts, and outputs excellent qualitative reconstructions. In some cases, some end-to-end methods retain robustness as well.
+
+Figures 2c & 5c show PSNR and SSIM scores obtained on reconstructed abdominal scans obtained from [1] at different acceleration factors. This represents both an anatomy and sampling pattern shift, and it can be seen that our method, MoDL, and the $\ell _ { 1 }$ -Wavelet algorithm retain their competitive advantage, while the ConvDecoder and E2E-VarNet suffer severe performance losses. Figure 3 further shows a qualitative comparison of a reconstructed abdominal scan at $R = 4$ , with highlighted artifacts. Appendix E shows another abdomen scan.
+
+Finally, Figures 2d & 5d show PSNR and SSIM scores obtained on fastMRI knee reconstructions, while Figure 1 (bottom row) shows the accompanying qualitative plots. This anatomy is challenging especially because of the poor signal-to-noise ratio conditions, which can be seen even in the groundtruth image. It can be noticed that this is the most severe shift for all methods, but our approach still shows the best performance at $R = 2 , 4$ and a significantly lower variance. Appendix D shows more examples of knee reconstructions with and without fat suppression, and Figure 20 shows metrics on fat suppressed knees.
+
+# 4.3 Uncertainty Estimation
+
+Our method can also provide uncertainty estimates for each reconstructed pixel by running multiple truc, for n samplers. For a given observation sufficiently large. Now, using the $y$ , we can obtain independeonditional mean estimate $\widehat { x } _ { 1 } , \cdots , \widehat { x } _ { K } \sim$ $\mu ( \cdot | y )$ $K$ $\textstyle { \bar { x } } = \sum _ { i = 1 } ^ { K } { \widehat { x } } _ { i } / K$ compute the pixel-wise standard deviation $\sqrt { \textstyle \sum _ { i = 1 } ^ { K } | \widehat { x } _ { i } - \bar { x } | ^ { 2 } / K }$ , and this gives an estimate of the error in each pixel. As shown in $\mathrm { F i g ~ 4 }$ , the pixel-wise standard deviation is a good estimate of the ground truth error $| x ^ { * } - { \bar { x } } |$ . Additionally, notice that the reconstructions are able to recover fine details such as the annotated meniscus tear3 in Fig 4 and predict low uncertainty for these features.
+
+Figure 17 in Appendix D shows another example of an annotated meniscus tear. Figures 18 and 19 show comparisons with baselines on the same examples.
+
+# 4.4 Radiologist Study
+
+We have conducted a preliminary blind assessment of overall image quality with two board-certified radiologists and one faculty member who uses neuroimaging for their research. These experts were not involved in our research. We have found that our algorithm was ranked best for knee scans, and tied with the baselines for abdominal and brain scans, supporting our robustness claims in the paper. For more details, please see Appendix H.
+
+# 5 Limitations
+
+We reported PSNR and SSIM values as they are correlated with radiologist evaluation upto an extent, and our preliminary radiologist study in Section 4.4 suggests the feasibility of clinical adoption. These metrics do not capture the needs of real-world radiologists, and a more detailed study is required before the proposed techniques can be clinically adopted.
+
+Though promising, our initial results were still limited to fast spin-echo imaging only and all data were retrospectively under-sampled. Further study is required to demonstrate prospective performance in a larger body of heterogeneous MRI data. Our method also currently requires a high compute cost at inference time, as well as the need for a pre-trained generative model. Clinical use requires fast reconstruction in addition to fast scanning. Future work should investigate whether score-based models can be trained without a fully-sampled training set as well as investigate approaches to reducing computation time.
+
+Finally, there are potential issues related to discrimination. Specifically, it is possible that the quality of the reconstructed images varies across protected attributes, such as gender or race [57].
+
+# 6 Conclusions
+
+This paper reports the first successful application of the CSGM framework for robust multi-coil MR image reconstruction under realistic sampling conditions, and provides theoretical evidence for the robustness of posterior sampling. Our score-based model was trained on a small subset of brain MRI scans without any explicit information about the sampling scheme. This shows state-of-the-art performance under severe distributional shifts, making our model applicable in a wide range of clinical settings.
+
+Our method shows a considerable degree of generalization to out-of-distribution samples such as abdomen and knee MRI, even when trained exclusively on brain MRI. Notably, these scans were acquired using different MRI vendors with different pulse sequence parameters and at different institutions. We postulate that adding a small set of diverse training samples to our generative model could further improve robustness, and we hypothesize that these samples may not necessarily be restricted to MR images.
+
+The results presented in this work represent an important step to applying deep learning models in the clinic, as there is a natural variation in sampling, image orientation, receive coils, scanner hardware, and anatomy in clinical practice.
+
+# 7 Acknowledgements
+
+Ajil Jalal, Giannis Daras and Alex Dimakis have been supported by NSF Grants CCF 1763702, 1934932, AF 1901281, 2008710, 2019844, the NSF IFML 2019844 award as well as research gifts by Western Digital, Interdigital, WNCG and MLL, computing resources from TACC and the Archie Straiton Fellowship.
+
+Eric Price has been supported by NSF Award CCF-1751040 (CAREER), NSF Award CCF-2008868, and NSF IFML 2019844.
+
+Marius Arvinte and Jon Tamir have been supported by NSF IFML 2019844 award, ONR grant N00014-19-1-2590, NIH Grant U24EB029240, and an AWS Machine Learning Research Award.
+
+We thank the anonymous NeurIPS reviewers for their helpful and considerate feedback.
+
+Finally, we would like to thank the experts who graciously helped with our image assessment study.
+
+# 8 Funding Transparency Statements
+
+# 8.1 Funding (financial activities supporting the submitted work):
+
+Funding in direct support of this work: NSF Grants CCF 1763702, 1934932, AF 1901281, 2008710, 2019844, 1751040, 2008868, the NSF IFML 2019844 award, ONR grant N00014-19-1-2590, NIH Grant U24EB029240, and an AWS Machine Learning Research Award, as well as research gifts by Western Digital, Interdigital, WNCG and MLL, computing resources from TACC and the Archie Straiton Fellowship.
+
+# 8.2 Competing Interests (financial activities outside the submitted work):
+
+Additional revenues related to this work: Internship at Intel and Google.
+
+# References
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 3, 4, 4.3   
+(b) Did you describe the limitations of your work? [Yes] See Section 5   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 5   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] Yes we have read the ethics review guidelines.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] In the statements of Theorem 3.3, Theorem 3.4. (b) Did you include complete proofs of all theoretical results? [Yes] In Appendix B
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Our GitHub link contains all relevant information   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In Appendix G   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All quantitative plots have $9 5 \%$ confidence intervals.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Yes, in section 4 and Appendix G
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] Our algorithm uses the NCSNv2 generative model, NYU fastMRI dataset, and Stanford MRI datasets, all of which have been cited.   
+(b) Did you mention the license of the assets? [Yes] In Appendix G   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] To the best of our knowledge, we are the first to train generative models for complex valued MR scans, and we include this in the GitHub link in Appendix G   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We used the NYU fastMRI and Stanford MRI datasets, both of which have this covered in their terms of agreement.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] We used the NYU fastMRI and Stanford MRI datasets, both of which have been anonymized.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] We didn’t do any human subject experiments   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] We didn’t do any human subject experiments   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] We didn’t do any human subject experiments

parse/train/wHoIjrT6MMb/wHoIjrT6MMb_content_list.json

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+        "type": "text",
+        "text": "Robust Compressed Sensing MRI with Deep Generative Priors ",
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+        "type": "text",
+        "text": "Ajil Jalal∗ ECE, UT Austin ajiljalal@utexas.edu ",
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+        "text": "Marius Arvinte\\* ECE, UT Austin arvinte@utexas.edu ",
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+        "text": "Giannis Daras CS, UT Austin giannisdaras@utexas.edu ",
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+    {
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+        "text": "Eric Price CS, UT Austin ecprice@cs.utexas.edu ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alexandros G. Dimakis ECE, UT Austin dimakis@austin.utexas.edu ",
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+    {
+        "type": "text",
+        "text": "Jonathan I. Tamir ECE, UT Austin jtamir@utexas.edu ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "The CSGM framework (Bora-Jalal-Price-Dimakis’17) has shown that deep generative priors can be powerful tools for solving inverse problems. However, to date this framework has been empirically successful only on certain datasets (for example, human faces and MNIST digits), and it is known to perform poorly on out-of-distribution samples. In this paper, we present the first successful application of the CSGM framework on clinical MRI data. We train a generative prior on brain scans from the fastMRI dataset, and show that posterior sampling via Langevin dynamics achieves high quality reconstructions. Furthermore, our experiments and theory show that posterior sampling is robust to changes in the ground-truth distribution and measurement process. Our code and models are available at: https://github.com/utcsilab/csgm-mri-langevin. ",
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+        "text": "1 Introduction ",
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+        "text": "Compressed sensing [23, 15] has enabled reductions to the number of measurements needed for successful reconstruction in a variety of imaging inverse problems. In particular, it has led to shorter scan times for magnetic resonance imaging (MRI) [62, 90], and most MRI vendors have released products leveraging this framework to accelerate clinical workflows. Despite their successes, sparsitybased methods are limited by the achievable acceleration rates, as the sparsity assumptions are either hand-crafted or are limited to simple learned sparse codes [72, 73]. ",
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+    {
+        "type": "text",
+        "text": "More recently, deep learning techniques have been used as powerful data-driven reconstruction methods for inverse problems [49, 68]. There are two broad families of deep learning inversion techniques [68]: end-to-end supervised and distribution-learning approaches. End-to-end supervised techniques use a training set of measured images and deploy convolutional neural networks (CNNs) and other architectures to learn the inverse mapping from measurements to image. Network architectures that include both CNN blocks and the imaging forward model have grown in popularity, as they combine deep learning with the compressed sensing optimization framework, see e.g. [32, 3, 64]. End-to-end methods are trained for specific imaging anatomy and measurement models and show excellent performance in these tasks. However, reconstruction quality is known to suffer when applied out of distribution, and recently has been shown to severely degrade [4, 19] under certain types of natural measurement and anatomy perturbations. ",
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+        "type": "text",
+        "text": "In this paper we study deep learning inversion techniques based on distribution learning. These models are trained without reference to measurements, and so easily adapt to changes in the measurement process. The most common family of such techniques, known also as Compressed Sensing with Generative Models (CSGM) [13] uses pre-trained generative models as priors. Generative models are extremely powerful at representing image statistics and CSGM has been successfully applied to numerous inverse problems [13, 34] including non-linear phase retrieval [35], and improved with invertible models [6], sparsity based deviations [21], image adaptivity [42], and posterior sampling [79, 45]. These methods have only recently been applied to MRI and have not yet been shown to be competitive with supervised end-to-end methods. The very recent work [53] trains a StyleGAN for magnitude-only DICOM images but requires the presence of side-information and studies Gaussian, real-valued measurements for reconstruction. The deviation from the true MRI measurement model and the use of magnitude images are known to be problematic when evaluating performance [77]. Another work [54] trained an Invertible Neural Network on complex-valued single-coil MR images and showed very good performance in comparison to sparsity and GAN priors. Untrained and unamortized generators [37] have also been recently explored [19], showing promising results in some cases. Further, [17] studies the harder problem of learning a generative model for a class of images using only partial observations, as first proposed in AmbientGAN [14]. ",
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+        "type": "text",
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+        "text": "In this paper we train the first score-based generative model [80] for MR images. We show that we can faithfully represent MR images without any assumptions on the measurement system. As a consequence, we are able to reconstruct retrospectively under-sampled MRI data under a variety of realistic sampling schemes. We show that our reconstruction algorithm is competitive with end-to-end supervised training when the test-data are matched to the training data and that it is robust to various out-of-distribution shifts, while in some cases end-to-end methods significantly degrade. ",
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+    {
+        "type": "text",
+        "text": "1.1 Contributions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "• We successfully train a score-based deep generative model for complex-valued, T2-weighted brain MR images without any assumptions on the measurement scheme. When applied to multi-coil MRI reconstruction under the CSGM framework, we achieve competitive performance compared to end-to-end deep learning methods when the test-time data are sampled within distribution. ",
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+        "type": "text",
+        "text": "• We give evidence that posterior sampling should give high-quality reconstructions. First, we show that for any measurements (including the Fourier measurements in MRI) that posterior sampling with the correct prior is within constant factors of the optimal recovery method; second, even if the prior is wrong but gives $\\alpha$ mass to the true distribution, we show that posterior sampling for Gaussian measurements is nearly optimal with just an additive ${ \\cal O } ( \\log ( 1 / \\alpha ) )$ loss. ",
+        "bbox": [
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+        "text": "• We empirically show that our approach is robust to test-time distribution shifts including different sampling patterns and imaging anatomy. The former is unsurprising given that our model was trained without knowledge of the measurement scheme. As a consequence, our approach provides a degree of flexibility in choosing scan parameters – a common situation in routine clinical imaging. Perhaps surprisingly, the latter indicates that a specialized training set may offer sufficient regularization for a larger class of images. In contrast, we empirically show that end-to-end methods do not always enjoy the same robustness guarantees, in some cases leading to severe degradation in reconstruction quality when applied out-of-distribution. ",
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+        "type": "text",
+        "text": "• Our method can be used to obtain multiple samples from the posterior by running Langevin dynamics with different random initializations. This allows us to get multiple reconstructions which can be used to obtain confidence intervals for each reconstructed voxel and visualize our reconstruction uncertainty on a voxel-by-voxel resolution. Uncertainty quantification can be incorporated into end-to-end methods, e.g., using variational auto-encoders [24], but this requires changes to the architecture. Our method does not require any modification and multiple reconstruction samplers can be run in parallel. ",
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+    {
+        "type": "text",
+        "text": "Our main results are succinctly summarized in Figure 1: we achieve equivalent reconstruction performance using a reduced training set when evaluated in-distribution and are robust when evaluated out-of-distribution. ",
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+    {
+        "type": "text",
+        "text": "1.2 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Generative priors have shown great utility to improving compressed sensing and other inverse problems, starting with [13], who generalized the theoretical framework of compressed sensing and restricted eigenvalue conditions [85, 23, 12, 15, 40, 11, 10, 25] for signals lying on the range of a deep generative model [29, 55, 81]. Lower bounds in [51, 61, 48] established that the sample complexities in [13] are order optimal. The approach in [13] has been generalized to tackle different inverse problems [47, 35, 7, 71, 60, 63, 74, 9], and different reconstruction algorithms [21, 50, 69, 27, 26, 64, 37, 38, 18]. The complexity of optimization algorithms using generative models have been analyzed in [28, 39, 58, 36]. Our prior work shows that posterior sampling is instance-optimal for compressed sensing [45], and satisfies certain fairness guarantees without explicit information about protected sensitive groups [46]. ",
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+    {
+        "type": "image",
+        "img_path": "images/2271a0cb7f02f9550b6dba08cb9ce9313a3b43e69cf78b71f113305da794db8e.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of reconstruction methods for in-distribution, sampling-shift, and anatomy-shift images. All methods and hyperparameters were optimized on T2-weighted brain scans with a vertical sampling mask, and tested at higher accelerations, horizontal masks, and on knee & abdomen scans. Our reconstructions are competitive with state-of-the-art methods, and introduce fewer artifacts out of distribution. All measurements are multicoil k-space from the NYU fastMRI dataset and the supervised baselines are trained from scratch on MVUE targets for a fair comparison. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
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+        "type": "text",
+        "text": "Using compressed sensing for multi-coil MRI reconstruction has led to a rich body of work in the past two decades [62, 20, 87, 75]. See [22] and the recent special issue [44] for an overview of these methods. Classical approaches impose sparsity in a well-chosen basis, such as the wavelet domain [62], or apply shallow learning that leverages low-level redundancy in the images [72, 73, 93]. Recent research has demonstrated the superior performance of deep neural networks for MR image reconstruction [76, 32, 3, 82, 83]. A broad class of approaches is represented by end-to-end unrolled methods, which use deep networks as learned data priors in the image [3, 32, 82] or k-space domain [84]. Recent work has also investigated the performance of untrained methods [89, 38] for MR reconstruction and has shown competitive results. A much less explored line of research is MR image reconstruction with generative priors. The work in [67] proposes a CSGM-like algorithm that finetunes an entire pre-trained generator that requires a carefully tuned optimization algorithm during inference. ",
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+        "type": "image",
+        "img_path": "images/aa8365114dade2d73b532b23e5dd1bac6e3a03cbc1f245453109db8cadcc4644.jpg",
+        "image_caption": [
+            "Figure 2: Average test PSNR in various scenarios, across a range of acceleration factors $R$ . Higher $R$ indicates a smaller number of acquired measurements. All methods and hyperparameters were optimized on brains with an equispaced vertical mask. Our approach mostly shows the best performance and lowest reconstruction variance both in- and out-of-distribution at test-time. Shaded regions indicate $9 5 \\%$ confidence intervals. Note that we trained baselines on MVUE images and hence these numerical values should not be compared with those in literature trained on RSS images (see Appendix A.1 for a more detailed discussion). "
+        ],
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+    {
+        "type": "text",
+        "text": "2 System Model and Algorithm ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 Multi-coil Magnetic Resonance Imaging ",
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+        "type": "text",
+        "text": "MRI is a medical imaging modality that makes measurements using an array of radio-frequency coils placed around the body. Each coil is spatially sensitive to a local region, and measurements are acquired directly in the spatial frequency, or $k$ -space, domain. To decrease scan time, reduce operating costs, and improve patient comfort, a reduced number of $\\mathbf { k }$ -space measurements are acquired in clinical use and reconstructed by incorporating explicit or implicit knowledge of the spatial sensitivity maps [78, 70, 30]. Formally, the vector of measurements $\\dot { y } _ { i } \\in \\mathbb { C } ^ { L }$ acquired by the $\\mathrm { i ^ { t h } }$ coil can be characterized by the forward model [70]: ",
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+    {
+        "type": "equation",
+        "img_path": "images/29b1692fef51364c3db80de63aa8662b2f4c15ab37ecaf07478309721f64fba8.jpg",
+        "text": "$$\ny _ { i } = P F S _ { i } x ^ { * } + w _ { i } , \\quad i = 1 , . . . , N _ { c } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { x } ^ { * } \\in \\mathbb { C } ^ { N }$ is the image containing $N$ pixels, $S _ { i }$ is an operator representing the point-wise multiplication of the $\\mathrm { i ^ { t h } }$ coil sensitivity map, $F$ is the spatial Fourier transform operator, $P$ represents the $\\mathbf { k }$ -space sampling operator, and we assume $w _ { i } \\sim \\dot { \\mathcal { N } } _ { c } \\left( 0 , \\sigma ^ { 2 } I \\right)$ for simplicity. Importantly, note that the same under-sampling operator is applied to all $N _ { c }$ coils. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The acceleration factor $R$ denotes the degree of under-sampling in the $k$ -space domain, i.e., $R = N / L$ Due to the multiple coils, the measurements may not be compressive for small $R$ . However, due to redundancy between the coils, the measurements are compressive for moderate values of $R$ (even if $N _ { c } \\cdot L > N )$ [41]. Also note that we use the true acceleration factor $R$ , and this does not match the values in fastMRI [56] 2 on certain sampling patterns. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given multi-coil measurements $y$ , sensitivity maps represented by $S$ and the sampling operator $P$ , the goal of MR image reconstruction is to estimate the underlying image variable $x ^ { * }$ . Prior work formulates this as a regularized optimization problem: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/9ebc91ed6c1378a1a7b084de2493d405f5149f9e1a1118564f7cf957eb81f657.jpg",
+        "text": "$$\n\\underset { x } { \\arg \\operatorname* { m i n } } \\| y - A x \\| _ { 2 } ^ { 2 } + \\lambda Q ( x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where we use the operator $A \\in \\mathbb { C } ^ { M \\times N }$ ( with $M = N _ { c } \\cdot L )$ ) to subsume the discrete approximation to all linear effects, and $Q$ is a suitably chosen functional prior for the image variable $x$ . For example, to enforce a sparsity prior, one can penalize the $\\ell _ { 1 }$ norm in the wavelet representation of $x$ [62]. More recent approaches involve learned regularization terms parameterized by deep neural networks [76, 32, 3]. These models are typically trained end-to-end using a fixed training set and certain assumptions about the sampling operator. In the sequel, we present how score-based generative models can be combined with the posterior sampling [45] mechanism to reformulate (2) and achieve good quality reconstructions without any a priori assumptions about the sampling scheme. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When k-space is fully sampled at the Nyquist rate and no regularization is applied, the solution to (2) corresponds to the minimum-variance unbiased estimator (MVUE) of $x ^ { * }$ , denoted by $\\hat { x } _ { \\mathrm { M V U E } }$ [70]. Given fully sampled $\\mathbf { k }$ -space data, this estimate can act as a reference image for evaluating reconstruction error as well as for end-to-end training. Alternatively, a reference image called the root-sum-of-squares (RSS) estimate can be formed by taking the inverse Fourier transform of each coil and subsequently applying the $\\ell _ { 2 }$ norm for each pixel across the coil dimension, i.e. $\\begin{array} { r } { \\hat { x } _ { \\mathrm { R S S } } = \\sqrt { \\sum _ { i = 1 } ^ { N _ { c } } \\left| \\left( F ^ { H } y _ { i } \\right) \\right| ^ { 2 } } } \\end{array}$ , where $F ^ { H }$ is the Hermitian transpose of $F$ (here the inverse DFT). Although the RSS estimate is a biased estimator, it is often used as it does not make any assumptions about the sensitivity maps, which are not explicitly measured by the MRI system. However, even if solving (2) results in perfect recovery of $x ^ { * }$ , there will be a bias when comparing the result to $\\scriptstyle { \\hat { x } } _ { \\mathrm { R S S } }$ and thus the RSS and MVUE cannot be directly compared numerically. ",
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+    {
+        "type": "text",
+        "text": "2.2 Posterior Sampling ",
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+        "type": "text",
+        "text": "The algorithm we consider is posterior sampling [45]. That is, given an observation of the form $y = A x ^ { * } + w$ , where $y \\in \\mathbb { C } ^ { M }$ , $A \\in \\mathbb { C } ^ { M \\times N }$ , $w \\stackrel { - } { \\sim } \\mathcal { N } _ { c } ( 0 , \\sigma ^ { 2 } I )$ , and $x ^ { * } \\sim \\mu$ , the posterior sampling recovery algorithm outputs $\\widehat { x }$ according to the posterior distribution $\\mu ( \\cdot | y )$ . ",
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+        "type": "text",
+        "text": "In order to sample from the posterior, we use Langevin Dynamics [8]. Assuming we have access to $\\nabla _ { x } \\log \\mu ( x | y )$ , we can sample from $\\mu ( x | y )$ by running noisy gradient ascent: ",
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+    {
+        "type": "equation",
+        "img_path": "images/b70c7754778d3cec0378156f0ede78ee2cdbc338a0f39fc6727feec9cba3daf4.jpg",
+        "text": "$$\nx _ { t + 1 } \\gets x _ { t } + \\eta _ { t } \\nabla _ { x _ { t } } \\log \\mu ( x _ { t } | y ) + \\sqrt { 2 \\eta _ { t } } \\ \\zeta _ { t } , \\quad \\zeta _ { t } \\sim \\mathcal { N } ( 0 , 1 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Prior work [8] has shown that as $t \\to \\infty$ and $\\eta _ { t }  0$ , Langevin dynamics will correctly sample from $\\mu ( x | y )$ . In practice, vanilla Langevin Dynamics are slow to converge. Hence, the work in [79] proposes annealed Langevin Dynamics, where the marginal distribution of $x$ at iteration $t$ is modelled as $\\mu _ { t } = \\mu * \\mathcal { N } ( 0 , \\beta _ { t } ^ { 2 } )$ and the generative model is trained to estimate the score function $f ( x _ { t } ; \\beta _ { t } ) : = \\nabla _ { x _ { t } } \\log ( ( \\mu * \\mathcal { N } ( 0 , \\beta _ { t } ^ { 2 } ) ( x _ { t } ) )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the distribution of $y | x ^ { * }$ is Gaussian in Eqn (2), we obtain $\\begin{array} { r } { \\nabla _ { x _ { t } } \\log \\mu ( y | x _ { t } ) = \\frac { A ^ { H } ( y - A x _ { t } ) } { \\sigma ^ { 2 } } } \\end{array}$ . We find that it is also helpful to anneal this term, and we set it to AH (y−Axt)σ2+γ2 , where γt → 0 is a decreasing sequence. An application of Bayes’ rule gives: $\\begin{array} { r } { \\nabla _ { x _ { t } } \\log \\mu ( x _ { t } | y ) = f ( x _ { t } ; \\beta _ { t } ) + \\frac { A ^ { H } ( y - A x _ { t } ) } { \\sigma ^ { 2 } + \\gamma _ { t } ^ { 2 } } , } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "Putting everything together, our final algorithm is: for $x _ { 0 } \\sim \\mathcal { N } _ { c } ( 0 , I )$ and for all $t = 0 , \\cdots , T - 1$ , ",
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+    {
+        "type": "equation",
+        "img_path": "images/ded09d7532b80b482b08bc4a5c530417020d93a37403e5fb93cdaeaa5cac8967.jpg",
+        "text": "$$\nx _ { t + 1 }  x _ { t } + \\eta _ { t } ( f ( x _ { t } ; \\beta _ { t } ) + \\frac { A ^ { H } ( y - A x _ { t } ) } { \\gamma _ { t } ^ { 2 } + \\sigma ^ { 2 } } ) + \\sqrt { 2 \\eta _ { t } } \\ \\zeta _ { t } , \\quad \\zeta _ { t } \\sim \\mathcal { N } ( 0 ; I ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Note that the parameters $T , \\{ \\beta _ { t } \\} _ { t = 0 } ^ { T - 1 }$ were fixed during training of the generative model, and hence the only hyperparameters during inference are $\\{ \\eta _ { t } \\} _ { t = 0 } ^ { T - 1 } , \\sigma$ and $\\{ \\gamma _ { t } \\} _ { t = 0 } ^ { T - 1 }$ . Scripts in our codebase describe hyperparameter values used in our experiments. ",
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+    {
+        "type": "text",
+        "text": "3 Theoretical Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Background and Notation. We first introduce background and notation required for our theoretical results. $\\| \\cdot \\|$ refers to the $\\ell _ { 2 }$ norm. In this section alone, for simplicity of exposition, we will assume that all matrices and vectors are real valued. ",
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+    {
+        "type": "text",
+        "text": "For two probability distributions $\\mu , \\nu$ on some normed space $\\Omega$ , and for any $q \\geq 1$ , the Wasserstein$q$ [91, 5] and Wasserstein- $\\infty$ [16] distances are defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/35e518ee0c97c76140f0bb6a7b11ec72e19574cb305f4f6c785ea0cbf2b4557a.jpg",
+        "text": "$$\n\\mathcal { W } _ { q } ( \\mu , \\nu ) : = \\operatorname* { i n f } _ { \\gamma \\in \\Pi ( \\mu , \\nu ) } \\left( \\bigoplus _ { ( u , v ) \\sim \\gamma } [ \\| u - v \\| ^ { q } ] \\right) ^ { 1 / q } , \\quad \\mathcal { W } _ { \\infty } ( \\mu , \\nu ) : = \\operatorname* { i n f } _ { \\gamma \\in \\Pi ( \\mu , \\nu ) } \\left( \\gamma _ { ( u , v ) \\in \\Omega ^ { 2 } } ^ { - \\mathrm { e s s } } \\| u - v \\| \\right) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\Pi ( \\mu , \\nu )$ denotes the set of joint distributions whose marginals are $\\mu , \\nu$ . The above definition says that if $\\mathcal { W } _ { \\infty } ( \\mu , \\nu ) \\leq \\varepsilon$ , and $( u , v ) \\sim \\gamma$ , then $\\| u - v \\| \\leq \\varepsilon$ almost surely. ",
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+    {
+        "type": "text",
+        "text": "The $( \\varepsilon , \\delta ) \\cdot$ −approximate covering number [45], is defined as the smallest number of $\\varepsilon$ -radius balls required to cover $1 - \\delta$ mass under a distribution. ",
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+        "type": "text",
+        "text": "Definition 3.1 $( \\varepsilon , \\delta )$ -approximate covering number). Let $\\mu$ be a distribution on $\\mathbb { R } ^ { N }$ . For some parameters $\\varepsilon > 0 , \\delta \\in [ 0 , 1 ]$ , the $( \\varepsilon , \\delta )$ -approximate covering number of $\\mu$ is defined as ",
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+        "type": "equation",
+        "img_path": "images/1dd825bd3421d010a912014d47f3af9dc57b4c5477f8887452c494794b0e3c89.jpg",
+        "text": "$$\n\\mathrm { C o v } _ { \\varepsilon , \\delta } ( \\mu ) : = \\operatorname* { m i n } \\left\\{ k : \\mu \\left[ \\cup _ { i = 1 } ^ { k } B ( x _ { i } , \\varepsilon ) \\right] \\geq 1 - \\delta , x _ { i } \\in \\mathbb { R } ^ { N } \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $B ( x , \\varepsilon )$ is the $\\ell _ { 2 }$ ball of radius $\\varepsilon$ centered at $x$ . ",
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+        "type": "text",
+        "text": "Distributional robustness under Gaussian measurements. First, we consider mismatch between the ground-truth distribution, denoted by $\\mu$ , and the generator distribution, denoted by $\\nu$ . Prior work [45] has shown that if (i) $\\mathcal { W } _ { q } ( \\mu , \\nu ) \\leq \\varepsilon$ for some $q \\geq 1$ and (ii) we are given $M \\geq O ( \\log \\mathrm { C o v } _ { \\varepsilon , \\delta } ( \\mu ) )$ Gaussian measurements, then posterior sampling with respect to $\\nu$ will recover $x ^ { * } \\sim \\mu$ up to an error of $\\varepsilon / \\delta ^ { 1 / q }$ with probability $1 - \\delta$ . Closeness in Wasserstein distance is a reasonable assumption in certain examples, such as when $\\mu$ is the distribution of celebrity faces and $\\nu$ is the distribution of a generator trained on FlickrFaces [52]. However, this assumption is unsatisfactory when we consider distributions of abdominal and brain MR scans, for example, since images of these anatomies look entirely different. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We define the following weaker notion of divergence between distributions. Informally, this new definition tells us that $\\nu$ and $\\mu$ are “close” if they can each be split into components which are close in $\\mathcal { W } _ { \\infty }$ distance, such that the close components contain a sufficiently large fraction under $\\nu$ and $\\mu$ . Formally, this is defined as: ",
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+        "type": "text",
+        "text": "Definition 3.2 $( ( \\delta , \\alpha ) – \\mathcal { W } _ { \\infty }$ divergence). For two probability distributions $\\nu$ and $\\mu ,$ , and parameters $\\delta , \\alpha \\in [ 0 , 1 ]$ , the $( \\delta , \\alpha ) \\ – \\mathcal { W } _ { \\infty }$ divergence is defined as ",
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+        "type": "equation",
+        "img_path": "images/0ddba17a0ff889534246acd5cb1545fe7028d1786ab1b44e8e32398c7b613bea.jpg",
+        "text": "$$\n\\begin{array} { r l } & { ( \\delta , \\alpha ) {  { \\mathcal W } } _ { \\infty } ( \\mu , \\nu ) : = \\operatorname* { i n f } \\{ \\varepsilon \\geq 0 : } \\\\ & { \\exists \\mu ^ { \\prime } , \\mu ^ { \\prime \\prime } , \\nu ^ { \\prime } , \\nu ^ { \\prime \\prime } \\in {  { \\mathcal W } } ( {  { \\mathbb R } } ^ { N } ) \\ s . t . \\ \\mu = ( 1 - \\delta ) \\mu ^ { \\prime } + \\delta \\mu ^ { \\prime \\prime } , \\nu = ( 1 - \\alpha ) \\nu ^ { \\prime } + \\alpha \\nu ^ { \\prime \\prime } , \\mathcal W _ { \\infty } ( \\mu ^ { \\prime } , \\nu ^ { \\prime } ) = \\varepsilon . \\} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lemma B.1 highlights that this is a strict generalization of Wasserstein distances, in the sense that closeness in Wasserstein distance implies closeness in this new divergence. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Since the $( \\delta , \\alpha ) \\ – \\mathcal { W } _ { \\infty }$ divergence is a generalization of Wasserstein distances, it is not clear that the main Theorem in [45] holds for distributions that are close in this new divergence. The following result shows a rather surprising fact: if $( \\delta , \\alpha ) \\ – \\mathcal { W } _ { \\infty } ( \\mu , \\nu ) \\leq \\varepsilon$ then posterior sampling with $M =$ $\\begin{array} { r } { O \\left( \\log \\left( \\frac { 1 } { 1 - \\alpha } \\right) + \\log \\mathrm { C o v } _ { \\varepsilon , \\delta } ( \\mu ) \\right) } \\end{array}$ measurements will still succeed with probability $\\ge 1 - O ( \\delta )$ . ",
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+        "type": "text",
+        "text": "Theorem 3.3. Let $\\delta , \\alpha \\in [ 0 , 1 ]$ , and $\\varepsilon > 0$ be parameters. Let $\\mu , \\nu$ be arbitrary distributions over $\\mathbb { R } ^ { N }$ satisfying $( \\delta , \\alpha ) \\ – \\mathcal { W } _ { \\infty } ( \\mu , \\nu ) \\leq \\varepsilon$ . Let $x ^ { * } \\sim \\mu$ and suppose $y = A x ^ { * } + w$ , where $A \\in \\mathbb { R } ^ { M \\times N }$ and $w \\in \\dot { \\mathbb { R } } ^ { M }$ are i.i.d. Gaussian normalized such that $A _ { i j } \\sim \\mathcal { N } ( 0 , 1 / M )$ and $w _ { i } \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } / M )$ , with $\\sigma \\gtrsim \\varepsilon$ . Given $y$ and the fixed matrix $A$ , let $\\widehat { x }$ be the output of posterior sampling with respect to $\\nu$ . ",
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+    {
+        "type": "text",
+        "text": "Then for $\\begin{array} { r } { M \\geq O \\left( \\log \\left( \\frac { 1 } { 1 - \\alpha } \\right) + \\operatorname* { m i n } ( \\log \\mathrm { C o v } _ { \\sigma , \\delta } ( \\mu ) , \\log \\mathrm { C o v } _ { \\sigma , \\delta } ( \\nu ) ) \\right) } \\end{array}$ , there exists a universal constant $c > 0$ such that with probability at least $1 - e ^ { - \\Omega ( M ) }$ over $A , w$ , ",
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+        "img_path": "images/58a5abdc99094d4283011d361873f698f4de0998ffe998e2a1c844ad72262e4b.jpg",
+        "text": "$$\n\\operatorname* { P r } _ { x ^ { * } \\sim \\mu , \\widehat { x } \\sim \\nu ( \\cdot | y ) } \\left[ \\| x ^ { * } - \\widehat { x } \\| \\geq c ( \\varepsilon + \\sigma ) \\right] \\leq \\delta + e ^ { - \\Omega ( M ) } .\n$$",
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+        "type": "image",
+        "img_path": "images/90078d3b474935e152b912aec68c52df9e50dcb9e83b5141f2339a17391f80a8.jpg",
+        "image_caption": [
+            "Figure 3: Comparative reconstructions of a 2D abdominal scan with uniform random under-sampling in the horizontal direction at $R = 4$ . None of the methods were trained to reconstruct abdomen MRI. Our method uses a score-based generative model trained on brain images (as explained) and obtains good reconstructions. The red arrows indicate missing details or artifacts in the kidney structure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For our running example of $\\nu$ being a generator trained on brain scans, and $\\mu$ the distribution of abdominal scans, we can set $\\nu ^ { \\prime }$ to be the distribution of our generator restricted to abdominal scans, and we can let $\\mu ^ { \\prime }$ be the distribution restricted to “inliers” in $\\mu$ . This shows that even if our generator places an exponentially small probability mass(i.e., $1 - \\alpha \\ll 1 ,$ ) on the set of abdominal scans, we can still recover abdominal scans with a polynomial additive increase in the number of measurements (i.e., $\\log ( 1 / ( 1 - \\alpha ) )$ ). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Near-optimality under arbitrary measurement processes. The previous result required Gaussian matrices to handle the distribution shift. Our next result shows that for an arbitrary measurement process, and assuming that there is no distribution shift between the generator and the ground truth distribution, posterior sampling is almost the best algorithm for this fixed measurement process. This result also shows that posterior sampling is good with respect to any metric. ",
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+    {
+        "type": "text",
+        "text": "Theorem 3.4. Let $d ( \\cdot , \\cdot )$ be an arbitrary metric over $\\mathbb { R } ^ { N } \\times \\mathbb { R } ^ { N }$ . Let $x ^ { * } \\sim \\mu$ and let $y = \\mathcal { A } ( x ^ { * } )$ be measurements generated from $x ^ { * }$ for some arbitrary forward operator $\\mathcal { A } : \\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { M }$ . Then if there exists an algorithm that uses y as inputs and outputs $x ^ { \\prime }$ such that ",
+        "bbox": [
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+    {
+        "type": "equation",
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+        "text": "$$\nd ( x ^ { * } , x ^ { \\prime } ) \\leq \\varepsilon { \\mathit { w i t h } } p r o b a b i l i t y \\ 1 - \\delta ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "then posterior sampling $\\widehat { x } \\sim \\mu ( \\cdot | y )$ will satisfy ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "equation",
+        "img_path": "images/2dfd04d7ca1ab69153f765dd53831d0231d84a0cfe1c58235156a309c4a8b4fd.jpg",
+        "text": "$$\nd ( x ^ { * } , \\widehat { x } ) \\leq 2 \\varepsilon w i t h p r o b a b i l i t y \\ \\geq 1 - 2 \\delta .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Remark on combining these results. Our theoretical results above show that posterior sampling is (1) highly robust to distribution shift under Gaussian measurements, and (2) accurate with arbitrary measurements without distribution shift. A natural hope would be to combine these two results and show that it is robust to distribution shift under Fourier measurements. Unfortunately, this is not true for general distributions: for example, if $\\mu$ and $\\nu$ are both random distributions over Fourier-sparse signals, then Fourier measurements will usually give zero information about the signal, so cannot convince the sampler to sample near $\\mu$ rather than $\\nu$ . ",
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+    {
+        "type": "text",
+        "text": "4 Experimental Results ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We perform retrospective under-sampling in all experiments, i.e., given fully-sampled $\\mathbf { k }$ -space measurements from the NYU fastMRI [56, 94] and Stanford MRI [1] datasets, we apply sampling masks and evaluate the performance of all considered algorithms on the reconstructed data. Depending on scan parameters (e.g., 3D scans for the Stanford knee data in Appendix F), we appropriately slice and sample the data in the proper dimension so as to not commit any inverse crime [31, 77]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first highlight that an advantage of the proposed approach is the invariance to the sampling scheme during training. In contrast, this is a design choice that must be made for supervised end-to-end methods, which here were trained on equispaced, vertical sampling masks, following the fastMRI 2020 challenge guidelines [94, 66]. As our results show, this affords us a significant degree of robustness across a wide distribution of sampling masks during inference. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We train a score-based model, NCSNv2 [80], on a small subset of scans from the NYU fastMRI brain dataset. Specifically, we train using T2-weighted images at a field strength of 3 Tesla for a total of 14,539 2D training slices. We calculate the MVUE from the fully sampled data and use the ESPIRiT algorithm [87, 43] applied to the fully-sampled central portion of $\\mathbf { k }$ -space to estimate the sensitivity maps. The backbone network for our model is a RefineNet [59]. Since the generator’s output is expected to be complex-valued, we treat the real and imaginary parts as separate image channels. Details about the architectures are given in Appendix G. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use an $\\ell _ { 1 }$ -Wavelet regularized reconstruction algorithm [62] as a parallel imaging and compressed sensing baseline. This aims to solve the optimization problem given in (2) with $\\mathbf { \\bar { Q } } ( x ) = | | \\mathbf { \\bar { W } } x | | _ { 1 }$ where $W$ is a 2D Wavelet transform. We use the publicly available implementation from the BART toolbox [88, 86] and optimize the regularization hyper-parameter using the same subset of samples from the brain dataset that was used to train our method. We find that $\\lambda = 0 . 0 1$ performs the best on the training data and use this value for all experiments. We consider three different deep learning baselines: MoDL [3], E2E-VarNet [82], and the ConvDecoder architecture [19]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We train the MoDL and E2E-VarNet baselines from scratch on the same training dataset as our method, at acceleration factors $R = \\{ 3 , 6 \\}$ and equispaced under-sampling, with a supervised SSIM loss on the magnitude MVUE image, for 40 and 15 epochs, respectively, using a batch size of 1. For the ConvDecoder baseline, we use the architecture for brain data in [19] that outputs a complex image estimate and optimize the number of fitting iterations on a subset of samples from the training data. We find that 10000 iterations are sufficient to reach a stable average performance at $R = 3$ . Put together, all of our baselines are tailored to estimate the complex image $x$ , thus all comparisons are fair. We evaluate reconstruction performance using the complex MVUE of the fully sampled data as a reference image and measure the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) [92] between the absolute values of the reconstruction and ground-truth MVUE images. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 In-Distribution Performance ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this experiment, we test all models using the same forward model that matches the training conditions for the baselines: vertical, equispaced sampling patterns. Examples of various sampling patterns are shown in Appendix C. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 1 (top three rows) shows qualitative results and Figures 2a & 5a respectively show PSNR & SSIM values, for the case where there is no mismatch between the training and inference sampling patterns. As the baselines were trained to maximize SSIM at $R = 3 ~ \\& ~ 6$ , we see that they achieve better SSIM scores than us at these accelerations, although there is clear aliasing in the baselines at $R = 6$ . We achieve better PSNR values at these accelerations, which supports the claim that our method does not overfit to a particular metric (Theorem 3.4). This also highlights the importance of qualitative evaluations in medical image reconstruction and the limitations of existing image quality metrics [65]. From the third row of Figure 1, and Figures 2a & 5a, we notice that our method surpasses baselines at higher accelerations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We find that $\\ell _ { 1 }$ -Wavelet suffers both qualitatively and quantitatively at high acceleration factors, while the ConvDecoder is also a competitive architecture, but incurs a large computational cost. When benchmarked on an NVIDIA RTX 2080Ti GPU, our method takes 16 minutes and $0 . 9 5 \\mathrm { G B }$ of memory to reconstruct a high-resolution brain scan, whereas the ConvDecoder takes longer than 80 minutes and $6 . 6 \\mathrm { G B }$ of memory. While our method is limited by the inference time and is not in the range of end-to-end models (where reconstruction takes at most on the order of seconds and $3 . 5 \\mathrm { G B }$ of memory), multiple scans can be reconstructed in parallel due to the reduced memory footprint. ",
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+    {
+        "type": "text",
+        "text": "4.2 Out-of-Distribution Performance ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Test-time sampling pattern shifts. Here we consider shifts in the forward sampling operator at test-time, while still evaluating on the same anatomy as the training conditions. We measure robustness by evaluating the average incurred performance loss when the sampling pattern changes. Recall that our proposed approach does not use any explicit information about the sampling pattern $P$ during training, hence we anticipate the highest degree of robustness. ",
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+        "img_path": "images/b018ec33241411896dc3fc3986cc59d405d15f8a24b43f8fcf25f05147bb0981.jpg",
+        "image_caption": [
+            "Figure 4: Our method successfully recovers fine details and can provide an estimate of the reconstruction error. The left column shows a knee from the fastMRI dataset, along with an annotated meniscus tear (indicated by red arrow in zoomed inset). Given measurements at an acceleration factor of $R = 4$ , we obtain 48 independent reconstructions via posterior sampling. The second column shows the pixel-wise average of reconstructions, the third column shows the pixel-wise standard deviation, and the fourth column shows the magnitude of the error between the ground truth and the mean reconstruction. Note that our generative prior has never seen such pathology, as it was trained on T2-weighted brain scans. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Figure 1 (fourth row) shows qualitative reconstructions when the measurements are obtained from an equispaced, horizontal sampling mask, with an acceleration factor $R = 3$ . It can be observed that the reconstructions output by E2E-VarNet show aliasing artifacts. Based on the statistical results in Figure 2b & 5b, our method retains its performance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, this experiment reveals that MoDL is more robust to this type of mask shift when compared to E2E-VarNet, even though it uses a smaller network. This is explained by the fact that E2E-VarNet does not use external sensitivity map estimates, but uses a deep neural network for endto-end map estimation. While this improves performance on in-distribution samples, the performance drop is strong evidence that accurate sensitivity map estimation is vital for robust generalization, and both our proposed approach and MoDL benefit from the external ESPIRiT algorithm, which is compatible with different sampling patterns. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We do note that retrospectively flipping the horizontal and vertical sampling direction is not necessarily representative of prospective sampling in the horizontal direction due to the discrete nature of the phase encoding direction in MRI, and this may contribute to the higher scores compared to the vertical mask experiments. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Test-time anatomy shifts. We now consider the more difficult problem of reconstructing different anatomies than the ones seen during. This was previously investigated in [19], which concluded that all methods suffer a drastic shift due to the various changes in scan parameters between body parts. In contrast to prior work, our main finding is that the proposed score-based model retains a significant degree of robustness under these shifts, and outputs excellent qualitative reconstructions. In some cases, some end-to-end methods retain robustness as well. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figures 2c & 5c show PSNR and SSIM scores obtained on reconstructed abdominal scans obtained from [1] at different acceleration factors. This represents both an anatomy and sampling pattern shift, and it can be seen that our method, MoDL, and the $\\ell _ { 1 }$ -Wavelet algorithm retain their competitive advantage, while the ConvDecoder and E2E-VarNet suffer severe performance losses. Figure 3 further shows a qualitative comparison of a reconstructed abdominal scan at $R = 4$ , with highlighted artifacts. Appendix E shows another abdomen scan. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, Figures 2d & 5d show PSNR and SSIM scores obtained on fastMRI knee reconstructions, while Figure 1 (bottom row) shows the accompanying qualitative plots. This anatomy is challenging especially because of the poor signal-to-noise ratio conditions, which can be seen even in the groundtruth image. It can be noticed that this is the most severe shift for all methods, but our approach still shows the best performance at $R = 2 , 4$ and a significantly lower variance. Appendix D shows more examples of knee reconstructions with and without fat suppression, and Figure 20 shows metrics on fat suppressed knees. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 Uncertainty Estimation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our method can also provide uncertainty estimates for each reconstructed pixel by running multiple truc, for n samplers. For a given observation sufficiently large. Now, using the $y$ , we can obtain independeonditional mean estimate $\\widehat { x } _ { 1 } , \\cdots , \\widehat { x } _ { K } \\sim$ $\\mu ( \\cdot | y )$ $K$ $\\textstyle { \\bar { x } } = \\sum _ { i = 1 } ^ { K } { \\widehat { x } } _ { i } / K$ compute the pixel-wise standard deviation $\\sqrt { \\textstyle \\sum _ { i = 1 } ^ { K } | \\widehat { x } _ { i } - \\bar { x } | ^ { 2 } / K }$ , and this gives an estimate of the error in each pixel. As shown in $\\mathrm { F i g ~ 4 }$ , the pixel-wise standard deviation is a good estimate of the ground truth error $| x ^ { * } - { \\bar { x } } |$ . Additionally, notice that the reconstructions are able to recover fine details such as the annotated meniscus tear3 in Fig 4 and predict low uncertainty for these features. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 17 in Appendix D shows another example of an annotated meniscus tear. Figures 18 and 19 show comparisons with baselines on the same examples. ",
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+        "text": "4.4 Radiologist Study ",
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+        "text": "We have conducted a preliminary blind assessment of overall image quality with two board-certified radiologists and one faculty member who uses neuroimaging for their research. These experts were not involved in our research. We have found that our algorithm was ranked best for knee scans, and tied with the baselines for abdominal and brain scans, supporting our robustness claims in the paper. For more details, please see Appendix H. ",
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+        "text": "5 Limitations ",
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+        "text": "We reported PSNR and SSIM values as they are correlated with radiologist evaluation upto an extent, and our preliminary radiologist study in Section 4.4 suggests the feasibility of clinical adoption. These metrics do not capture the needs of real-world radiologists, and a more detailed study is required before the proposed techniques can be clinically adopted. ",
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+        "text": "Though promising, our initial results were still limited to fast spin-echo imaging only and all data were retrospectively under-sampled. Further study is required to demonstrate prospective performance in a larger body of heterogeneous MRI data. Our method also currently requires a high compute cost at inference time, as well as the need for a pre-trained generative model. Clinical use requires fast reconstruction in addition to fast scanning. Future work should investigate whether score-based models can be trained without a fully-sampled training set as well as investigate approaches to reducing computation time. ",
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+        "text": "Finally, there are potential issues related to discrimination. Specifically, it is possible that the quality of the reconstructed images varies across protected attributes, such as gender or race [57]. ",
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+        "text": "6 Conclusions ",
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+    {
+        "type": "text",
+        "text": "This paper reports the first successful application of the CSGM framework for robust multi-coil MR image reconstruction under realistic sampling conditions, and provides theoretical evidence for the robustness of posterior sampling. Our score-based model was trained on a small subset of brain MRI scans without any explicit information about the sampling scheme. This shows state-of-the-art performance under severe distributional shifts, making our model applicable in a wide range of clinical settings. ",
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+    {
+        "type": "text",
+        "text": "Our method shows a considerable degree of generalization to out-of-distribution samples such as abdomen and knee MRI, even when trained exclusively on brain MRI. Notably, these scans were acquired using different MRI vendors with different pulse sequence parameters and at different institutions. We postulate that adding a small set of diverse training samples to our generative model could further improve robustness, and we hypothesize that these samples may not necessarily be restricted to MR images. ",
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+        "type": "text",
+        "text": "The results presented in this work represent an important step to applying deep learning models in the clinic, as there is a natural variation in sampling, image orientation, receive coils, scanner hardware, and anatomy in clinical practice. ",
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+        "type": "text",
+        "text": "7 Acknowledgements ",
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+    {
+        "type": "text",
+        "text": "Ajil Jalal, Giannis Daras and Alex Dimakis have been supported by NSF Grants CCF 1763702, 1934932, AF 1901281, 2008710, 2019844, the NSF IFML 2019844 award as well as research gifts by Western Digital, Interdigital, WNCG and MLL, computing resources from TACC and the Archie Straiton Fellowship. ",
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+    {
+        "type": "text",
+        "text": "Eric Price has been supported by NSF Award CCF-1751040 (CAREER), NSF Award CCF-2008868, and NSF IFML 2019844. ",
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+    {
+        "type": "text",
+        "text": "Marius Arvinte and Jon Tamir have been supported by NSF IFML 2019844 award, ONR grant N00014-19-1-2590, NIH Grant U24EB029240, and an AWS Machine Learning Research Award. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We thank the anonymous NeurIPS reviewers for their helpful and considerate feedback. ",
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+    {
+        "type": "text",
+        "text": "Finally, we would like to thank the experts who graciously helped with our image assessment study. ",
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+    {
+        "type": "text",
+        "text": "8 Funding Transparency Statements ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "8.1 Funding (financial activities supporting the submitted work): ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Funding in direct support of this work: NSF Grants CCF 1763702, 1934932, AF 1901281, 2008710, 2019844, 1751040, 2008868, the NSF IFML 2019844 award, ONR grant N00014-19-1-2590, NIH Grant U24EB029240, and an AWS Machine Learning Research Award, as well as research gifts by Western Digital, Interdigital, WNCG and MLL, computing resources from TACC and the Archie Straiton Fellowship. ",
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+        "type": "text",
+        "text": "8.2 Competing Interests (financial activities outside the submitted work): ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Additional revenues related to this work: Internship at Intel and Google. ",
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+    {
+        "type": "text",
+        "text": "References ",
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Synthesis Lectures on Image, Video, and Multimedia Processing, 2(1):1–156, 2006.   \n[93] Bihan Wen, Saiprasad Ravishankar, Luke Pfister, and Yoram Bresler. Transform learning for magnetic resonance image reconstruction: From model-based learning to building neural networks. IEEE Signal Processing Magazine, 37(1):41–53, 2020.   \n[94] Jure Zbontar, Florian Knoll, Anuroop Sriram, Tullie Murrell, Zhengnan Huang, Matthew J. Muckley, Aaron Defazio, Ruben Stern, Patricia Johnson, Mary Bruno, Marc Parente, Krzysztof J. Geras, Joe Katsnelson, Hersh Chandarana, Zizhao Zhang, Michal Drozdzal, Adriana Romero, Michael Rabbat, Pascal Vincent, Nafissa Yakubova, James Pinkerton, Duo Wang, Erich Owens, C. Lawrence Zitnick, Michael P. Recht, Daniel K. Sodickson, and Yvonne W. Lui. fastMRI: An open dataset and benchmarks for accelerated MRI. 2018. ",
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