Add files using upload-large-folder tool

parse/train/HJgZrsC5t7/HJgZrsC5t7_content_list.json

@@ -0,0 +1,1595 @@
+[
+    {
+        "type": "text",
+        "text": "IMPROVING ON-POLICY LEARNING WITH STATISTICAL REWARD ACCUMULATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            671,
+            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": "Deep reinforcement learning has obtained significant breakthroughs in recent years. Most methods in deep-RL achieve good results via the maximization of the reward signal provided by the environment, typically in the form of discounted cumulative returns. Such reward signals represent the immediate feedback of a particular action performed by an agent. However, tasks with sparse reward signals are still challenging to on-policy methods. In this paper, we introduce an effective characterization of past reward statistics (which can be seen as long-term feedback signals) to supplement this immediate reward feedback. In particular, value functions are learned with multi-critics supervision, enabling complex value functions to be more easily approximated in on-policy learning, even when the reward signals are sparse. We also introduce a novel exploration mechanism called “hot-wiring” that can give a boost to seemingly trapped agents. We demonstrate the effectiveness of our advantage actor multi-critic (A2MC) method across the discrete domains in Atari games as well as continuous domains in the MuJoCo environments. A video demo is provided at https://youtu.be/zBmpf3Yz8tc and source codes will be made available upon paper acceptance. ",
+        "bbox": [
+            233,
+            266,
+            766,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            334,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Advances in deep learning have mobilized the research community to adopt deep reinforcement learning (RL) agents for challenging control problems, typically in complex environments with raw sensory state-spaces. Breakthroughs by Mnih et al. (2015) show RL-agents can reach abovehuman performance in Atari 2600 games, and AlphaGo Zero Silver et al. (2017) becomes the world champions on the game of Go. Still, training RL agents is non-trivial. Off-policy methods typically require days of training to obtain competitive performance, while on-policy methods could be trapped in local minima. Recent techniques featuring on-policy learning Mnih et al. (2016); Schulman et al. (2017); Wu et al. (2017) have shown promising results in stabilizing the learning processes, enabling an agent to solve a variety of tasks in much less time. In particular, the state-of-the-art on-policy ACKTR agent by Wu et al. (2017) shows improved sample efficiency with the help of Kronecker-factored (K-Fac) approximate curvature for natural gradient updates, resulting in stable and effective model updates towards a more promising direction. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            714
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, tasks with sparse rewards remain challenging to on-policy methods. An agent could take massive amount of exploration before reaching non-zero rewards; and as the agent learns on-policy, the sparseness of reward feedback (receiving all-zero rewards from most actions performed by the agent) could be malicious and render an agent to falsely predict all states in an environment towards a value of zero. As there does not exist a universal criterion for measuring “task sparseness”, we show an ad-hoc metric in Figure 1 in an attempt to provide intuition. For example, we observe that the ACKTR agent is unable to receive sufficient non-zero immediate rewards that can provide instructive agent updates in Atari games “Freeway” and “Enduro”, resulting in failures when solving these two games. Moreover, if ACKTR gets drawn to and trapped in unfavorable states (as in games like Boxing and WizardOfWor), it could again take long hours of exploration before the agent can get out of the local minima. Such evidence shows that on-policy agent could indeed suffer from the insufficiencies of guidance provided by the exclusive immediate reward signals from the environment. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduce an effective auxiliary reward signal in tasks with sparse rewards to remedy the deficiencies of learning purely from standard immediate reward feedbacks. As on-policy agents may take many explorations before reaching non-zero immediate rewards, we argue that we can leverage past reward statistics to provide more instructive feedbacks to agents in the same environment. To this end, we propose to characterize the past reward statistics in order to gauge the “long-term” performance of an agent (detailed in Section 4). After performing an action, an agent will receive a long-term reward signal describing its past performance upon this step, as well as the conventional immediate reward from the environment. To effectively characterize the long-term performance of the agent, we take into considerations both the crude amount of rewards and the volatility of rewards received in the past, where highly volatile distributions of long-term rewards are explicitly penalized. This enables complex value functions to be more easily approximated in multi-critics supervision. We find in practice that by explicitly penalizing highly volatile long-term rewards while maximizing the expectation of short-term rewards, the learning process and the overall performance are improved regarding both sample efficiency and final rewards. We further propose a “hot-wiring” exploration mechanism that can boost seemingly trapped agent in the earlier stage of learning. By leveraging the characterization of long/short-term reward statistics, our proposed advantage actor multi-critic model (A2MC) shows significantly improved performance on the Atari 2600 games and the MuJoCo tasks as compared to the state-of-the-art on-policy methods. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/cd753a39ce5a48461924628e1e3afe7243006032133b48f3e336125002681ed3.jpg",
+        "image_caption": [
+            "Figure 1: Performance of A2MC on Atari games trained with 15 million timesteps. Our method has a winning rate of $5 5 . 3 \\%$ among all the Atari games tested, as compared to the ACKTR. Our A2MC learns quickly in some of the hardest games for on-policy methods, such as “Boxing”, “Enduro”, “Freeway”, “Robotank” and “WizardOfWor”. The sparseness of a game is defined as the sparseness of average rewards $\\mathbf { x }$ obtained by ACKTR within the first $n = 1 0 ^ { 6 }$ timesteps by $\\begin{array} { r } { \\varphi ( \\mathbf { x } ) ^ { \\mathbf { \\tilde { \\alpha } } } = \\left( \\sqrt { n } - \\frac { \\| \\mathbf { x } \\| _ { 1 } } { \\| \\mathbf { x } \\| _ { 2 } } \\right) \\tilde { \\mathbf { \\alpha } } ( \\sqrt { n } - 1 ) } \\end{array}$ . A higher value of sparseness indicates sparser rewards. A relative performance margin (in terms of final reward) larger than $1 0 \\%$ is deemed as winning / losing. The shaded region denotes the standard deviation over 2 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            108,
+            805,
+            332
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            491,
+            825,
+            714
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            737,
+            344,
+            753
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Reward shaping and pseudo-rewards: To tackle the challenge in tasks with rarely observed rewards, pseudo-rewards maximization is adopted in earlier works Konidaris & Barto (2009); Silver & Ciosek (2012). Auxiliary vision tasks (e.g., learning pixel changes or network features) are adopted in the off-policy UNREAL agent Jaderberg et al. (2016) in order to facilitate learning better feature representations, particularly for sparse reward environments. Another direction of effort involves directly engineering a better reward function or shaping the reward signal. Andrychowicz et al. (2017) enhances off-policy learning by re-producing informative reward in hindsight for sequences of actions that do not lead to success previously. The HRA approach Van Seijen et al. (2017) exploits domain knowledge to define a set of environment-specific rewards based on reward categories. And the winning approach that learns playing “Doom” Lample & Chaplot (2017) shows promising success in the FPS game that carefully crafting the task rewards would indeed be beneficial. In contrast to heuristically defining vision-related auxiliary tasks, our proposed A2MC agent learns from the characterization of intrinsic past reward statistics obtainable from any environment; and different from the hybrid architecture pertaining to Ms. Pacman only and the reward shaping settings tailored specifically to ”Doom”, our proposed reward characterization mechanism is generic and our A2MC generalizes well to a variety of tasks without the need to engineer a decomposition of problemspecific environment rewards. Moreover, the capability of the proposed method to further boost reward shaping is evidenced in our case study on playing Doom (see Appendix F). ",
+        "bbox": [
+            174,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-agents: The multi-agent approaches Lanctot et al. (2017); Lowe et al. (2017); Jin et al. (2018) present another promising direction for learning. They propose to train multiple agents in parallel when solving a task, and to combine multiple action-value functions with a centralized action-value function. The multi-critics supervision in our proposed A2MC model can be seen as a form of joint-task or multi-task learning Teh et al. (2017) for both long-term and short-term rewards. ",
+        "bbox": [
+            174,
+            208,
+            825,
+            279
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On-policy v.s. Off-policy: Our empirical results based on learning the characterization of long/shortterm reward statistics also echo the effectiveness of a recently proposed off-policy reinforcement learning framework Bellemare et al. (2017) that features a distributional variant of Q-learning, wherein the value functions are learned to match the distribution of standard immediate returns. Also, Wang et al. (2016) shows that applying experience replay to on-policy methods can further enhance learning stability. Schulman et al. (2016) proposes a variant of advantage function using eligibility traces that provides both low-variance and low-bias gradient estimates. These works are orthogonal to our approach can potentially be combined with the proposed characterization of past reward statistics to further enhance learning performance. While our extensive experiments (see also Appendix E and Appendix F) show promising results of our approach in both on- and off-policy frameworks, we focus on “on-policy” methods (i.e., those that do not involve off-policy trajectories or experience replay) as in $\\mathrm { W u }$ et al. (2017) in the main text in order to systematically evaluate the potential of our proposed reward mechanism within the scope of this work. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            465
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            488,
+            323,
+            503
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider the standard reinforcement learning setting where an agent interacts with an environment over a number of discrete time step. At each time step $t$ , the agent receives an environment state $s _ { t }$ , then executes an action $a _ { t }$ based on policy $\\pi _ { t }$ . The environment produces reward $r _ { t }$ and next state $s _ { t + 1 }$ , according to which the agent gets feedback of its immediate action and will decide its next action $a _ { t + 1 }$ . The process $< { \\bf S } , { \\bf A } , { \\bf R } , { \\bf S } >$ , typically considered as a Markov Decision Process, continues until a terminal state $s _ { T }$ upon which the environment resets itself and produces a new episode. Under conventional settings, the return is calculated as the discounted summation of rewards $r _ { t }$ accumulated from time step $t$ onwards $\\begin{array} { r } { R _ { t } = \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } r _ { t + k } } \\end{array}$ . The goal of the agent is to maximize the expected return from each state $s _ { t }$ while following policy $\\pi$ . Each policy has a corresponding action-value function defined as $Q ^ { \\pi } ( s , a ) = \\mathbb { E } [ R _ { t } | s _ { t } = s , a _ { t } = a ; \\pi ]$ . Similarly, each state $s \\in S$ under policy $\\pi$ has a value function defined as: $V ^ { \\pi } ( s ) = \\mathbb { E } [ R _ { t } | s _ { t } = s ]$ . In value-based approaches (e.g., Q-learning Mnih et al. (2015)), function approximator $Q ( s , a ; \\theta )$ can be used to approximate the optimal action value function $Q ^ { * } ( s , a )$ . This is conventionally learned by iteratively minimizing the below loss function: ",
+        "bbox": [
+            173,
+            520,
+            826,
+            715
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdd934fbecb0d5427ee4d48f6748c5378952e4782dc12219900f950e1a877213.jpg",
+        "text": "$$\nL ( \\theta ) = \\mathbb { E } [ ( y _ { t } ^ { t a r g e t } - Q ( s _ { t } , a _ { t } ; \\theta ) ) ^ { 2 } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            715,
+            619,
+            734
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ytargett = rt + γ maxa0 Q(st+1, a0; θ) and st+1 is the next state following state st. ",
+        "bbox": [
+            171,
+            741,
+            758,
+            757
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In policy-based approaches (e.g., policy gradient methods), the optimal policy $\\pi ^ { * } ( a | s )$ is approximated using the approximator $\\pi ( a | s ; \\theta )$ . The policy approximator is then learned by gradient ascent on $\\nabla _ { \\boldsymbol { \\theta } } \\mathbb { E } [ h _ { t } ] \\approx \\dot { \\nabla _ { \\boldsymbol { \\theta } } } \\log \\pi ( a _ { t } | s _ { t } ; \\boldsymbol { \\theta } ) \\dot { R } _ { t }$ . The REINFORCE method Williams (1992) further incorporates a baseline $b ( s _ { t } )$ to reduce the variance of the gradient estimator: $\\nabla _ { \\theta } \\mathbb { E } [ R _ { t } ] _ { R E I N F O R C E } \\approx$ $\\nabla _ { \\theta } \\log \\pi ( a _ { t } | s _ { t } ; \\theta ) \\dot { ( R _ { t } - b ( s _ { t } ) ) }$ ",
+        "bbox": [
+            174,
+            762,
+            825,
+            835
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In actor-critic based approaches, the variance reduction further evolves into the advantage function $A ( s _ { t } , a _ { t } ) = Q ( s _ { t } , a _ { t } ) - V ( s _ { t } )$ in Mnih et al. (2016), where the action value $Q ^ { \\pi } ( s _ { t } , a _ { t } )$ is approximated by $R _ { t }$ and $b ( s _ { t } )$ is replaced by $V ^ { \\pi } ( s _ { t } )$ , deriving the advantage actor-critic architecture where actor-head $\\pi ( \\cdot | s )$ and the critic-head $V ( s )$ share low-level features. The gradient update rule w.r.t. the action-head is $\\nabla _ { \\theta } \\log \\pi ( a _ { t } | s _ { t } ; \\theta ) ( R _ { t } - V ( s _ { t } ; \\theta ) )$ . The gradient update w.r.t. the critic-head is: $\\nabla _ { \\boldsymbol { \\theta } } ( R _ { t } - V ( s _ { t } ; \\boldsymbol { \\theta } ) ) ^ { 2 }$ , where $R _ { t } = r _ { t } + \\gamma V ( s _ { t + 1 } )$ . ",
+        "bbox": [
+            173,
+            840,
+            825,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/691a77d1436be9adc6c816257080ddb1982e633dec02a7c34a27ed2085021986.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the proposed variability-weighted reward (VWR). The first row shows the raw reward sequence (blue) while the second row presents the post-processed sequence $\\vec { \\mathcal { R } }$ (green) and the zero-variability reference $\\vec { \\mathcal { R } } ^ { z e r o }$ (orange), and $\\mathcal { R } _ { H }$ is calculated as a reflection of how high the immediate reward is. The third row shows the volatility statistics of $\\delta _ { \\mathcal { R } }$ , representing how varied past rewards were. We curated 3 hypothetical reward sequences – (a) highly varied sequence with low immediate reward, resulting in the lowest VWR; (b) highly varied sequence with high immediate reward, leading to a relatively high VWR; (c) stable sequence with high immediate reward, achieving the best VWR. More examples can be found in the Appendix A. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            104,
+            810,
+            320
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 CHARACTERIZATION OF PAST REWARD STATISTICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            633,
+            483
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The conventional reward $r _ { t }$ received from the environment at time step $t$ after an action $a _ { t }$ is performed represents the immediate reward regarding this particular action. This “immediacy” could be interpreted as a short-term horizon of how the agent is doing, i.e., evaluating the agent via judging its actions by immediate rewards. We argue that the deficiencies of learning solely from immediate rewards mainly come from this limitation that the agent is learning from one single type of exclusive short-term feedback. ",
+        "bbox": [
+            173,
+            498,
+            825,
+            582
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As the goal of providing reward feedback to an agent is to inform the agent of its performance, we seek to find an auxiliary performance metric that can measure whether the agent is performing consistently well. Inspired by the formulation of Sharpe Ratio $\\begin{array} { r } { ( \\mathbb { E } [ r ] \\times \\frac { 1 } { \\sigma _ { r } } ) } \\end{array}$ in evaluating the long-term performance of porfolio strategies where the return $\\mathbb { E } [ r ]$ is inversely weighted by the risk $\\sigma _ { r }$ , an effective characterization of historical reward statistics should take into account at least two factors, namely 1) how high the immediate reward is and 2) how varied past rewards were, bringing the desired notion of “risk-adjusted return” as in Sharpe (1994). ",
+        "bbox": [
+            173,
+            588,
+            826,
+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 VARIABILITY-WEIGHTED REWARD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            705,
+            457,
+            719
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To this end, we follow insights behind Dowd (2000); Sharpe (1994) and define a variability-weighted characterization of past rewards. This is illustrated in Figure 2. We consider a historical sequence of $T$ rewards upon timestep $t$ (looking backward $T - 1$ timesteps): $\\vec { \\mathbf { r } } = \\left[ r _ { t - ( T - 1 ) } . . . , r _ { t - 2 } , r _ { t - 1 } , r _ { t } \\right]$ . In order to evaluate how high and varied the reward sequence is, a few steps of pre-processing $\\mathcal { G }$ is applied, denoted as $\\vec { \\mathcal { R } } = \\mathcal { G } ( \\vec { \\bf r } )$ . Specifically, we first derive the reward change at each timestep (similar to the “differential return” concept in Sharpe (1994)) with $d _ { n } = r _ { n } - r _ { n - 1 }$ : ",
+        "bbox": [
+            173,
+            731,
+            825,
+            818
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d1a41e8d46f16485988e1e846429a2a98dca9d0e35f49628981c04a400babf5.jpg",
+        "text": "$$\n\\vec { \\mathbf { d } } = [ d _ { t - ( T - 1 ) } , d _ { t - ( T - 2 ) } , \\dots , d _ { t } ] = [ r _ { t - ( T - 1 ) } , r _ { t - ( T - 2 ) } - r _ { t - ( T - 1 ) } , \\dots , r _ { t } - r _ { t - 1 } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            823,
+            781,
+            843
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then we re-order the sequence by flipping 1 with $f _ { n } = d _ { t + 1 - n }$ : ",
+        "bbox": [
+            173,
+            848,
+            593,
+            864
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1ca19f978de9502729114ada4eb16aebd622ec6d1b4c6e739e10be181f0620f.jpg",
+        "text": "$$\n\\vec { \\mathbf { f } } = [ f _ { 1 } , f _ { 2 } , \\dotsc , f _ { T } ] = [ d _ { t } , d _ { t - 1 } , \\dotsc , d _ { t - ( T - 1 ) } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            868,
+            658,
+            888
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next we append $f _ { 0 } = 1$ to the head of sequence $\\vec { \\mathbf { f } }$ and take the normalized cumulative sum to obtain the post-processed reward sequence as $\\begin{array} { r } { \\vec { \\mathcal { R } } = [ \\mathcal { R } _ { 0 } , \\mathcal { R } _ { 1 } , \\ldots , \\mathcal { R } _ { T } ] = \\frac { 1 } { T + 1 } [ f _ { 0 } , f _ { 0 } + f _ { 1 } , \\ldots , \\sum _ { i = 0 } ^ { T } f _ { i } ] } \\end{array}$ . Under such processing, numerical instability (see Eq. 4) when all rewards in the sequence are zero can be alleviated, while the averaging term $\\\\frac { 1 } { T + 1 }$ mitigates the effect of introducing the artificial $f _ { 0 }$ . ",
+        "bbox": [
+            173,
+            102,
+            825,
+            165
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The resulting $\\vec { \\mathcal { R } }$ is a reward sequence with $\\begin{array} { r } { \\mathcal { R } _ { T } - \\mathcal { R } _ { 0 } = \\frac { 1 } { T + 1 } r _ { t } } \\end{array}$ , and $\\begin{array} { r } { \\mathcal { R } _ { n } - \\mathcal { R } _ { n - 1 } = \\frac { 1 } { T + 1 } ( r _ { t + 1 - n } - } \\end{array}$ $r _ { t - n } )$ . Therefore, the difference between $\\mathcal { R } _ { T }$ and $\\mathcal { R } _ { 0 }$ represents the immediate reward and the whole sequence $\\vec { \\mathcal { R } }$ reflects the volatility of past rewards. In Figure 2, three examples of processed sequence are presented in the second row with the corresponding raw rewards shown in the first row. We account for how high the immediate reward is by defining the relative percentage log total return as: ",
+        "bbox": [
+            173,
+            172,
+            825,
+            251
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/427ecef02fe5e4c69fa5d7dcf959c9999480044bf1edb915dc4de36f0b6a0a0b.jpg",
+        "text": "$$\n\\mathcal { R } _ { H } = 1 0 0 \\times \\left( e ^ { \\frac { 1 } { T } \\ln \\frac { \\mathcal { R } _ { T } } { \\mathcal { R } _ { 0 } } } - 1 \\right) = \\frac { { \\mathcal { R } _ { T } } ^ { 1 / T } - { \\mathcal { R } _ { 0 } } ^ { 1 / T } } { { \\mathcal { R } _ { 0 } } ^ { 1 / T } } \\times 1 0 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            256,
+            691,
+            295
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To account for how varied past rewards were, we first define a smooth zero-variability reference as: $\\vec { \\mathcal { R } } ^ { z e r o } = [ \\mathcal { R } _ { 0 } ^ { z e r o } , \\mathcal { R } _ { 1 } ^ { z e r o } , \\ldots , \\mathcal { R } _ { T } ^ { z e r o } ] = \\mathcal { R } _ { 0 } [ e ^ { 0 \\times \\widetilde { \\mathcal { R } } } , e ^ { 1 \\times \\widetilde { \\mathcal { R } } } , \\ldots , e ^ { T \\widetilde { \\mathcal { R } } } ]$ with $\\begin{array} { r } { \\widetilde { \\mathcal { R } } = \\frac { 1 } { T } \\ln \\frac { \\mathcal { R } _ { T } } { \\mathcal { R } _ { 0 } } } \\end{array}$ , represent a smooth monotonic reference sequence from $\\mathcal { R } _ { 0 }$ to $\\mathcal { R } _ { T }$ . Then we define the reward differential $\\delta _ { \\mathcal { R } }$ as the differential reward versus its zero-variability reference as $\\begin{array} { r } { \\delta _ { \\mathcal { R } } ( n ) = \\frac { \\mathcal { R } _ { n } - \\mathcal { R } _ { n } ^ { z e r o } } { \\mathcal { R } _ { n } ^ { z e r o } } } \\end{array}$ , whose statistics are sketched in the third row of Figure 2. With maximally allowed volatility as $\\sigma _ { m a x }$ , the variability weights can be defined as: $\\begin{array} { r } { \\omega = 1 - \\big [ { \\frac { \\sigma ( \\delta _ { \\mathcal { R } } ) } { \\sigma _ { m a x } } } \\big ] ^ { \\tau } } \\end{array}$ , where $\\sigma ( \\cdot )$ is the standard deviation and $\\tau$ controls the rate to penalize highly volatile reward distribution. Finally we can derive the variability-weighted past reward indicator $r ^ { v w r }$ for the characterization of past reward statistics: ",
+        "bbox": [
+            173,
+            300,
+            825,
+            428
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/253d2ea817aaea969bbad50eaf18b1beb7163a9bd883c86f027b1988ecc9f45a.jpg",
+        "text": "$$\nr ^ { v w r } = \\left. \\begin{array} { c c } { \\mathcal { R } _ { H } ( 1 - [ \\frac { \\sigma ( \\delta _ { \\mathcal { R } } ) } { \\sigma _ { m a x } } ] ^ { \\tau } ) } & { \\mathrm { i f } \\sigma ( \\delta _ { \\mathcal { R } } ) < \\sigma _ { m a x } , \\mathcal { R } _ { T } > 0 } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            434,
+            692,
+            470
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The formulation of Equation 5 share principled themes as in Sharpe (1994) and Dowd (2000): ",
+        "bbox": [
+            173,
+            477,
+            790,
+            492
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. Dowd (2000) compares the newly obtained $\\mathbf { S R } ^ { n e w }$ with the previous $\\mathbf { S R } ^ { o l d }$ in choosing new assets; we derive $\\mathcal { R } _ { H }$ in Eq. 4 by comparing the latest reward $\\mathcal { R } _ { T }$ with $\\mathcal { R } _ { 0 }$ to explicitly encourage the agent to aim for reward improvements in “choosing new actions”; 2. Both the Sharpe Ratio (SR) and Eq. 5 involve “variability weights” to adjust for the unit risk of return $\\mathbb { E } [ \\mathcal { R } ]$ Sharpe (1994) (i.e., $\\scriptstyle { \\frac { 1 } { \\sigma _ { r } } }$ for SR and $\\begin{array} { r } { 1 - \\big [ \\frac { \\sigma ( \\delta \\mathcal { R } ) } { \\sigma _ { m a x } } \\big ] ^ { \\tau } } \\end{array}$ [ σ(δR) ]τ for rvwr ); 3. Whereas Dowd (2000) introduces the concept of “minimum required return” based on the elasticity of value at risk (VaR), we consider the maximum tolerance level $\\sigma _ { m a x }$ with elasticity controlled by $\\tau$ for improved learning stability of $r ^ { v w r }$ (see also Appendix H). ",
+        "bbox": [
+            210,
+            503,
+            825,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Example computed values of $r ^ { v w r }$ for the characterization of different reward statistics are shown in Figure 2 and we show strong empirical results (in Section 6) to confirm the validity and robustness of the proposed formulation in multiple reinforcement learning domains. ",
+        "bbox": [
+            174,
+            636,
+            826,
+            679
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 MULTI-CRITIC ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            695,
+            433,
+            710
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A higher value of $r ^ { v w r }$ indicates better agent performance as the result of the historical sequence of actions. The same set of optimization procedures for conventional value function (i.e., via maximization of immediate reward signal $r$ ) update can be applied accordingly. The actual returns computed from both the “long-term” and “short-term” rewards are discounted by the same factor $\\gamma$ In particular, for standard $N$ -step look-ahead approaches, we have: ",
+        "bbox": [
+            174,
+            720,
+            825,
+            791
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bfd5d11b867e90455afb9c5795897bbe6941eb272481c3dd5ccc7494f377e426.jpg",
+        "text": "$$\nR _ { t } ^ { \\mathrm { { s h o r t . e r m } } } = \\sum _ { n = 0 } ^ { N - 1 } \\gamma ^ { n } r _ { t + n } + \\gamma ^ { N } V ( s _ { t + N } ) , \\ : \\ : \\ : R _ { t } ^ { \\mathrm { l o n g . e r m } } = \\sum _ { n = 0 } ^ { N - 1 } \\gamma ^ { n } r _ { t + n } ^ { v w r } + \\gamma ^ { N } V ^ { v w r } ( s _ { t + N } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            797,
+            777,
+            842
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Similar to the standard state value function $V ( s )$ , we further define $V ^ { v w r } ( s )$ as the value function w.r.t the variability-weighted reward $r ^ { v w r }$ . These value functions form multiple critics judging a given state $s$ . The gradients w.r.t. the critics then become: ",
+        "bbox": [
+            174,
+            856,
+            825,
+            898
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/942d31dd344bb42e1547ebe501d78899219b1ac8ad6fa39b5ecb2b64aba0533e.jpg",
+        "text": "$$\n\\nabla _ { \\theta ^ { \\mathrm { s h o r i c m } } } [ ( R _ { t } ^ { \\mathrm { s h o r t - e r m } } - V ( s _ { t } ; \\theta ^ { \\mathrm { s h o r t - e r m } } ) ) ^ { 2 } ] + \\nabla _ { \\theta ^ { \\mathrm { l o r g - t e r m } } } [ ( R _ { t } ^ { \\mathrm { l o n g - t e r m } } - V ^ { v w r } ( s _ { t } ; \\theta ^ { \\mathrm { l o n g - t e r m } } ) ) ^ { 2 } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            199,
+            906,
+            779,
+            925
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd2a20fd68d11f839066eb2b4cb98f37deca225e98926d444892f991d6494b10.jpg",
+        "image_caption": [
+            "Figure 3: Performance of A2MC on Atari games. “Hot-Wiring” exploration makes the agent easier to figure out how to play challenging games like “Robotank” and “WizardOfWor”, and for most games, it provides a better initial state for the agent to start off at a game and hence to obtain better final results. The number in figure legend shows the average reward among the last 100 episodes and the percentage shows the performance margin as compared to ACKTR. The shaded region denotes the standard deviation over 2 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            104,
+            816,
+            359
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the standard grading clipping approach can be applied in Eq. 7 for enhanced stability. More advanced methods for estimating $R _ { t } ^ { \\mathrm { s h o r t - t e r m } }$ and $R _ { t } ^ { \\mathrm { { l o n g - t e r m } } }$ above, such as the online variant of generalized advantage estimation (GAE) using eligibility traces Schulman et al. (2016) can be adopted in place of Eq. 6, as shown below (see also Appendix $\\mathbf { G }$ ): ",
+        "bbox": [
+            176,
+            455,
+            823,
+            515
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb1b8494d87a4ed83938b0f7b62c6811d50ea334bbfc84498118b8b1e5182cba.jpg",
+        "text": "$$\n\\begin{array} { r l } & { A _ { t } ^ { \\mathrm { s h o r t . t e r m } } = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } ( \\gamma \\lambda ) ^ { n } \\delta _ { t + n } ^ { v w r } , \\mathrm { w i t h } \\delta _ { t } = r _ { t } + \\gamma V ( s _ { t + 1 } ) - V ( s _ { t } ) } \\\\ & { A _ { t } ^ { \\mathrm { l o n g . t e r m } } = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } ( \\gamma \\lambda ) ^ { n } \\delta _ { t + n } ^ { v w r } , \\mathrm { w i t h } \\delta _ { t } ^ { v w r } = r _ { t } ^ { v w r } + \\gamma V ^ { v w r } ( s _ { t + 1 } ) - V ^ { v w r } ( s _ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            516,
+            753,
+            599
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the generalized estimator of the advantage function $A _ { t } ^ { \\mathrm { s h o r t - t e r m } }$ and $A _ { t } ^ { \\mathrm { l o n g - t e r m } }$ allows a trade-off of bias $\\nu . s .$ . variance using the parameter $0 \\leq \\lambda \\leq 1$ , similar to the $\\mathrm { T D } ( \\lambda )$ approach for eligibility traces. We show the effectiveness of the proposed characterization of past reward statistics in multiple advantage actor-critic frameworks (i.e., ACKTR and PPO), where the two different value functions can share the same low-level feature representation, enabling a single agent to learn multiple critics parameterized by $\\theta ^ { j } , j \\in \\{ \\mathrm { s h o r t - t e r m } , \\mathrm { l o n g - t e r m } \\}$ . (See also Appendix I for the full algorithm). ",
+        "bbox": [
+            173,
+            602,
+            825,
+            689
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 HOT-WIRE $\\epsilon$ -EXPLORATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            707,
+            437,
+            723
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Being handed a game-stick, a human most likely would try out all the available buttons on it to see which particular button entails whatever actions on the game screen, hence receiving useful feedbacks. Inspired by this, we propose to hot-wire the agent to perform an identical sequence of randomly chosen actions in the N-step look-ahead during the initial stage (randomly pressing down a game-stick button for a while): ",
+        "bbox": [
+            173,
+            737,
+            825,
+            808
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9e044e8dcdca9627fdc25fd8a0c10fbf649b1d0ef193a2d713811abf0e0d9598.jpg",
+        "text": "$$\na _ { t + k } = \\left\\{ \\begin{array} { l l } { { \\mathrm { ~ a ~ r a n d o m ~ a c t i o n ~ i d e n t i c a l ~ f o r ~ a l l ~ k ~ } } } & { { \\mathrm { w i t h ~ p r o b ~ } \\epsilon } } \\\\ { { \\pi ( a _ { t + k } | s _ { t + k } ) ~ \\mathrm { f o r } ~ k = 0 , 1 , 2 , . . . , N - 1 } } & { { \\mathrm { w i t h ~ p r o b ~ } 1 - \\epsilon } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            809,
+            728,
+            844
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We show that by enabling the “hot-wiring” mechanism2, a seemingly trapped agent can be boosted to learn to quickly solve problems where rewards can only be triggered by particular action sequences, as shown in games like “Robotank” and “WizardOfWor” in Figure 3. ",
+        "bbox": [
+            174,
+            847,
+            826,
+            888
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            326,
+            117
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use the same network architecture and natural gradient optimization method as in the ACKTR model Wu et al. (2017). We set $\\sigma _ { m a x } = 1 . 0$ , $\\tau = 2 . 0$ and $T = 2 0$ in the computation of variabilityweighted reward (see Appendix C for hyperparameter studies). For hot-wiring exploration, we choose $\\epsilon = 0 . 2 0$ and initial stage to be first $\\scriptstyle { \\frac { 1 } { 4 0 } }$ of the total training period for all experiments. Other hyperparameters such as learning rate and gradient clipping remain the same as in the ACKTR model Wu et al. (2017), in addition to adopting GAE Schulman et al. (2016) for a stronger ACKTR baseline (see Sec 4.2). We first present results of evaluating the proposed A2MC model in two standard benchmarks, the discrete Atari experiments and the continuous MuJoCo domain. Then we show ablation studies on the robustness of the hyper-parameters involved as well as evaluating the extensibility of the proposed long/short-term reward characterizations to other on-policy methods. Further extensions to off-policy domains are presented in Appendix E and Appendix F. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            286
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1 ATARI 2600 GAMES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            303,
+            361,
+            316
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We follow standard evaluation protocol to evaluate A2MC in a variety of Atari game environments (starting with 30 no-op actions). We train our models for 15 million timesteps for each game environment and score each game based on the average episode rewards obtained among the last 100 episodes as in Wu et al. (2017). The learning results on 12 Atari games are shown in Figure 3 where we also included an ablation experiment of A2MC without hot-wiring. We observe that on average A2MC improves upon ACKTR in terms of final performance under the same training budget. Our A2MC is able to consistently improve agent performance based on the proposed characterization of reward statistics, hence the agent is able to get out of local minima in less time (higher sample efficiency) compared to ACKTR. The complete learning results on all games are attached in the Appendix B. ",
+        "bbox": [
+            173,
+            329,
+            825,
+            467
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2ed308461e3b8e633582fe7e03463a635268184cc2bfcfdb25c6abb428826abe.jpg",
+        "table_caption": [
+            "Table 1: Comparison of average episode rewards at the end of 50 million timesteps in Atari experiments. The reward scores and the first episodes reaching human-level performance Mnih et al. (2015) are reported as in Wu et al. (2017). A2MC is able to solve games that are challenging to ACKTR and also retain comparable performance in the rest of games. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"2\">ACKTR</td><td colspan=\"2\">A2MC</td></tr><tr><td>Domain</td><td>Human Level</td><td>Rewards</td><td>Episode</td><td>Rewards</td><td>Episode</td></tr><tr><td>Asteroids</td><td>47388.7</td><td>34171.0</td><td>N/A</td><td>830232.5</td><td>11314</td></tr><tr><td>Beamrider</td><td>5775.0</td><td>13581.4</td><td>3279</td><td>13564.3</td><td>3012</td></tr><tr><td>Boxing</td><td>12.1</td><td>1.5</td><td>N/A</td><td>99.1</td><td>158</td></tr><tr><td>Breakout</td><td>31.8</td><td>735.7</td><td>4097</td><td>411.4</td><td>3664</td></tr><tr><td>Double Dunk</td><td>-16.4</td><td>-0.5</td><td>742</td><td>21.3</td><td>544</td></tr><tr><td>Enduro</td><td>860.5</td><td>0.0</td><td>N/A</td><td>3492.2</td><td>730</td></tr><tr><td>Freeway</td><td>29.6</td><td>0.0</td><td>N/A</td><td>32.7</td><td>1058</td></tr><tr><td>Pong</td><td>9.3</td><td>20.9</td><td>904</td><td>19.4</td><td>804</td></tr><tr><td>Q-bert</td><td>13455.0</td><td>21500.3</td><td>6422</td><td>25229.0</td><td>7259</td></tr><tr><td>Robotank</td><td>11.9</td><td>16.5</td><td>-</td><td>25.7</td><td>4158</td></tr><tr><td>Seaquest</td><td>20182.0</td><td>1776.0</td><td>N/A</td><td>1798.6</td><td>N/A</td></tr><tr><td>Space Invaders</td><td>1652.0</td><td>19723.0</td><td>14696</td><td>11774.0</td><td>11064</td></tr><tr><td>Wizard of Wor</td><td>4756.5</td><td>702</td><td>N/A</td><td>7471.0</td><td>8119</td></tr></table>",
+        "bbox": [
+            214,
+            536,
+            785,
+            767
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We further expand the training budget and continue learning the games until 50 million timesteps as in Wu et al. (2017). As shown in Table 1, our A2MC model can solve games like “Boxing”, “Freeway” and “Enduro” that are challenging for the baseline ACKTR model. For a full picture of model performance in Atari games, A2MC has a human-level performance rate of $7 4 . 5 \\%$ (38 out of 51 games) in the Atari benchmarks, compared to $6 3 . 6 \\%$ reached by ACKTR. Individual game scores for all the Atari games are reported in the Appendix B. ",
+        "bbox": [
+            173,
+            768,
+            826,
+            853
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.2 CONTINUOUS CONTROL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            869,
+            382,
+            883
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the evaluations on continuous control tasks simulated in MuJoCo environment, we first follow $\\mathrm { W u }$ et al. (2017) and tune a different set of hyper-parameters from Atari experiments. Specifically, all ",
+        "bbox": [
+            174,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a06f904da805d4c7acb1d86247f9e70b0ea241d1cad9dbdc73b865f57ba7291.jpg",
+        "image_caption": [
+            "Figure 4: Performance on the MuJoCo benchmark. A2MC is also competitive on MuJoCo continuous domain when compared to ACKTR. The shaded region denotes std over 3 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            104,
+            816,
+            277
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "MuJoCo experiments are trained with a larger batch size of 2500. The results of eight MuJoCo environments trained for 1 million timesteps are shown in Figure 4. We observe that A2MC also performs well in all MuJoCo continuous control tasks. In particular, A2MC has brought significant improvement on the tasks of HalfCheetah, Swimmer and Walker2d (see Table 2). ",
+        "bbox": [
+            174,
+            319,
+            825,
+            375
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To test the robustness of A2MC, we perform another set of evaluations on MuJoCo tasks by keeping an identical set of hyper-parameters used in the Atari experiments. Figure 7 in Appendix C shows this ablation result. We observe that even under sub-optimal hyper-parameters, our A2MC model can still learn to solve the MuJoCo control tasks in the long run. Moreover, it is less prone to overfitting when compared to ACKTR under such “stress testing”. Additional hyper-parameter studies can be found in Appendix C. ",
+        "bbox": [
+            174,
+            381,
+            825,
+            465
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also evaluate a multi-critics variant of the proximal policy optimization (PPO) model on the MuJoCo tasks with our proposed long/short-term rewards. In particular, we observe that our proposed variability-weighted reward generalizes well with the vanilla PPO, and our multi-critics PPO variant (MC-PPO) shows more favorable performance, as shown in Table 2. Specifically, MC-PPO shows the best performance on Hopper and Walker- $_ { 2 d }$ among all models under the 1-million timesteps training budget. Both of our multi-critics variants (A2MC and MC-PPO) have won 6 out of the 8 MuJoCo tasks with relative performance margins (percentages in parentheses) larger than $2 5 \\%$ (see Table 2). ",
+        "bbox": [
+            173,
+            472,
+            825,
+            570
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/1e2bae610559ebc2f49d3e1e65b4075eea701056682ef5ff8818f3d110903b1a.jpg",
+        "table_caption": [
+            "Table 2: Average episode rewards obtained among the last 10 episodes upon 1 million timesteps of training in MuJoCo experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>GAMES</td><td>ACKTR</td><td colspan=\"2\">Our A2MC</td><td>PPO</td><td colspan=\"2\">Our MC-PPO</td></tr><tr><td>Ant</td><td>1671.6</td><td>2216.1</td><td>(32.5%)</td><td>411.4 (± 107.7)</td><td>618.9</td><td>(50.4%)</td></tr><tr><td>HalfCheetah</td><td>1676.2</td><td>2696.6</td><td>(60.8%)</td><td>1433.7 (± 83.9)</td><td>2473.4</td><td>(72.5%)</td></tr><tr><td>Hopper</td><td>2259.1</td><td>2835.7</td><td>(25.5%)</td><td>2055.8 (± 274.6)</td><td>3131.3</td><td>(52.3%)</td></tr><tr><td>InvertedDoublePendulum 6295.4</td><td></td><td>7872.6</td><td>(25.0%)</td><td>4454.1 (± 1098.1)</td><td>7648.7</td><td>(71.7%)</td></tr><tr><td>InvertedPendulum</td><td>1000.0</td><td>957.2</td><td>(-4.2%)</td><td>839.7 (± 127.1)</td><td>777.4</td><td>(-7.4%)</td></tr><tr><td>Reacher</td><td>-4.2</td><td>-3.9</td><td>(0.4%)</td><td>-5.47 (± 0.3)</td><td>-10.3</td><td>(-8.5%)</td></tr><tr><td>Swimmer</td><td>43.2</td><td></td><td>187.4 (333.7%)</td><td>79.1 (± 31.2)</td><td>102.9</td><td>(30.2%)</td></tr><tr><td>Walker2d</td><td>1090.8</td><td>2405.9 (120.5%)</td><td></td><td>2300.8 (± 397.6)</td><td>3718.1</td><td>(61.6%)</td></tr><tr><td>Win—Fair—Lose</td><td>N/A</td><td>6-2-0</td><td></td><td>N/A</td><td>6—2-0</td><td></td></tr></table>",
+        "bbox": [
+            174,
+            616,
+            823,
+            777
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            795,
+            318,
+            810
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduce an effective auxiliary reward signal to remedy the deficiencies of learning solely from the standard environment rewards. Our proposed characterization of past reward statistics results in improved learning and higher sample efficiencies for on-policy methods, especially in challenging tasks with sparse rewards. Experiments on both discrete tasks in Atari environment and MuJoCo continuous control tasks validate the effectiveness of utilizing the proposed long/short-term reward statistics for on-policy methods using multi-critic architectures. This suggests that expanding the form of reward feedbacks in reinforcement learning is a promising research direction. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            287,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. In Advances in Neural Information Processing Systems, pp. 5048–5058, 2017. ",
+        "bbox": [
+            174,
+            126,
+            823,
+            169
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Marc G Bellemare, Will Dabney, and Remi Munos. A distributional perspective on reinforcement ´ learning. arXiv preprint arXiv:1707.06887, 2017. ",
+        "bbox": [
+            173,
+            176,
+            823,
+            205
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kevin Dowd. Adjusting for risk:: An improved sharpe ratio. International review of economics & finance, 9(3):209–222, 2000. ",
+        "bbox": [
+            171,
+            213,
+            823,
+            243
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matteo Hessel, Joseph Modayil, Hado Van Hasselt, Tom Schaul, Georg Ostrovski, Will Dabney, Dan Horgan, Bilal Piot, Mohammad Azar, and David Silver. Rainbow: Combining improvements in deep reinforcement learning. 2018. ",
+        "bbox": [
+            174,
+            251,
+            823,
+            294
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Max Jaderberg, Volodymyr Mnih, Wojciech Marian Czarnecki, Tom Schaul, Joel Z Leibo, David Silver, and Koray Kavukcuoglu. Reinforcement learning with unsupervised auxiliary tasks. arXiv preprint arXiv:1611.05397, 2016. ",
+        "bbox": [
+            176,
+            303,
+            823,
+            345
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Junqi Jin, Chengru Song, Han Li, Kun Gai, Jun Wang, and Weinan Zhang. Real-time bidding with multi-agent reinforcement learning in display advertising. arXiv preprint arXiv:1802.09756, 2018. ",
+        "bbox": [
+            171,
+            353,
+            825,
+            383
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "George Konidaris and Andrew G Barto. Skill discovery in continuous reinforcement learning domains using skill chaining. In Advances in neural information processing systems, pp. 1015–1023, 2009. ",
+        "bbox": [
+            171,
+            390,
+            825,
+            421
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Guillaume Lample and Devendra Singh Chaplot. Playing fps games with deep reinforcement learning. In AAAI, 2017. ",
+        "bbox": [
+            173,
+            428,
+            825,
+            458
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Marc Lanctot, Vinicius Zambaldi, Audrunas Gruslys, Angeliki Lazaridou, Julien Perolat, David Silver, Thore Graepel, et al. A unified game-theoretic approach to multiagent reinforcement learning. In Advances in Neural Information Processing Systems, pp. 4193–4206, 2017. ",
+        "bbox": [
+            176,
+            465,
+            823,
+            510
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ryan Lowe, Yi Wu, Aviv Tamar, Jean Harb, OpenAI Pieter Abbeel, and Igor Mordatch. Multi-agent actor-critic for mixed cooperative-competitive environments. In Advances in Neural Information Processing Systems, pp. 6382–6393, 2017. ",
+        "bbox": [
+            176,
+            517,
+            823,
+            560
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015. ",
+        "bbox": [
+            176,
+            568,
+            823,
+            612
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning, pp. 1928–1937, 2016. ",
+        "bbox": [
+            176,
+            619,
+            823,
+            662
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. High-dimensional continuous control using generalized advantage estimation. ICLR, 2016. ",
+        "bbox": [
+            173,
+            670,
+            823,
+            700
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. ",
+        "bbox": [
+            173,
+            708,
+            823,
+            737
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "William F Sharpe. The sharpe ratio. Journal of portfolio management, 21(1):49–58, 1994. ",
+        "bbox": [
+            171,
+            744,
+            766,
+            761
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "David Silver and Kamil Ciosek. Compositional planning using optimal option models. arXiv preprint arXiv:1206.6473, 2012. ",
+        "bbox": [
+            173,
+            768,
+            821,
+            797
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676):354, 2017. ",
+        "bbox": [
+            176,
+            806,
+            823,
+            849
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018. ",
+        "bbox": [
+            173,
+            858,
+            823,
+            873
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yee Teh, Victor Bapst, Wojciech M Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. In Advances in Neural Information Processing Systems, pp. 4499–4509, 2017. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Harm Van Seijen, Mehdi Fatemi, Joshua Romoff, Romain Laroche, Tavian Barnes, and Jeffrey Tsang. Hybrid reward architecture for reinforcement learning. In NIPS, pp. 5392–5402, 2017. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ziyu Wang, Victor Bapst, Nicolas Heess, Volodymyr Mnih, Remi Munos, Koray Kavukcuoglu, and Nando de Freitas. Sample efficient actor-critic with experience replay. arXiv preprint arXiv:1611.01224, 2016. ",
+        "bbox": [
+            174,
+            141,
+            823,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. In Reinforcement Learning, pp. 5–32. Springer, 1992. ",
+        "bbox": [
+            171,
+            193,
+            825,
+            222
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yuhuai Wu, Elman Mansimov, Roger B Grosse, Shun Liao, and Jimmy Ba. Scalable trust-region method for deep reinforcement learning using kronecker-factored approximation. In Advances in neural information processing systems, pp. 5285–5294, 2017. ",
+        "bbox": [
+            178,
+            231,
+            823,
+            272
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            254,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A EFFECTS OF FLIPPING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            137,
+            395,
+            155
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "While introducing the variability-weighted reward, a flipping operation is conducted in the preprocessing of the reward sequence as formulated in Eq. (3). In Figure 5 and 6, we construct 4 reward sequences to show that the flipping operation can further penalize the oscillation in the recent past rewards while encourage recent stable rewards. (a1, a2, b1, b2) share the same value of immediate reward at $t = 9$ and thus the $\\mathcal { R } _ { H }$ of all reward sequences are the same. Therefore, the variability-weighted reward only depends on the volatility statistics of $\\delta _ { R }$ , i.e., how varied past rewards were. ",
+        "bbox": [
+            173,
+            170,
+            826,
+            239
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8c96c2bfb2986a8131744bff4f87fb64d6ad7d6ab6465c97ddc8d0c4bc5483f.jpg",
+        "image_caption": [
+            "Figure 5: Calculation without flipping. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            256,
+            482,
+            535
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/03f318fa74dc2f857ea0e3562e3d4f2c6896c003afc641c8b8400191574d409c.jpg",
+        "image_caption": [
+            "Figure 6: Calculation with flipping. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            517,
+            256,
+            799,
+            535
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            569,
+            826,
+            598
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Without flipping. In Figure 5, sequence $( a l )$ and $( a 2 )$ are mirror symmetrical to the $y$ -axis, and the only difference between them is that the recent past rewards $( t = 5 , 6 , 7 , 8 )$ ) of $( a 2 )$ are more stable than (a1). Intuitively, we want to encourage stable past rewards like $( a 2 )$ while penalizing oscillation in (a1). As presented in the third row of Figure 5, the $r ^ { v w r }$ difference of $( a I )$ and $( a 2 )$ is less than 1 without flipping in the pre-processing. ",
+        "bbox": [
+            173,
+            604,
+            825,
+            675
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "With flipping. In Figure 6, (b1, b2) exactly have the same reward sequence as $( a l , a 2 )$ , respectively. However, flipping is performed as a step of pre-processing, largely increasing the $r ^ { v w r }$ gap (from less than 1 to nearly 4) between the two constructed sequences. Comparing $( b l , b 2 )$ with $( a l , a 2 )$ , the post-processed sequences $\\vec { \\mathcal { R } }$ (shown in green) become centrosymmetric to those without flipping. Specifically, the recent reward drops at $t = 6 , 7 , 8$ are reflected as high values at the beginning of $\\vec { \\mathcal { R } }$ as shown in $( b l )$ , while oscillations long ago are transformed into high values at the end of $\\vec { \\mathcal { R } }$ as presented in $( b 2 )$ . When compared to the zero-variability reference (shown in orange), which is designed as an exponential function, the flipping leads to a higher variability for the former sequence while a lower variability for the latter one, enlarging the $r ^ { v w r }$ gap between those two sequences. ",
+        "bbox": [
+            173,
+            681,
+            826,
+            813
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B COMPLETE RESULTS IN ATARI 2600 GAMES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            576,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We show the learning curves for 15 million timesteps on all Atari games in Figure 12 and in Table 3 we show the complete results of training til 50 million timesteps. report the mean episode reward as in Wu et al. (2017). Entries with $\\sim$ indicates approximated value as retrieved from learning figures published by Wu et al. (2017). Results from other models are taken from Wu et al. (2017) and Mnih et al. (2015). We show that A2MC has reached a human-level performance rate of $7 4 . 5 \\%$ (38 out of 51 games) as compared to $6 3 . 6 \\%$ reached by ACKTR. The relative performance margin of A2MC as compared to ACKTR is also shown. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            231
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C HYPER-PARAMETER STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            251,
+            455,
+            267
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The proposed variability-weighted reward mechanism considers the volatility of rewards by keeping a $T$ -step history of agent’s performance. The hyper-parameter $T = 2 0$ is empirically chosen to be the same as the look-ahead parameter $N$ in standard on-policy methods, so as to keep the same period $\\textstyle T = N = 2 0$ ) in “T-step history” and “N-step forward”. And $\\sigma _ { m a x } = 1$ is chosen as the maximum of the observed volatility based on statistics in the $\\mathrm { T }$ history rewards of the ACKTR models. As parameter choices could be vital, we perform an additional ablation study shown below. The result shows that the performance of A2MC is robust across different parameters of choice and is not too sensitive to changes on either of the hyper-params. ",
+        "bbox": [
+            173,
+            282,
+            825,
+            395
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/6ade0ecef6c2b5d9349f04a023ea9f5b2696c56a34037d2a33e639d95a140ad1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Games</td><td>ACKTR</td><td>A2MC w/</td><td>T=20 Omax=1</td><td>T=10 Omax=1</td><td>T=10 0max=2</td><td>T=40 Omax=1</td><td>T=40 Omax=2</td></tr><tr><td>Boxing</td><td>1.23</td><td></td><td>99.19</td><td>94.76</td><td>98.51</td><td>99.18</td><td>98.07</td></tr><tr><td>Jamesbond</td><td>409.50</td><td></td><td>453.50</td><td>438.50</td><td>470.00</td><td>442.25</td><td>457.75</td></tr><tr><td>Wizard of Wor</td><td>744.50</td><td></td><td>5448.00</td><td>5601.00</td><td>5363.50</td><td>2528.50</td><td>3287.50</td></tr></table>",
+        "bbox": [
+            178,
+            397,
+            820,
+            477
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/162bf3e0520ec61e2d4440c63fcb2a71cb0c9e75a49eabbe055eb3df783697fa.jpg",
+        "image_caption": [
+            "Figure 7: “Stress testing” ablation study on the MuJoCo continuous benchmark using hyperparameters tuned in Atari discrete control. Although this set of hyperparameters is suboptimal for the MuJoCo continuous control tasks, A2MC still obtain reasonable performance in the long run and it is less prone to overfitting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            500,
+            816,
+            671
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/d8a6d65006ae7a4e435a5a42e4b9f11f39b7fb4e310a57557618adc84fab20b3.jpg",
+        "table_caption": [
+            "Table 3: Raw scores across all games, starting with 30 no-op actions. Scores are reported by averaging the last 500 episodes upon 50 million timesteps of training as in Wu et al. (2017). A relative margin comparing A2MC to ACKTR is shown. Scores from other models are taken from Wu et al. (2017) and Mnih et al. (2015). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>GAME</td><td>Human</td><td>DQN</td><td>DDQN Prior. Duel</td><td></td><td>ACKTR OurA2MC</td><td></td><td>(Margin)</td></tr><tr><td>Alien</td><td>7127.7</td><td>1620</td><td>3747.7</td><td>3941</td><td>3197.1</td><td>2986.3</td><td>-6.6%</td></tr><tr><td>Amidar</td><td>1719.5</td><td>978</td><td>1793.3</td><td>2296.8</td><td>1059.4</td><td>2040.1</td><td>92.6%</td></tr><tr><td>Assault</td><td>742.0</td><td>4280.4</td><td>5393.2</td><td>11477</td><td>10777.7</td><td>9892.4</td><td>-8.2%</td></tr><tr><td>Asterix</td><td>8503.3</td><td>4359</td><td>17356.5</td><td>375080</td><td>31583.0</td><td>32671.0</td><td>3.4%</td></tr><tr><td>Asteroids</td><td>47388.7</td><td>1364.5</td><td>734.7</td><td>1192.7</td><td>34171.6</td><td>828931.6</td><td>2325.8%</td></tr><tr><td>Atlantis</td><td>29028.1</td><td>279987</td><td>106056</td><td>395762</td><td>3433182.0</td><td>2886274.0</td><td> -15.9%</td></tr><tr><td>Bankheist</td><td>753.1</td><td>455</td><td>1030.6</td><td>1503.1</td><td>1289.7</td><td>1290.6</td><td>0.1%</td></tr><tr><td>Battlezone</td><td>37187.5</td><td>29900</td><td>31700</td><td>35520</td><td>8910.0</td><td>10570.0</td><td>18.6%</td></tr><tr><td>Beamrider</td><td>16926.5</td><td>8627.5</td><td>13772.8</td><td>30276.5</td><td>13581.4</td><td>13715.6</td><td>1.0%</td></tr><tr><td>Berzerk</td><td>2630.4</td><td>585.6</td><td>1225.4</td><td>3409</td><td>927.2</td><td>974.0</td><td>5.0%</td></tr><tr><td>Bowling</td><td>160.7</td><td>50.4</td><td>68.1</td><td>46.7</td><td>24.3</td><td>31.6</td><td>30.0%</td></tr><tr><td>Boxing</td><td>12.1</td><td>88</td><td>91.6</td><td>98.9</td><td>1.5</td><td>93.5</td><td>6344.8%</td></tr><tr><td>Breakout</td><td>30.5</td><td>385.5</td><td>418.5</td><td>366</td><td>735.7</td><td>420.6</td><td>-42.8%</td></tr><tr><td>Centipede</td><td>12017.0</td><td>4657.7</td><td>5409.4</td><td>7687.5</td><td>7125.3</td><td>12096.5</td><td>69.8%</td></tr><tr><td>Choppercommand</td><td>9882.0</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~8000</td><td>12149.0</td><td>~42.5%</td></tr><tr><td>Crazyclimber</td><td>35829.4</td><td>110763</td><td>117282</td><td>162224</td><td>150444.0</td><td>152439.0</td><td>1.3%</td></tr><tr><td>Demonattack</td><td>1971.0</td><td>12149.4</td><td>58044.2</td><td>72878.6</td><td>274176.7</td><td>361807.1</td><td>32.0%</td></tr><tr><td>Doubledunk</td><td>-16.4</td><td>-6.6</td><td>-5.5</td><td>-12.5</td><td>-0.5</td><td>20.6</td><td>3907.5%</td></tr><tr><td>Enduro</td><td>860.5</td><td>729</td><td>1211.8</td><td>2306.4</td><td>0.0</td><td>3550.6</td><td>8%</td></tr><tr><td>Fishingderby</td><td>-38.7</td><td>-4.9</td><td>15.5</td><td>41.3</td><td>33.7</td><td>38.4</td><td>13.9%</td></tr><tr><td>Freeway</td><td>29.6</td><td>30.8</td><td>33.3</td><td>33</td><td>0.0</td><td>32.7</td><td>0%</td></tr><tr><td>Frostbite</td><td>4335.0</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~280</td><td>293.7</td><td>~5.1%</td></tr><tr><td>Gopher</td><td>2412.5</td><td>8777.4</td><td>14840.8</td><td>104368.2</td><td>47730.8</td><td>86101.4</td><td>80.4%</td></tr><tr><td>Gravitar</td><td>2672.0</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~300</td><td>995.0</td><td>-2.9%</td></tr><tr><td>Icehockey</td><td>0.9</td><td>-1.9</td><td>-2.7</td><td>-0.4</td><td>-4.2</td><td>-2.1</td><td>16.3%</td></tr><tr><td>Jamesbond</td><td>302.8</td><td>768.5</td><td>1358</td><td>812</td><td>490.0</td><td>545.0</td><td>11.2%</td></tr><tr><td>Kangaroo</td><td>3035.0</td><td>7259</td><td>12992</td><td>1792</td><td>3150.0</td><td>11269.0</td><td>257.7%</td></tr><tr><td>Krull</td><td>2665.5</td><td>8422.3</td><td>7920.5</td><td>10374.4</td><td>9686.9</td><td>10245.4</td><td>5.8%</td></tr><tr><td>Kungfumaster</td><td>22736.3</td><td>26059</td><td>29710</td><td>48375</td><td>34954.0</td><td>39773.0</td><td>13.8%</td></tr><tr><td>Mspacman</td><td>15693.0</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~3500</td><td>5006.1</td><td>~34.5%</td></tr><tr><td>Namethisgame</td><td>4076.0</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~12500</td><td>12569.9</td><td>~0.6%</td></tr><tr><td>Phoenix</td><td>7242.6</td><td>8485.2</td><td>12252.5</td><td>70324.3</td><td>133433.7</td><td>221288.3</td><td>65.8%</td></tr><tr><td>Pitfall</td><td>6463.7</td><td>-286.1</td><td>-29.9</td><td>0</td><td>-1.1</td><td>-2.5</td><td>-0.3%</td></tr><tr><td>Pong</td><td>14.6</td><td>20.9</td><td>21</td><td>20.9</td><td>20.9</td><td>19.7</td><td>-5.9%</td></tr><tr><td>Privateeye</td><td>69571.0</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~560</td><td>507.0</td><td>-9.5%</td></tr><tr><td>Qbert</td><td>13455.0</td><td>13117.3</td><td>15088.5</td><td>18760.3</td><td>23151.5</td><td>24075.8</td><td>4.0%</td></tr><tr><td>Riverraid</td><td>17118.0</td><td>7377.6</td><td>14884.5</td><td>20607.6</td><td>17762.8</td><td>18671.9</td><td>5.1%</td></tr><tr><td>Roadrunner</td><td>7845.0</td><td>39544</td><td>44127</td><td>62151</td><td>53446.0</td><td>50071.0</td><td>-6.3%</td></tr><tr><td>Robotank</td><td>11.9</td><td>63.9</td><td>65.1</td><td>27.5</td><td>16.5</td><td>26.5</td><td>60.5%</td></tr><tr><td>Seaquest</td><td>42054.7</td><td>5860.6</td><td>16452.7</td><td>931.6</td><td>1776.0</td><td>1805.6</td><td>1.7%</td></tr><tr><td>Solaris</td><td>12326.7</td><td>3482.8</td><td>3067.8</td><td>133.4</td><td>2368.6</td><td>2277.2</td><td>-3.9%</td></tr><tr><td>Spaceinvaders</td><td>1668.7</td><td>1692.3</td><td>2525.5</td><td>15311.5</td><td>19723.0</td><td>13544.2</td><td>-31.3%</td></tr><tr><td>Stargunner</td><td>10250.0</td><td>54282</td><td>60142</td><td>125117</td><td>82920.0</td><td>89616.0</td><td>8.1%</td></tr><tr><td>Tennis</td><td>-8.9</td><td>N/A</td><td>N/A</td><td>N/A</td><td>~-12</td><td>-4.7</td><td>~20.4%</td></tr><tr><td>Timepilot</td><td>5229.2</td><td>4870</td><td>8339</td><td>7553</td><td>22286.0</td><td>21992.0</td><td>-1.3%</td></tr><tr><td>Tutankham</td><td>167.6</td><td>68.1</td><td>218.4</td><td>245.9</td><td>314.3</td><td>193.7</td><td>-38.4%</td></tr><tr><td>Upndown</td><td>11693.2</td><td>9989.9</td><td>22972.2</td><td>33879.1</td><td>436665.8</td><td>563659.3</td><td>29.1%</td></tr><tr><td>Videopinball</td><td>17667.9</td><td>196760.4</td><td>309941.9</td><td>479197</td><td>100496.0</td><td>127452.4</td><td>26.8%</td></tr><tr><td>Wizardofwor YarsRevenge</td><td>4756.5</td><td>2704</td><td>7492</td><td>12352</td><td>702.0 125169.0</td><td>7864.0 143141.5</td><td>1020.2% 14.4%</td></tr></table>",
+        "bbox": [
+            178,
+            171,
+            820,
+            922
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D EXTENSION TO MULTI-CRITIC PPO (MC-PPO) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            607,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The learning results of the proposed MC-PPO model on the MuJoCo tasks are shown in Figure 8. MC-PPO shows the best performance on Hopper and Walker- $_ { 2 d }$ among all models under the 1-million timesteps training budget. Both of our multi-critics variants (A2MC and MC-PPO) have won 6 out of the 8 MuJoCo tasks with relative performance margins (percentages in parentheses) larger than $2 5 \\%$ . ",
+        "bbox": [
+            173,
+            133,
+            826,
+            189
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/47b1442f99fadd4e266ec71e7ffe602454c89c06ae82dbdc9f5bc465cc0f2b12.jpg",
+        "image_caption": [
+            "Figure 8: Performance on the MuJoCo continuous control benchmarks using PPO-based methods. Our proposed long/short-term reward characterization can be extended to the PPO method, i.e., the proposed multi-critic variant of PPO (MC-PPO). The shaded region denotes the standard deviation over 3 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            202,
+            825,
+            377
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E EXTENSION TO OFF-POLICY METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            535,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Methods involving experience replay belong to the family of off-policy methods as they learn from off-policy trajectories. They were considered to be beyond the scope of this work, as we set out to improve the family of “on-policy” methods and we try to present as complete the analyses as possible (on both Atari and MuJoCo) in the main text. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            189
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Notwithstanding this, we have been actively exploring the potential of applying the proposed reward mechanism with off-policy methods (in particular, on the strong baseline Rainbow Hessel et al. (2018). For consistencies in comparisons, all hyperparameters (e.g., learning rate, distributional atoms, noisy net $\\sigma _ { 0 }$ ) are kept identical as in Hessel et al. (2018) except that we used a smaller replay buffer size of 50,000 for both the baseline and our method (due to limited compute). Moreover, we use the same experiment settings as in Sec 6 and we have NOT further tuned any parameters in VWR. We show preliminary results at 10 million time steps on Atari games in Figure 9 and we observe it is promising that introducing the proposed characterization of variability-weighted reward mechanism improves off-policy methods as well. ",
+        "bbox": [
+            174,
+            195,
+            825,
+            321
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The robustness of our proposed reward mechanism across both on-policy and off-policy frameworks suggests that the concept of “risk-adjusted return” Sharpe (1994) should apply in reinforcement learning in general, as it brings the desired property in faciliating better sample efficiency and learning stability. Given limited time and computing resources we are not able to present a full analysis on all the off-policy frameworks as we did for the on-policy methods within this paper (since training off-policy models takes significantly longer time). Potentially we aim to have the complete results in an additional paper in our future works. ",
+        "bbox": [
+            173,
+            329,
+            825,
+            425
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/49950388de336cc09e187e4bf97c4e8d8151b5317eb9c6ae2d692622fb39fa69.jpg",
+        "image_caption": [
+            "Figure 9: Performance of applying the variability-weighted reward to the Rainbow model on the Atari benchmark. We observe that introducing the proposed reward characterizations significantly expediate the learning in games such as “Jamesbond” and “NameThisName”, while showing consistent improvement towards the rest. The shaded region denotes the standard deviation over 2 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            450,
+            812,
+            777
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F CASE STUDY: PLAYING DOOM WITH REWARD SHAPING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            101,
+            679,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "It is worth investigating whether the proposed auxiliary reward signal VWR can work “side-by-side” with carefully shaped rewards specific to some particular game scenario – for example, the FPS game Doom Lample & Chaplot (2017). As our proposed reward characterization is generic in design and orthogonal to reward shaping, we aim to validate that the concept of risk-adjusted return and variability weights can be equally applied under such shaping settings. ",
+        "bbox": [
+            174,
+            133,
+            826,
+            204
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To this end, we adopt the off-policy agent “Arnold” Lample & Chaplot (2017) with experience replay as our baseline and we calculate VWR (see Section 4]) based on the historical sequence of the shaped rewards defined in Lample & Chaplot (2017) (See the Table 4). For VWR parameters, we set $\\sigma _ { m a x } = 5$ since the maximum (minimum) attainable reward is $5 . 0 \\ : ( - 5 . 0 )$ under such reward shaping3. The rest of the game setup and bot numbers are defaulted to the code released by Lample & Chaplot (2017). ",
+        "bbox": [
+            174,
+            210,
+            825,
+            294
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/872c069e17fb0a0b16b5169cdd6906c92712179ea040d6c145f3a06bf76f374f.jpg",
+        "table_caption": [
+            "Table 4: Reward shaping settings as in Arnold Lample & Chaplot (2017) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Type</td><td>Base /Dist Kill</td><td></td><td></td><td></td><td> Suicide Death Injured Use ammo Weapon /Ammo /Medkit /Armor</td></tr><tr><td>Value</td><td>0.0 5.0 -5.0</td><td>-5.0 -1.0</td><td></td><td>-0.2</td><td>1.0 / 1.0/ 1.0 /1.0</td></tr></table>",
+        "bbox": [
+            173,
+            334,
+            828,
+            368
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We follow the evaluation criterion of Track-1 in ViZDoom AI Competition 2016 using “Frags per episode”, i.e., the number of kills minus the number of suicides for the agent in one round of game (higher is better). The result under 50 training hours is shown in Figure 10 and we consistently observe that the Arnold agent can be significantly boosted with the help of VWR. This confirms that our proposed reward characterization is able to bring further improvements on top of both reward shaping and experience replay methods across domains. ",
+        "bbox": [
+            173,
+            383,
+            825,
+            467
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/219e99cb434509ef5c07a242db4012a9338561375cc58691a61c438f2aa4e136.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">(a) Game statistics in 50 hours</td></tr><tr><td>After 24 hours</td><td>Arnold</td><td>Arnod + VWR</td></tr><tr><td>Kills</td><td>105</td><td>183</td></tr><tr><td>Frags</td><td>87</td><td>173</td></tr><tr><td>K/D ratio</td><td>1.48</td><td>2.08</td></tr><tr><td>After 50 hours</td><td></td><td></td></tr><tr><td>Kills</td><td>116</td><td>244</td></tr><tr><td>Frags</td><td>113</td><td>223</td></tr><tr><td>K/D ratio</td><td>2.00</td><td>2.65</td></tr></table>",
+        "bbox": [
+            184,
+            506,
+            485,
+            715
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/db55751aa06be9678b9b3cbbc236a6556fe7f569ecf88f6d40f94b718c84ff35.jpg",
+        "image_caption": [
+            "Figure 10: Doom - Limited Deathmatch (Track-1) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            496,
+            523,
+            810,
+            753
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "(b) Learning results averaged over 2 random seeds ",
+        "bbox": [
+            501,
+            507,
+            802,
+            520
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "G ABLATION STUDY: VWR V.S. ELIGIBILITY TRACE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            630,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Eligibility traces $\\mathrm { T D } ( \\lambda )$ is widely used in bridging TD algorithms to Monte Carlo (MC) methods.   \nEssentially, the discounted cumulative return can be formulated by not just toward “any n-step” return (using n-step look ahead), but toward any average of n-step look-ahead returns Sutton & Barto (2018).   \nThe online variant of generalized advantage estimation using eligibility traces (GAE) Schulman et al.   \n(2016) confirms that on-policy methods can benefit from $\\mathrm { T D } ( \\lambda )$ learning. ",
+        "bbox": [
+            174,
+            133,
+            826,
+            204
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "For the proposed variability-weighted reward, the design theme is to look explicitly backward and to assess the past performance of the agent via the “risk-adjusted return” concept. These two mechanisms can be combined seamlessly via Eq. 8 and our empirical results suggest VWR brings further improvements on top of eligibility traces. ",
+        "bbox": [
+            174,
+            210,
+            825,
+            266
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "As VWR and eligibility traces are thematically similar in some sense, we further perform an ablation study to contrast the contributions brought by VWR. As shown in Figure 11, we compare three different settings: (1) ACKTR $^ +$ GAE, (2) ACKTR $^ +$ vwr and (3) $\\mathbf { A C K T R } + \\mathbf { G A E } + \\mathbf { v w r }$ (i.e., the proposed A2MC). We observe that on average VWR brings greater improvements compared to eligibility traces, and the combination of both (i.e., A2MC) results in consistently good performance across the Atari testbed. ",
+        "bbox": [
+            173,
+            272,
+            825,
+            356
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/07967da42241a720cc821748d81b04ae5e04c2e52c7cff56b270943ae05ee01a.jpg",
+        "image_caption": [
+            "Figure 11: Ablation study of separately applying the (1) the eligibility traces (GAE) and (2) variabilityweighted reward (VWR) to the ACKTR model on the Atari benchmark. We observe that the combination of both (i.e., A2MC) results in consistently good performance across the Atari testbed. The shaded region denotes the standard deviation over 2 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            366,
+            821,
+            626
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "H THE SHARPE RATIO ITSELF ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            441,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We have explored other forms of reward that fits the general idea of introducing variability weights to the reward shaping mechanism. One example is the “Sharpe Ratio” itself, which is defined as $\\begin{array} { r } { r ^ { S R } \\ = \\ \\frac { \\mathbb { E } [ r ] } { \\sigma ( r ) } } \\end{array}$ . In our initial studies, we found it only improved upon the baseline marginally, as rSR could end up emphasizing on penalizing high-variations and it might discourage the agent too intensively (see Figure below). Thats why we have sought an alternative formulation using the proposed $r ^ { v w r }$ and found that $A 2 M C _ { V \\bar { W } R } ~ > ~ A 2 M \\bar { C _ { S R } } ~ > ~ A C K T R$ . An example highlighting the vwr benefit is provided in Appendix A and a more thorough survey on key components in reward designs/formulations will be included in our future works. ",
+        "bbox": [
+            173,
+            132,
+            826,
+            252
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "I ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            286,
+            308,
+            303
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The learning algorithm of A2MC is shown in Algorithm 1. ",
+        "bbox": [
+            173,
+            318,
+            558,
+            333
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/253d4905e0b1e4bda4a2e88e4a8861653c091bb27f95f978262ed8fdd7f93806.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            663,
+            256,
+            816,
+            377
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Advantage Actor Multi-Critic Learning (A2MC) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            390,
+            581,
+            406
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize parameters: $\\theta _ { a } , \\theta _ { v } ^ { j } , j \\in \\{ \\mathrm { s h o r t - t e r m } , \\mathrm { l o n g - t e r m } \\}$   \n2: Initialize look-ahead steps: $N$ , step counter: $T = 0$ , maximum step: $T _ { m a x }$   \n3: Initialize hot-wire probability: $\\epsilon$   \n4: Initialize environment: Env   \n5: Initialize reward history: \\~r   \n6: repeat   \n7: Reset gradients: $d \\theta \\gets 0$ and $d \\theta _ { v } ^ { j } \\gets 0 , j \\in \\{ \\mathrm { s h o r t } \\mathrm { - t e r m } , \\mathrm { l o n g } \\mathrm { - t e r m } \\}$   \n8: Get state: $s _ { t } \\gets E n v$   \n9: $f l a g = 1$ , $a _ { r a n d }$ is uniformly sampled in action space with probability $\\epsilon$ , otherwise $f l a g = 0$   \n10: for $t = 0 : N - 1$ do   \n11: Perform $a _ { t }$ according to policy $\\pi ( a _ { t } | s _ { t } ; \\theta _ { a } )$ if not f lag else $a _ { t } = a _ { r a n d }$   \n12: Received reward $r _ { t }$ and new state $s _ { t + 1 }$ , append $r _ { t }$ to $\\vec { \\bf r }$   \n13: Calculate $r _ { t } ^ { v w r }$ from $\\vec { \\mathbf { r } }$ based on Eq. (2-7)   \n14: $T \\gets T + \\dot { 1 }$   \n15: end for   \n16: $R ^ { \\mathrm { s h o r t - t e r m } } = V ( s _ { N } ; \\theta _ { v } ^ { \\mathrm { s h o r t - t e r m } } )$   \n17: $R ^ { \\mathrm { { l o n g - t e r m } } } = V ( s _ { N } ; \\theta _ { v } ^ { \\mathrm { { l o n g - t e r m } } } )$   \n18: for $i = N - 1$ to 0 step $- 1$ do   \n19: $\\begin{array} { l } { { R _ { } ^ { \\mathrm { s h o r t - t e r m } }  r _ { i } + \\bar { \\gamma } R ^ { \\mathrm { s h o r t - t e r m } } } } \\\\ { { R _ { } ^ { \\mathrm { l o n g - t e r m } }  r _ { i } ^ { v w r } + \\gamma R ^ { \\mathrm { l o n g - t e r m } } } } \\end{array}$   \n20:   \n21: Advantange gradients wrt $\\begin{array} { r } { \\theta _ { a } : d \\theta _ { a } \\gets d \\theta _ { a } + \\nabla _ { \\theta _ { a } } \\log \\pi ( a _ { i } | s _ { i } ; \\theta _ { a } ) \\sum _ { j } ( R ^ { j } - V ( s _ { i } ; \\theta _ { v } ^ { j } ) ) } \\end{array}$   \n22: for $j \\in$ {short-term, long-term} do   \n23: Accumulate gradients wrt $\\dot { \\theta } _ { v } ^ { j } : d \\theta _ { v } ^ { j }  d \\theta _ { v } ^ { j } + \\partial ( R ^ { j } - V ( s _ { i } ; \\theta _ { v } ^ { j } ) ) ^ { 2 } / \\partial \\theta _ { v } ^ { j }$   \n24: end for   \n25: end for   \n26: until $T \\geq T _ { m a x }$ ",
+        "bbox": [
+            173,
+            412,
+            826,
+            781
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/9fd25f5ccbc9cf96954f6fdd9cbc44a1bd0277357795eda6b54b4793ba59308d.jpg",
+        "image_caption": [
+            "Figure 12: Performance of A2MC on Atari games. The number in figure legend shows the average reward among the last 100 episodes upon 15 million timesteps and the percentage shows the performance margin as compared to ACKTR. The shaded region denotes the standard deviation over 2 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            142,
+            815,
+            804
+        ],
+        "page_idx": 18
+    }
+]

parse/train/rHCzkRd0UK/rHCzkRd0UK.md

@@ -0,0 +1,489 @@
+# Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+The ability to compare and align related datasets living in heterogeneous spaces plays an increasingly important role in machine learning. The Gromov-Wasserstein (GW) formalism can help tackle this problem. Its main goal is to seek an assignment (more generally a coupling matrix) that can register points across otherwise incomparable datasets. As a non-convex and quadratic generalization of optimal transport (OT), GW is NP-hard. Yet, heuristics are known to work reasonably well in practice, the state of the art approach being to solve a sequence of nested regularized OT problems. While popular, that heuristic remains too costly to scale, with cubic complexity in the number of samples $n$ . We show in this paper how a recent variant of the Sinkhorn algorithm can substantially speed up the resolution of GW. That variant restricts the set of admissible couplings to those admitting a low rank factorization as the product of two sub-couplings. By updating alternatively each sub-coupling, our algorithm computes a stationary point of the problem in quadratic time with respect to the number of samples. When cost matrices have themselves low rank, our algorithm has time complexity ${ \mathcal { O } } ( n )$ . We demonstrate the efficiency of our method on simulated and real data.
+
+# 17 1 Introduction
+
+18 The ever increasing interest for Gromov-Wasserstein... Several problems in machine learning   
+19 involve comparing families of points that live in heterogeneous spaces. This situation arises typically   
+20 when realigning two distinct sets of feature representations obtained from the similar source. Recent   
+21 applications to single-cell genomics [15] and NLP [12, 1] provide two cases in point: Thousands   
+22 of cells taken from the same tissue are split in two groups, each group is processed with a different   
+23 experimental protocol, resulting in two distinct sets of heterogeneous feature vectors; Thousands of   
+24 word embeddings for two languages are learned independently. In both cases, one expects to find   
+25 a meaningful way to register points across sets living in heteregeneous spaces, since they contain   
+26 similar overall information. That realignment is usually carried out using the Gromov-Wasserstein   
+27 (GW) machinery proposed by Mémoli [26] and Sturm [36], which seeks a relaxed assignment matrix   
+28 that is as “close” to an isometry as possible, using a quadratic score to quantify that closeness. GW   
+29 has a lot of practical appeal: It has been used in supervised learning [41], generative modeling [7],   
+30 domain adaptation [9], structured prediction [37], quantum chemistry [27] and alignment layers [17].   
+31 ... despite its cubic cost. Because it is an NP-hard problem, these applications rely on approximating   
+32 GW, typically by solving a sequence of OT problems using entropic regularization. This heuristic is   
+33 efficient yet costly, since it requires $\mathcal { O } ( n ^ { 3 } )$ operations to register two sets of $n$ samples, a price that is   
+34 paid when re-instantiating each OT problem. Our goal is to reduce substantially that complexity by   
+35 exploiting low-factorization of both parameters (data) and variable (relaxed assignment) matrices in   
+36 the GW problem, while maintaining state of the art performance in applications.   
+37 Wasserstein: from cubic to linear complexity. A comparatively simpler problem is the registration   
+38 of two populations embedded in the same space. This corresponds to the classic optimal transport   
+39 (OT) problem, which has received considerable attention in ML [28]. OT has found applications   
+40 in computer vision [29], NLP [24], single cell tracking [33] or multi-task regression in neuro  
+41 imaging [22]. While the OT problem is originally cast as a linear program, with a ${ \overline { { O ( n ^ { 3 } \log ( n ) } } } )$ cost,   
+42 many of these works rely on solving instead a penalized OT problem using Sinkhorn’s algorithm [34,   
+43 13]. In its most naive implementation, the Sinkhorn has quadratic complexity [2]. Recent works   
+44 achieve $O ( n )$ complexity by targeting the matrix-vector updates in Sinkhorn’s algorithm using   
+45 low-rank approximations of the data kernel matrix [4, 3, 31]. This idea can be further improved by   
+46 imposing the low-rank constraint on the optimization variables of the original OT problem [19], to   
+47 modify Sinkorn’s steps by enforcing a low rank factorization of the coupling variable [32].   
+48 Gromov-Wasserstein: from NP-hard to linear approximations. The GW problem replaces   
+49 the linear objective function in OT by a non-convex quadratic objective. Much like OT is a re  
+50 laxation of the optimal assignment problem, GW can be seen as a relaxation of the quadratic   
+51 assignment problem (QAP). Both GW and QAP are NP-hard to solve [8]. In practice, iteratively   
+52 minimizing a linearization of that quadratic objective using Sinkhorn works surprisingly well [20, 35].   
+53 This method corresponds to a mirror-descent scheme [27],   
+54 and in the special case of Euclidean distance matrices, the   
+55 loss is concave and it can be also interpreted as a bi-linear   
+56 relaxation [23]. In the most general case, this results in an   
+57 $O ( n ^ { 4 } )$ algorithm (the objective is a quadratic function of a   
+58 $n \times n$ relaxed assignment matrix), that is reduced to $O ( n ^ { 3 } )$   
+59 when using separable losses [27], a price that remains too   
+60 high for several ML applications. It is possible to replace   
+61 the GW distance by cheaper yet only distantly related prox  
+62 ies, such as lower bounds based on OT [26] (see also [30])   
+63 or sliced projections [38]. Whether GW can be efficiently   
+64 sped up remains an open question. We propose in this   
+65 work a novel approach that leverages, as done recently for   
+66 OT, low-rank methods. A very recent line of works attacks   
+67 this problem by quantizing first the two input spaces to   
+68 solve a GW problem of reduced size, thus effectively pro  
+69 ducing an ad-hoc low-rank coupling [11]. A nice feature   
+70 of this approach is that it maintains the triangular inequal  
+71 ity and provides a valid upper-bound on the GW distance.   
+72 Related approaches which also approximate GW distance   
+73 using clustering methods (possibly in a recursive way)   
+74 are [6] and [40]. We take in this paper a direct approach:   
+75 instead of separating clustering and GW resolution in 2   
+76 independent steps, we propose do address them simulta  
+77 neously: our method seeks the least-costly (in GW sense)   
+78 coupling with a low rank constraint, as illustrated in Fig. 1.   
+79 Contributions We introduce the low-rank-GW problem, by imposing a low rank constraint on   
+80 feasible couplings. This method works hand-in-hand with entropic regularization and leads to a   
+81 Sinkhorn-like algorithm. Because of its exclusive reliance on matrix-vector products, the method   
+82 streams well on GPUs. This method can also leverage low-rank factorizations of the input data   
+83 matrices to further reduce the complexity of each iteration to reach linear time. Numerical evaluations   
+84 on simulated and real datasets show that this low-rank approximation maintains the favorable   
+85 property of entropic-regularized GW (namely its ability to compute “good” local minima) for a linear   
+86 computational price, thus paving the way for larger scale uses of GW in ML.
+
+![](images/75f07794a62ec3be032add0ebc59f95a5f74a2fe87bba8885a69e073ad67fc1a.jpg)  
+Figure 1: Top row: we compute the GW coupling between two curves in 2D and 3D, with $n = m = 1 0 0 0 0$ points. These points are endowed with the squared L2 distance. Bottom row: coupling obtained with the SoTA entropic approach [20, 27], compared with our linear method with rank $r = 1 0$ . See Appendix D.1 for more details.
+
+# 87 2 Background on the Gromov-Wasserstein Framework
+
+88 Comparing measured metric spaces. Let $( \mathcal { X } , d _ { \mathcal { X } } )$ and $( \mathcal { V } , d _ { \mathcal { V } } )$ be two metric spaces, and $\mu$ and   
+89 $\nu$ two discrete probability measures on $\mathcal { X }$ and $\mathcal { V }$ , respectively. We write $\textstyle \mu : = \sum _ { i = 1 } ^ { n } a _ { i } \delta _ { x _ { i } }$ and   
+90 $\begin{array} { r } { \nu : = \sum _ { i = j } ^ { m } b _ { j } \delta _ { y _ { j } } } \end{array}$ where $n , m \geq 1 , a , b$ are two histograms in the probability simplicies $\Delta _ { n } , \Delta _ { m }$ of   
+91 respective size $n$ and $m$ , and $( x _ { 1 } , \ldots , x _ { n } )$ , $\left( y _ { 1 } , \ldots , y _ { m } \right)$ are two families in $\mathcal { X }$ and $\mathcal { V }$ . For $q \geq 1$ ,   
+92 let us also denote $A : = ( d _ { \mathcal { X } } ^ { q } ( x _ { i } , x _ { i ^ { \prime } } ) ) _ { 1 \leq i , i ^ { \prime } \leq n } \in \mathbb { R } ^ { n \times n }$ and $B : = ( d _ { \mathcal { V } } ^ { q } ( x _ { j } , x _ { j ^ { \prime } } ) ) _ { 1 \leq i , i ^ { \prime } \leq m } \in \mathbb { R } ^ { m \times m }$   
+93 X Ytwo pairwise cost matrices between the points in the respective supports of $\mu$ and $\nu$ . The Gromov  
+94 Wasserstein (GW) discrepancy between two discrete metric measure spaces $( \mu , d _ { \mathcal { X } } )$ and $( \nu , d _ { 3 } )$ is   
+95 the solution of the following non-convex quadratic problem, instantiated here for simplicity as a   
+96 function of $( a , A )$ and $( b , B )$ , which contain all the information that is needed:
+
+$$
+\mathbf { G } \mathbf { W } ( ( a , A ) , ( b , B ) ) = \operatorname* { m i n } _ { P \in \Pi _ { a , b } } \mathcal { E } _ { A , B } ( P ) , \mathrm { w h e r e ~ } \Pi _ { a , b } : = \{ P \in \mathbb { R } _ { + } ^ { n \times m } | P \mathbf { 1 } _ { m } = a , P ^ { T } \mathbf { 1 } _ { n } = b \} ,
+$$
+
+97 and the energy $\mathcal { E } _ { A , B }$ is a quadratic function parameterized by a loss $L : \mathbb { R } \times \mathbb { R } \to \mathbb { R }$ :
+
+$$
+\mathcal { E } _ { A , B } ( P ) : = \sum _ { i , j , i ^ { \prime } , j ^ { \prime } } L ( A _ { i , i ^ { \prime } } , B _ { j , j ^ { \prime } } ) P _ { i , j } P _ { i ^ { \prime } , j ^ { \prime } } \ .
+$$
+
+98 A typical choice of the loss is the $L ^ { p }$ distance $L ( a , b ) = | a - b | ^ { p }$ with $p \geq 1$ . In that case, [26]   
+99 proves that $\mathrm { G W } ^ { 1 / p }$ defines a distance on the space of metric measure spaces quotiented by measure  
+100 preserving isometries. When $p = 2$ , as we consider from now on, the GW objective can be evaluated   
+101 efficiently using the marginal constraints imposed on $P$ , as follows [27]:
+
+$$
+\begin{array} { r } { \mathcal { E } _ { A , B } ( P ) = \langle A ^ { \odot 2 } a , a \rangle + \langle B ^ { \odot 2 } b , b \rangle - 2 \langle A P B , P \rangle . } \end{array}
+$$
+
+02 Indeed, (3) can be computed efficiently in $\mathcal { O } ( n ^ { 2 } m + n m ^ { 2 } )$ operations, using only matrix/matrix multiplications, instead of the 103 $\mathcal { O } ( n ^ { 2 } m ^ { 2 } )$ complexity of the naive evaluation of (2).
+
+104 Entropic Gromov-Wasserstein. The original GW problem (1) can be regularized using an entropic   
+105 term [20, 35, 27], leading to the following problem:
+
+$$
+\mathbf { G } \mathbf { W } _ { \varepsilon } ( ( a , A ) , ( b , B ) ) = \operatorname* { m i n } _ { P \in \Pi _ { a , b } } \mathcal { E } _ { A , B } ( P ) - \varepsilon H ( P ) ,
+$$
+
+106 where $\begin{array} { r } { H ( P ) ~ : = ~ - \sum _ { i , j } P _ { i , j } ( \log ( P _ { i , j } ) ~ - ~ 1 ) } \end{array}$ is the entropy of $P$ . By applying a Mirror   
+107 Descent (MD) scheme with respect to the KL divergence and by choosing the step-size to   
+108 be $\gamma ~ = ~ 1 / \varepsilon$ , Peyré et al. [27] provide a simple algorithm which consists in solving a se  
+109 quence of regularized OT problem as presented in Algorithm 1. Indeed, each KL pro  
+110 jection in Algorithm 1 can be computed efficiently thanks to the Sinkhorn algorithm [13].   
+112 Computational complexity. Given a cost matrix $C$ , the   
+113 KL projection of $K _ { \varepsilon }$ onto the polytope $\textstyle \prod ( a , b )$ , where   
+114 ${ \mathrm { K L } } ( { \bar { P } } , { \bar { Q } } ) = \langle P , \log ( P / Q ) - 1 \rangle$ , is carried out in the inner   
+115 loop of Algo. 1 using the Sinkhorn algorithm, through   
+116 matrix-vector products. This quadratic complexity (in   
+117 red) is dominated by the cost of updating matrix $C$ at each   
+118 iteration in Algorithm 1, which requires $\mathcal { O } ( n ^ { 2 } m + n m ^ { 2 } )$   
+119 algebraic operations (cubic, in violet). As noted above,   
+120 evaluating the objective $\mathcal { E } _ { A , B } ( \boldsymbol { P } )$ has the same order. In   
+121 the following we show that by considering a low rank   
+122 exact decomposition (or approximation) of the distance   
+123 matrices, the cubic cost of reupdating $C$ and subsequently   
+124 evaluating $\mathcal { E } _ { A , B }$ can be brought down to quadratic.
+
+# Algorithm 1 Entropic-GW
+
+# 25 3 Exploiting a Low-Rank Factorization for Cost Matrices
+
+Exact factorization of cost matrices. In this section we consider the case where the cost matrices $A$ and $B$ admit a low-rank factorization. More precisely, we make the following assumption.
+
+Assumption 1. Assume that $A$ and $B$ admit a low-rank factorization, that is there exists $A _ { 1 } , A _ { 2 } \in$ $\mathbb { R } ^ { n \times d }$ and $B _ { 1 } , B _ { 2 } \in \mathbb { R } ^ { m \times d ^ { \prime } }$ such that $A = A _ { 1 } A _ { 2 } ^ { T }$ and $B = B _ { 1 } B _ { 2 } ^ { T }$ , where $d \ll n , d ^ { \prime } \ll m$ .
+
+130 A case in point is when both $A$ and $B$ are squared Euclidean distance matrices, with a sample size   
+131 that is larger than ambient dimension. This case is highly relevant, covering many applications of OT   
+132 to ML. The $d \ll n$ assumption is also likely to hold for most applications, since cases where $d \gg n$   
+133 are known to pose challenges to the estimation of OT [16, 39]. Writing $X = [ x _ { 1 } , \ldots , x _ { n } ] \in \mathbb { R } ^ { d \times n }$ , if
+
+134 $A = \left[ \Vert x _ { i } - x _ { j } \Vert _ { 2 } ^ { 2 } \right] _ { i , j }$ , then one has, writing $z = ( X ^ { \odot 2 } ) ^ { T } \mathbf { 1 } _ { d } \in \mathbb { R } ^ { n }$ that $A = z \mathbf { 1 } _ { n } ^ { T } + \mathbf { 1 } _ { n } z ^ { T } - 2 X ^ { T } X$ . Therefore by denoting 135 $A _ { 1 } = [ z , \mathbf { 1 } _ { n } , - \sqrt { 2 } X ^ { T } ] \in \mathbb { R } ^ { n \times ( d + 2 ) }$ and $A _ { 2 } = [ \mathbf { 1 } _ { n } , z , \sqrt { 2 } X ^ { T } ] \in \mathbb { R } ^ { n \times ( d + 2 ) }$ 136 we obtain the factorization above.
+
+137 Under Assumption 1, the complexity of Algo. 1 is downgraded to quadratic in sample size: the two   
+138 operations that make Algo. 1 cubic lie in the updates of the cost and the computation of the objective.   
+139 Observe that for any given $P \in \mathbb { R } ^ { n \times m }$ , one can compute at each iteration
+
+$$
+\begin{array} { r } { C = - 4 A _ { 1 } A _ { 2 } ^ { T } P B _ { 1 } B _ { 2 } ^ { T } } \end{array}
+$$
+
+140 in $n m ( d + d ^ { \prime } ) + d d ^ { \prime } ( n + m )$ algebraic operations. Moreover thanks to the reformulation of   
+141 $\mathcal { E } _ { A , B } ( \boldsymbol { P } )$ given in (3), one can compute it in quadratic time as well. Indeed writing $G _ { 1 } : =$   
+142 $A _ { 1 } ^ { T } P B _ { 2 }$ and $G _ { 2 } : = A _ { 2 } ^ { T } P B _ { 1 }$ , both in $\mathbb { R } ^ { d \times d ^ { \prime } }$ , one has $\langle A P B , P \rangle = \mathbf { 1 } _ { d } ^ { T } ( G _ { 1 } \odot G _ { 2 } ) \mathbf { 1 } _ { d ^ { \prime } }$ . Com  
+143 puting $G _ { 1 } , G _ { 2 }$ given $P$ requires only $2 ( n m d + m d d ^ { \prime } )$ , and computing their dot product adds   
+144 $d d ^ { \prime }$ algebraic operations. The overall complexity to compute $\mathcal { E } _ { A , B } ( { \cal P } )$ is $\mathcal { O } ( n m d + m d d ^ { \prime } )$ .   
+146 General distance matrices. When the original   
+147 cost matrices $A$ , are not low-rank but describe   
+148 distances, we propose to use a recent body of   
+149 work that output their low-rank approximation   
+150 in linear time [5, 21]. These algorithms produce,   
+151 for any distance matrix $D \in \bar { \mathbb { R } } ^ { n \times m }$ and $\gamma > 0$ ,   
+152 matrices $D _ { 1 } \in \mathbb { R } ^ { n \times d }$ , $D _ { 2 } \in \mathbb { R } ^ { m \times d }$ in $\mathcal { O } ( ( m +$   
+153 $\scriptstyle n ) \mathtt { p o l y } ( { \frac { d } { \gamma } } ) )$ algebraic operations such that with   
+154 probability at least 0.99 one has
+
+# Algorithm 2 Quadratic Entropic-GW
+
+$$
+\| D - D _ { 1 } D _ { 2 } ^ { T } \| _ { F } ^ { 2 } \leq \| D - C _ { d } \| _ { F } ^ { 2 } + \gamma \| D \| _ { F } ^ { 2 }
+$$
+
+155 where $C _ { d }$ denotes the best rank- $d$ approximation   
+156 to $D$ . We fall back on this approach to obtain a   
+157 low-rank factorization of a distance matrix in lin  
+158 ear time whenever needed, aware that this incurs   
+159 an additional approximation. See Appendix B   
+160 for more details.
+
+# end
+
+$$
+\begin{array} { r l } & { c _ { 1 } \gets \langle A ^ { \odot 2 } a , a \rangle + \langle B ^ { \odot 2 } b , b \rangle \quad \mathcal { O } ( \mathrm { n r } } \\ & { G _ { 2 } \gets A _ { 2 } ^ { T } P B _ { 1 } \quad \mathrm { n m d ~ + ~ m d d } ^ { \flat } , } \\ & { G _ { 1 } \gets A _ { 1 } ^ { T } P B _ { 2 } \quad \mathrm { n m d ~ + ~ m d d } ^ { \flat } } \\ & { c _ { 2 } \gets - 2 \mathbf { 1 } _ { d } ^ { T } ( G _ { 1 } \odot G _ { 2 } ) \mathbf { 1 } _ { d ^ { \prime } } \quad \mathcal { O } ( \mathrm { d d } ^ { \flat } ) } \\ & { \xi _ { A , B } ( P ) \gets c _ { 1 } + c _ { 2 } } \\ & { \mathbf { R e t u r n } \colon \mathcal { E } _ { A , B } ( P ) } \end{array}
+$$
+
+# 161 4 Imposing a Low Nonnegative Low-Rank for the Coupling
+
+In this section, we shift our attention to a different opportunity for speed-ups, without assuming that Assumption 1 holds: we regularize the GW problem problem by decomposing the coupling as a product of two low-rank couplings, in the footsteps of [18, 32], using the following definition:
+
+165 Definition 1. Given $M \in \mathbb { R } ^ { n \times m }$ , the nonnegative (NN) rank of $M$ is the smallest number of   
+166 nonnegative rank-one matrices into which the matrix can be decomposed additively:
+
+$$
+\operatorname { r k } _ { + } ( M ) : = \operatorname* { m i n } \left\{ q | M = \sum _ { i = 1 } ^ { q } R _ { i } , \forall i , \operatorname { r k } ( R _ { i } ) = 1 , R _ { i } \geq 0 \right\} .
+$$
+
+167 Following [18, 32], we propose to constrain GW, enforcing a rank $r$ on the coupling:
+
+$$
+\mathrm { G W } \mathrm { L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) : = \operatorname* { m i n } _ { P \in \Pi _ { a , b } ( r ) } \xi _ { A , B } ( P ) , \mathrm { ~ w h e r e ~ } \Pi _ { a , b } ( r ) : = \{ P \in \Pi _ { a , b } , \mathrm { r k } _ { + } ( P ) \leq r \} \ : .
+$$
+
+168 Note that the minimum is always attained as $\Pi _ { a , b } ( r )$ is compact and the objective is continuous.   
+169 In [32], the authors show that one can parameterize any coupling in $\Pi _ { a , b } ( r )$ as a product of two   
+170 low-rank couplings linked by a common marginal. For any $g \in \Delta _ { r } ^ { * }$ , the interior of $\Delta _ { r }$ , writing
+
+$$
+\Pi _ { a , g , b } : = \Bigl \{ P \in \mathbb { R } _ { + } ^ { n \times m } , P = Q \mathrm { d i a g } ( 1 / g ) { R } ^ { T } , Q \in \Pi _ { a , g } , \mathrm { a n d } R \in \Pi _ { b , g } \Bigr \} .
+$$
+
+one has that 171 $\begin{array} { r } { \bigcup _ { g \in \Delta _ { r } ^ { * } } \Pi _ { a , g , b } = \Pi _ { a , b } ( r ) } \end{array}$ . Therefore GW-LR introduced in (5) can be reformulated as 172 the following optimization problem
+
+$$
+\mathrm { G W - L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) = \operatorname* { m i n } _ { ( Q , R , g ) \in { \mathcal C } ( a , b , r ) } { \mathcal E } _ { A , B } ( Q \mathrm { d i a g } ( 1 / g ) R ^ { T } )
+$$
+
+173 where $\mathcal { C } ( a , b , r ) : = \mathcal { C } _ { 1 } ( a , b , r ) \cap \mathcal { C } _ { 2 } ( r )$ , with
+
+$$
+\begin{array} { r } { \mathcal { C } _ { 1 } ( a , b , r ) : = \Big \{ ( Q , R , g ) \in \mathbb { R } _ { + } ^ { n \times r } \times \mathbb { R } _ { + } ^ { m \times r } \times \big ( \mathbb { R } _ { + } ^ { * } \big ) ^ { r } \mathrm { s . t . } Q \mathbf { 1 } _ { r } = a , R \mathbf { 1 } _ { r } = b \Big \} , } \\ { \mathcal { C } _ { 2 } ( r ) : = \Big \{ ( Q , R , g ) \in \mathbb { R } _ { + } ^ { n \times r } \times \mathbb { R } _ { + } ^ { m \times r } \times \mathbb { R } _ { + } ^ { r } \mathrm { s . t . } Q ^ { T } \mathbf { 1 } _ { n } = R ^ { T } \mathbf { 1 } _ { m } = g \Big \} . } \end{array}
+$$
+
+174 Stabilization of the Method. [32] propose to stabilize the objective defined in (6) by adding to the   
+175 constraints a lower bound $\alpha$ on the weight vector $g$ such that $g \geq \alpha$ coordinate-wise. Indeed, as   
+176 a solution of (6) must satisfies $g > 0$ coordinate-wise, then for $\alpha$ sufficiently small, the solution   
+177 of the same problem where one adds the constraint $g \geq \alpha$ will remain the same. Therefore let us   
+178 introduce our new set of constraints $\mathcal { C } ( a , b , r , \alpha ) : = \mathcal { C } _ { 1 } ( a , b , r , \alpha ) \cap \mathcal { C } _ { 2 } ( r )$ where $\mathcal { C } _ { 1 } ( a , b , r , \alpha ) : =$   
+179 $\mathcal { C } _ { 1 } ( a , b , r ) \cap \{ ( Q , R , g ) \mid g \geq \alpha \}$ . Another way to stabilize the method is by considering a double   
+180 regularization scheme as proposed in [32] where in addition of constraining the nonnegative rank   
+181 of the coupling, we regularize the objective by adding an entropic term in $( Q , R , g )$ , which is to be   
+182 understood as that of the values of the three respective entropies evaluated for each term.
+
+$$
+\mathrm { G W } \mathrm { L R } _ { \varepsilon , \alpha } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) : = \operatorname* { m i n } _ { ( Q , R , g ) \in { \mathscr C } ( a , b , r , \alpha ) } { \mathscr E } _ { A , B } ( Q \mathrm { d i a g } ( 1 / g ) R ^ { T } ) - \varepsilon H ( ( Q , R , g ) ) ~ .
+$$
+
+183 Mirror Descent Scheme. As in [27], we propose to use a MD scheme with respect to the $\mathrm { K L }$   
+184 divergence to approximate $\mathbf { G } \mathbf { W } \mathbf { - L R } _ { \varepsilon , \alpha } ^ { ( r ) }$ in (7). More precisely, for any $\varepsilon \geq 0$ , the MD scheme leads   
+185 for all $k \geq 0$ to the following updates which require solving a convex barycenter problem per step:
+
+$$
+( Q _ { k + 1 } , R _ { k + 1 } , g _ { k + 1 } ) : = \underset { \zeta \in \mathcal { C } ( a , b , r , \alpha ) } { \mathrm { a r g m i n ~ } } \mathrm { K L } ( \zeta , K _ { k } )
+$$
+
+186 where $( Q _ { 0 } , R _ { 0 } , g _ { 0 } ) \in \mathcal { C } ( a , b , r )$ is an initial point such that $ { Q _ { 0 } } \ > \ 0$ and $\begin{array} { r l r } { R _ { 0 } } & { { } > } & { 0 } \end{array}$ ,   
+187 $P _ { k } : = Q _ { k } \mathrm { d i a g } ( 1 / g _ { k } ) R _ { k } ^ { T }$ $\mathring { \mathbf { \ i } } _ { k } , \ \mathbf { K } _ { k } : = \big ( \boldsymbol { K } _ { k } ^ { ( 1 ) } , \boldsymbol { K } _ { k } ^ { ( 2 ) } , \boldsymbol { K } _ { k } ^ { ( 3 ) } \big ) , \ \boldsymbol { K } _ { k } ^ { ( 1 ) } : = \mathrm { e x p } ( 4 \gamma A P _ { k } B R _ { k } \operatorname { d i a g } ( 1 /$   
+188 $( \gamma \varepsilon \mathrm { ~ - ~ } 1 ) \log ( Q _ { k } ) )$ , $\begin{array} { r c l } { K _ { k } ^ { ( z ) } } & { : = } & { \exp ( 4 \gamma B P _ { k } ^ { T } D Q _ { k } \mathrm { d i a g } ( 1 / g _ { k } ) - ( \gamma \varepsilon - 1 ) \log ( R _ { k } ) ) } \end{array}$ , $K _ { k } ^ { ( 3 ) } \ : =$   
+189 $\exp ( - 4 \gamma \omega _ { k } / g _ { k } ^ { 2 } - ( \gamma \varepsilon - 1 ) \log ( g _ { k } ) )$ with $[ \omega _ { k } ] _ { i } : = [ Q _ { k } ^ { T } A P _ { k } B R _ { k } ] _ { i , i }$ for all $i \in \{ 1 , \ldots , r \}$ and   
+190 $\gamma$ is a positive step size. Solving (8) can be done efficiently thanks to the Dykstra’s Algorithm as   
+191 showed in [32]. See Appendix $\textrm { C }$ for more details.
+
+Initialization. To initialize our algorithm, we adapt the First Lower Bound of [26] to our case of interest. More precisely, we show the following Proposition. See appendix A for the proof.
+
+194 Proposition 1. Let us denote $\tilde { x } = A ^ { \odot 2 } a \in \mathbb { R } ^ { n }$ , $\tilde { y } = B ^ { \odot 2 } b \in \mathbb { R } ^ { m }$ and $\tilde { C } = ( | \tilde { x } _ { i } - \tilde { y } _ { j } | ^ { 2 } ) _ { i , j } \in \mathbb { R } ^ { n \times m }$   
+195 Then for all $\varepsilon \geq 0$ and $r \geq 1$ we have,
+
+$$
+\mathrm { G W - L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) \geq \operatorname* { m i n } _ { ( Q , R , g ) \in { \mathcal C } ( a , b , r , \alpha ) } \langle \tilde { C } , Q \mathrm { d i a g } ( 1 / g ) R ^ { T } \rangle - \varepsilon H ( ( Q , R , g ) ) ~ .
+$$
+
+196 Note that the RHS of the inequality (9) is exactly the problem studied in [32] for which an algorithm   
+197 was proposed. Therefore to initialize our algorithm, we propose to use their approach. Note that here   
+198 the cost $\tilde { C }$ is the squared Euclidean distance between two families $\{ \tilde { x } _ { 1 } , \ldots , \tilde { x } _ { n } \}$ and $\{ \tilde { y } _ { 1 } , \dots , \tilde { y } _ { m } \}$   
+199 in 1-D which admits a low-rank factorization. Therefore we can apply the linear-time version of the   
+200 algorithm presented in [32] to compute the solution. Algorithm 3 summarizes our approach, where   
+201 $\mathcal { D } ( \cdot )$ denotes the operator extracting the diagonal of a square matrix.   
+202 Computational Cost. Computing the initialization goes through the computations of $\tilde { x }$ and $\tilde { y }$ which   
+203 requires $O ( n ^ { 2 } + m ^ { 2 } )$ algebraic operations. Moreover, applying the algorithm proposed in [32]   
+204 when the underlying cost is the squared Euclidean distances between two families in 1-D needs   
+205 only $\mathcal { O } ( ( n + m ) r )$ algebraic operations. Solving the barycenter problem as defined in (8) can be   
+206 207 done given $( K _ { k } ^ { ( 1 ) } , \dot { K } _ { k } ^ { ( 2 ) } , K _ { k } ^ { ( 3 ) } )$ ) Dykstra’s Algorithm. Indeed in [32, Algorithm, each iteration of their algorithm requires only $\mathcal { O } ( ( n + m ) r )$ show thatalgebraic   
+208 operations since it involves only matrix/vector multiplications. However computing the kernel   
+209 matrices $( K _ { k } ^ { ( 1 ) } , K _ { k } ^ { ( 2 ) } , K _ { k } ^ { ( 3 ) } )$ at each iteration of Algorithm 3 requires a quadratic complexity with   
+210 respect to the number of samples. Overall the proposed algorithm, while faster than the cubic   
+211 implementation proposed in [27], still needs $\mathcal { O } ( ( n ^ { 2 } + m ^ { 2 } ) r )$ operations per iteration. In the following   
+212 we will see that by combining both nonnegative low-rank constraints on the coupling and low-rank   
+213 approximations of the distance matrices, we can obtain a linear time algorithm with respect to the   
+214 number of samples which computes an approximation of the GW distance.
+
+$\mathrm { A l g o r i t h m } 3 \mathrm { L o w - R a n k } \mathrm { G W } , \mathrm { G W - L R } _ { \varepsilon , \alpha } ^ { ( r ) } ( ( a , A ) , ( b , B ) )$
+
+for $k = 1 , \dots$ do
+
+# end
+
+Return:
+
+215 Convergence of the mirror descent. Even if the objective (7) is not convex in $( Q , R , g )$ , we obtain   
+216 the non-asymptotic stationary convergence of the MD algorithm in this setting. For that purpose   
+217 we consider the same convergence criterion as the one proposed in [32] to obtain non-asymptotic   
+218 stationary convergence of the MD scheme defined as
+
+$$
+\Delta _ { \varepsilon , \alpha } ( \pmb { \xi } , \gamma ) : = \frac { 1 } { \gamma ^ { 2 } } ( \mathrm { K L } ( \pmb { \xi } , \mathcal { G } _ { \varepsilon , \alpha } ( \pmb { \xi } , \gamma ) ) + \mathrm { K L } ( \mathcal { G } _ { \varepsilon , \alpha } ( \pmb { \xi } , \gamma ) , \pmb { \xi } ) )
+$$
+
+where 219 $\begin{array} { r } { \mathcal { G } _ { \varepsilon , \alpha } ( \pmb { \xi } , \gamma ) : = \mathrm { a r g m i n } _ { \zeta \in \mathcal { C } ( a , b , r , \alpha ) } \{ \langle \nabla \mathcal { E } _ { A , B } ( \pmb { \xi } ) , \pmb { \zeta } \rangle + \frac { 1 } { \gamma } \mathrm { K L } ( \zeta , \pmb { \xi } ) \} } \end{array}$ . For any $1 / r \ge \alpha > 0$ , we 220 show in the following proposition the non-asymptotic stationary convergence of the MD scheme 221 applied to the problem (7). See Appendix A for the proof.
+
+222 Proposition 2. Let $\varepsilon \geq 0$ , $\textstyle { \frac { 1 } { r } } \geq \alpha > 0$ and $N \geq 1$ . By denoting $L _ { \varepsilon , \alpha } : = 2 7 ( \| A \| _ { 2 } \| B \| _ { 2 } / \alpha ^ { 4 } + \varepsilon )$ and by considering a constant stepsize in the MD scheme 223 $\begin{array} { r } { { \bf \Phi } ^ { \left( 8 \right) } \gamma = \frac { 1 } { 2 L _ { \varepsilon , \alpha } } } \end{array}$ , we obtain that
+
+$$
+\operatorname* { m i n } _ { 1 \leq k \leq N } \Delta _ { \varepsilon , \alpha } ( ( Q _ { k } , R _ { k } , g _ { k } ) , \gamma ) \leq \frac { 4 L _ { \varepsilon , \alpha } D _ { 0 } } { N } .
+$$
+
+where 224 $D _ { 0 } : = \mathscr { E } _ { A , B } ( Q _ { 0 } \mathrm { d i a g } ( 1 / g _ { 0 } R _ { 0 } ^ { T } ) - \mathbf { G } \mathbf { W } \mathbf { - } \mathbf { L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) )$ is the distance of the initial value 225 to the optimal one.
+
+Recall that for $\alpha$ sufficiently small, we have $\mathrm { { \bf G W - L R } } _ { \varepsilon , \alpha } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) = { \bf G W - L R } _ { \varepsilon } ^ { ( r ) } ( ( a , A ) , ( b , B ) )$ . Thus Proposition 2 show that our algorithm reach a stationary point of (7). In particular, if $\varepsilon = 0$ , the proposed algorithm converges towards a stationary point of (5).
+
+# 229 5 Double Low-rank Approach for Linear Time GW
+
+Almost all operations in Algorithm 3 are linear time, except for the three updates highlighted in red, involving $C _ { 1 }$ and $C _ { 2 }$ , and the computations of $\tilde { x } = A ^ { \odot \tilde { 2 } } a$ and $\tilde { y } = B ^ { \odot 2 } b$ as they still require a quadratic number of algebraic operations. When adding Assumption 1 from $\ S 3$ to the rank constrained approach from $\ S 4$ , we notice that the strengths of both approaches can work hand in hand, both in easier initial evaluations of $\tilde { x } , \tilde { y }$ , but, most importantly, at each new recomputation of a factorized linearization of the quadratic objective:
+
+Linear time outfactorization for rms. Because . Indeed, rema $A$ admits a that for a low-rank. Therefore $A ^ { \odot 2 }$ $\begin{array} { r } { \boldsymbol { x } , \boldsymbol { y } \in \mathbb { R } ^ { d } , \langle \boldsymbol { x } , \boldsymbol { y } \rangle ^ { 2 } = \sum _ { i , j = 1 } ^ { d } x _ { i } x _ { j } y _ { i } y _ { j } } \end{array}$
+
+by studying the rows of matrices $A _ { 1 } : = [ a _ { 1 } ^ { ( 1 ) } ; . . . ; a _ { n } ^ { ( 1 ) } ]$ and $A _ { 2 } : = [ a _ { 1 } ^ { ( 2 ) } ; . . . ; a _ { n } ^ { ( 2 ) } ]$ , if one writes $\psi ( \boldsymbol { x } ) : = \mathrm { V e c t } ( \boldsymbol { x } \boldsymbol { x } ^ { T } ) \in \mathbb { R } ^ { d ^ { 2 } }$ where $\mathrm { V e c t } ( \cdot )$ is the vectorization operation, we obtain that
+
+$$
+\begin{array} { r } { A ^ { \odot 2 } = \tilde { A } _ { 1 } \tilde { A _ { 2 } } ^ { T } \mathrm { ~ w h e r e ~ } \tilde { A _ { 1 } } = [ \psi ( a _ { 1 } ^ { ( 1 ) } ) , \dots , \psi ( a _ { n } ^ { ( 1 ) } ) ] ^ { T } , \tilde { A _ { 2 } } = [ \psi ( a _ { 1 } ^ { ( 2 ) } ) , \dots , \psi ( a _ { n } ^ { ( 2 ) } ) ] ^ { T } \ . } \end{array}
+$$
+
+In Algorithm 3, the line “Step 236 $( \star ) ^ { \dagger }$ can thus be replaced by $\tilde { x } \gets \tilde { A _ { 1 } } \tilde { A _ { 2 } } ^ { T } a$ and $\tilde { y } \gets \tilde { B _ { 1 } } \tilde { B _ { 2 } } ^ { T } b$ Note 237 that computing $\tilde { A } _ { 1 }$ given $A _ { 1 }$ requires only $\mathcal { O } ( n d ^ { 2 } )$ operations, so that this alternate code only takes 238 $\mathcal { O } ( n d ^ { 2 } ) \bar { + } \mathcal { O } ( \bar { m } ( d ^ { \prime } ) ^ { 2 } )$ operations.
+
+239 Linear time linearization of the GW objective. The linearization step, the critical step in Algo.1   
+240 that consists in updating $C$ at each iteration, consumes a substantial portion of the computational   
+241 budget of GW. Introducing the low-rank Sinkhorn approach makes this step quadratic in Algo.3; the   
+242 complexity of that step is also quadratic using the low-rank assumption on costs $A$ and $B$ , in Algo.2.   
+243 There is therefore an opportunity to marry both to speed-up that important step. We argue that this is   
+244 indeed what happens, in the sense that combining the two yields indeed linear time complexities in   
+245 sample sizes, by replacing in Algorithm 3, the lines “Step $( \star \star ) ^ { \flat }$ b y
+
+$$
+{ \cal C } _ { 1 } \gets - A _ { 1 } A _ { 2 } ^ { T } Q \mathrm { d i a g } ( 1 / g ) \quad \mathrm { a n d } \quad { \cal C } _ { 2 } \gets R ^ { T } B _ { 2 } B _ { 1 } ^ { T } .
+$$
+
+246 Note that this speed-up would not be achieved using other approaches that output a low rank   
+247 approximation of the transport plan [4, 3, 31]. The crucial obstacle to using these methods here is that   
+248 the cost matrix $C$ in GW is “synthetic“, in the sense that it is the output of a matrix product $A P B$   
+249 involving the very last transport $P$ . This stands in stark contrast with the requirements in [4, 3, 31]   
+250 that the kernel matrix corresponding to $K _ { \varepsilon } = e ^ { - C / \varepsilon }$ admits favorable properties, such as being p.s.d   
+251 or admitting an explicit (random or not) finite dimensional feature approximation. Since $C$ changes   
+252 at each iteration in Algo.1, they are not directly applicable.   
+53 Combining the results in $\ S 4$ with those from $\ S _ { \mathbf { B } }$ results in updates for $C _ { 1 }$ and $C _ { 2 }$ that only require   
+54 $\mathcal { O } ( n r d )$ and $\mathcal { O } ( m r d ^ { \prime } )$ operations.
+
+Linear time GW. Finally all the quadratic operations appearing in Algorithm (3) can be replaced by linear counterparts. The iterations that have not been modified had an overall complexity of $\bar { \mathcal { O } } ( m r ( r + d ^ { \prime } ) + \bar { n } r ( r + d ) )$ at each iteration. The initialization and linearization steps can now be performed in linear time, with respective complexity of respectively $\mathcal { O } ( n ( r + d ^ { 2 } ) + \mathrm { \dot { \ m } } ( ( d ^ { \prime } ) ^ { 2 } + r ) )$ and $\mathcal { O } ( ( n r ( r + d ) + m r ( r + d ^ { \prime } ) )$ .
+
+# 260 6 Experiments
+
+Our goal in this section is to demonstrate that, for a far smaller computational budget, the GW-LR approach is competitive with the direct entropic approach on datasets that are either synthesized to exhibit local clusters, or directly validated on a real high-dimensional dataset as well. Because both approaches have different hyperparameters, our goal is to stick to a realistic evaluation that stresses both optimality of solutions as a function of computational effort, as well as performance in real life applications. We start by investigating the sensitivity of hyperparamaters $\varepsilon$ and $\gamma$ on our method. Since GW is not convex, these may interact in unexpected ways. Experiments were run on a personal MacBook Pro 2019 laptop. We reused code from github.com/meyerscetbon/LOT, and downloaded genomics data from github.com/rsinghlab/SCOT.
+
+Benchmarks. We consider three synthetic problems and one real world problem to evaluate timeaccuracy trade-offs, and also compare the couplings obtained by our method and that of the entropic version [27]. More precisely, we compare the quadratic approach in GW-LR computed with algorithm (3) (and its linear time counterpat, Lin GW-LR as presented in $\ S 5$ ), with EntropicGW, the cubic implementation of [27] (as well as its quadratic counterpart, Quad Entropic-GW presented in Algo. 2). For GW-LR and Lin GW-LR, and in all experiments, we set the lower bound on entries of g to ↵ = 1010 .
+
+Initialization To initialize all algorithms with a common strategy, we adapted the first lower bound of [26, Def. 6.1] to the entropic case. In all experiments showing time-accuracy tradeoffs, we choose to use number of operations to provide platform independent quantities. Accuracy is measured by evaluating the ground-truth energy $\mathcal { E } _ { , B }$ (even in scenarios when the method uses a low rank approximation for $A , B$ at optimization time).
+
+![](images/ced3128aec5a3e2f1ee8b3a2ad54bff8f7dd8e662233522796a537c2ee35f7a0.jpg)  
+Figure 3: The number of cluster in each distribution is 10 and the number of samples is $n = m = 5 0 0 0$ . The ground cost is the Euclidean distance. As we can evaluate the distance between two arbitrary points, we can obtain in linear-time an efficient approximation of the distance matrices $A$ and $B$ as presented in 3. The rank of their factorizations is fixed to be $d = d ^ { \prime } = 1 0 0$ . GW-LR and EntropicGW corresponds to the case where the full matrices $A$ and $B$ are considered while Lin GW-LR and Quad Entropic-GW take as inputs the low-rank approximations of the distance matrices. We plot the time-accuracy tradeoff for multiple choices of $\gamma$ and rank $r$ defined as a fraction of $n$ . For Entropic-GW and Quad Entropic-GW, we set $\varepsilon = 1 / \gamma$ as proposed in [27]. Recall that for low-rank methods, we set $\varepsilon = 0$ .
+
+Sensitivity to $\gamma$ and $\varepsilon$ Here we aim at showing the dependence in both $\gamma$ and $\varepsilon$ of our proposed method. In Figure 2, we compare the GW loss obtained by our algorithm when varying $\varepsilon$ and $\gamma$ on two mixtures. We show that when $\varepsilon = 0$ , the proposed method manage to consistently obtain small GW loss whatever $\gamma$ is. 8 By allowing $\varepsilon > 0$ , the algorithm is able to reach even smaller GW loss, however, the choice of $\varepsilon$ depends highly on $\gamma$ . Therefore in the following experiments, we fix $\varepsilon = 0$ for our method. We also show the dependence in $\gamma$ and $\varepsilon$ of our method in other settings and observe similar behaviors. See Appendix D.2 for 4 more details.
+
+Remark 1. As shown in Figure 8 in Appendix D.2, allowing $\varepsilon > 0$ may also increase the speed of convergence of the algorithm. However choosing well $\varepsilon$ for a given $\gamma$ must be done carefully and we prefer in the following experiments to present the performance of our method in the simplest setting where $\varepsilon = 0$ .
+
+![](images/58a709d7ba81bc3cb25edfaf380271a8b7d1638e8c02cdcfc50025f0b4029cf8.jpg)  
+Figure 2: In this experiment, we consider two mixtures of (2 and 3) Gaussians in respectively 5-D and 10-D, sampled as discrete measures with $n = m = 5 0 0 0$ points, see more details on setup in Appendix D.2. The ground cost is the squared Euclidean distance, which provides an exact low-rank factorization of the cost as presented in $\ S \ O 3$ . Results on speed (in Appendix) are therefore obtained using Lin GW-LR. The nonnegative rank of the coupling is set to $r = 5 0 = n / 1 0 0$ . We plot the GW loss obtained by Lin GW-LR when varying $\epsilon$ for multiple choices of $\gamma$ . Both size and color have been used to quantify visually the value of the loss at that parameter pair. Occasional inversions are due to the nonconvex nature of the GW problem.
+
+01 Synthetic low-rank problem In this experiment   
+02 we aim at comparing the time-accuracy tradeoff of   
+03 the different methods when the underlying distribu  
+04 tions has a low-rank structure. For that purpose, we   
+05 consider two distributions in respectively 10-D and   
+06 15-D, where the support of each distributions is the   
+07 concatenation of clusters of points, and where the eu  
+08 clidean distance between the centroids of the clusters   
+09 is bigger than a threshold $\beta$ . Here we set $\beta = 1 0$ .   
+10 Both distributions are uniform, have the same number   
+311 of clusters and the same number of points in each cluster. Some illustrations of the simulated data   
+312 is provided in Appendix D.3. In Figure 3, when the underlying cost is the (not squared) Euclidean   
+313 distance, our methods manage to consistently obtain similar accuracy that the ones obtained by   
+314 entropic methods, with very low rank $r = n / 5 0 0$ , while being orders of magnitude faster. In Figure 4,   
+315 we also compare the time-accuracy tradeoffs in the more favorable case where the underlying cost   
+316 is the squared Euclidean distance and obtain similar results. We also show more experiments for   
+317 different number of clusters in Appendix D.3, leading to similar conclusions.
+
+![](images/205bfafdd7068632b78380a9723796c677156a099b7b76bf03b8ebfa0f53f9ad.jpg)  
+Figure 4: The number of clusters in each distribution is 5 and the number of samples considered here is $n = m = 1 0 0 0 0$ . The ground cost is the squared Euclidean distance. We compare Lin GW-LR and Quad Entropic-GW as we have an exact factorization of the matrices $A$ and $B$ . We plot the time-accuracy tradeoff when varying $\gamma$ for multiple choices of $r$ . For Quad Entropic-GW, we set $\varepsilon = 1 / \gamma$ and for Lin GW-LR we set $\varepsilon = 0$ .
+
+![](images/f357507bc64ee0a9d0b9ea5317bd8491b5252e6d657cf1bfa6c8e18bf3b3b8cf.jpg)  
+Figure 5: We plot, for each cells of the SNAREseq dataset, the FOSCTTM ranked in the increasing order for both GW-LR and Entropic-GW.
+
+![](images/44a0b85b7dc95af7583f8947b4baaddc578e9a97e8c9a0235263eef73478c8a8.jpg)  
+Figure 6: Plot of the time-accuracy tradeoff when varying $\gamma$ for multiple choices of rank $r$ on the SNAREseq dataset. For Entropic-GW we set $\varepsilon = 1 / \gamma$ , for GW-LR, we set $\varepsilon = 0$ .
+
+Experiments on Single Cell Genomics Data. We reproduce the single-cell alignment experiment introduced in [14]. The dataset consists in single-cell multi-omics data generated by co-assays. In that setup, the ground truth one–to-one correspondence information between cells is known, and can therefore be used to benchmark GW strategies. The dataset considered is the SNAREseq [10], with $n = m = 1 0 4 7$ . We apply the exact same pre-processing steps as proposed in [14] by computing intra-domain distance matrices $A$ and $B$ with a k-NN graphs based on correlations, to compute shortest path distance matrices. Note that in that case, one cannot obtain directly in linear time a low-rank factorization of $A$ and $B$ using [5, 21], since the shortest path distances need to be computed first. Therefore we only consider the quadratic GW-LR and the cubic Entropic-GW. In Figure 6, we compare the alignment performance through the “fraction of samples closer than the true match” (FOSCTTM) introduced in [25]. We see that both algorithm obtain similar performance. However, in Figure 5, we show that whatever the $\gamma$ chosen, GW-LR reaches better accuracy while being order of magnitude faster than Entropic-GW for a very small rank $r = 1 0$ .
+
+Conclusion. While the factorization introduced in [32] held the promise to speed up classic OT, we have shown in this work that it delivers an even larger impact when applied to the GW problem: Indeed, the combination of low-rank Sinkhorn factorization with-low rank cost matrices is the only one, to our knowledge, that ensures that the linearization step of the GW objective can be carried out with a linear complexity, throughout outer iterations. This linear complexity is comparable to that of the most recent OT solvers, yet still retains the appealing properties of the Entropic approach, such as stability and convergence to meaningful solutions.
+
+# References
+
+[1] Jean Alaux, Edouard Grave, Marco Cuturi, and Armand Joulin. Unsupervised hyperalignment for multilingual word embeddings. arXiv preprint arXiv:1811.01124, 2018.   
+[2] Jason Altschuler, Jonathan Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. arXiv preprint arXiv:1705.09634, 2017.   
+[3] Jason Altschuler, Francis Bach, Alessandro Rudi, and Jonathan Niles-Weed. Massively scalable sinkhorn distances via the nyström method, 2018.   
+[4] Jason Altschuler, Francis Bach, Alessandro Rudi, and Jonathan Weed. Approximating the quadratic transportation metric in near-linear time. arXiv preprint arXiv:1810.10046, 2018.   
+[5] Ainesh Bakshi and David P. Woodruff. Sublinear time low-rank approximation of distance matrices, 2018.   
+[6] Andrew J Blumberg, Mathieu Carriere, Michael A Mandell, Raul Rabadan, and Soledad Villar. Mrec: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data. arXiv preprint arXiv:2001.01666, 2020.   
+[7] Charlotte Bunne, David Alvarez-Melis, Andreas Krause, and Stefanie Jegelka. Learning generative models across incomparable spaces. arXiv preprint arXiv:1905.05461, 2019.   
+[8] Rainer E Burkard, Eranda Cela, Panos M Pardalos, and Leonidas S Pitsoulis. The quadratic assignment problem. In Handbook of combinatorial optimization, pages 1713–1809. Springer, 1998.   
+[9] Laetitia Chapel, Mokhtar Alaya, and Gilles Gasso. Partial optimal transport with applications on positive-unlabeled learning. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020.   
+[10] Song Chen, Blue B Lake, and Kun Zhang. High-throughput sequencing of the transcriptome and chromatin accessibility in the same cell. Nature biotechnology, 37(12):1452–1457, 2019.   
+[11] Samir Chowdhury, David Miller, and Tom Needham. Quantized gromov-wasserstein. arXiv preprint arXiv:2104.02013, 2021.   
+[12] Alexis Conneau, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, and Hervé Jégou. Word translation without parallel data. arXiv preprint arXiv:1710.04087, 2017.   
+[13] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in neural information processing systems, pages 2292–2300, 2013.   
+[14] Pinar Demetci, Rebecca Santorella, Björn Sandstede, William Stafford Noble, and Ritambhara Singh. Gromov-wasserstein optimal transport to align single-cell multi-omics data. bioRxiv, 2020. doi: 10.1101/2020.04.28.066787.   
+[15] Pinar Demetci, Rebecca Santorella, Bjorn Sandstede, William Stafford Noble, and Ritambhara Singh. Gromov-wasserstein optimal transport to align single-cell multi-omics data. BioRxiv, 2020.   
+[16] Richard Mansfield Dudley et al. Weak convergence of probabilities on nonseparable metric spaces and empirical measures on euclidean spaces. Illinois Journal of Mathematics, 10(1): 109–126, 1966.   
+[17] Danielle Ezuz, Justin Solomon, Vladimir G Kim, and Mirela Ben-Chen. Gwcnn: A metric alignment layer for deep shape analysis. In Computer Graphics Forum, volume 36, pages 49–57. Wiley Online Library, 2017.   
+[18] Aden Forrow, Jan-Christian Hütter, Mor Nitzan, Philippe Rigollet, Geoffrey Schiebinger, and Jonathan Weed. Statistical optimal transport via factored couplings, 2018.   
+[19] Aden Forrow, Jan-Christian Hütter, Mor Nitzan, Philippe Rigollet, Geoffrey Schiebinger, and Jonathan Weed. Statistical optimal transport via factored couplings. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 2454–2465. PMLR, 2019.   
+[20] Steven Gold and Anand Rangarajan. Softassign versus softmax: Benchmarks in combinatorial optimization. Advances in neural information processing systems, pages 626–632, 1996.   
+[21] Piotr Indyk, Ali Vakilian, Tal Wagner, and David Woodruff. Sample-optimal low-rank approximation of distance matrices, 2019.   
+[22] Hicham Janati, Thomas Bazeille, Bertrand Thirion, Marco Cuturi, and Alexandre Gramfort. Multi-subject meg/eeg source imaging with sparse multi-task regression. NeuroImage, page 116847, 2020.   
+[23] Hiroshi Konno. Maximization of a convex quadratic function under linear constraints. Mathematical programming, 11(1):117–127, 1976.   
+[24] Matt Kusner, Yu Sun, Nicholas Kolkin, and Kilian Q Weinberger. From word embeddings to document distances. In Proc. of the 32nd Intern. Conf. on Machine Learning, pages 957–966, 2015.   
+[25] Jie Liu, Yuanhao Huang, Ritambhara Singh, Jean-Philippe Vert, and William Stafford Noble. Jointly embedding multiple single-cell omics measurements. BioRxiv, page 644310, 2019.   
+[26] Facundo Mémoli. Gromov–wasserstein distances and the metric approach to object matching. Foundations of computational mathematics, 11(4):417–487, 2011.   
+[27] Gabriel Peyré, Marco Cuturi, and Justin Solomon. Gromov-wasserstein averaging of kernel and distance matrices. In International Conference on Machine Learning, pages 2664–2672, 2016.   
+[28] Gabriel Peyré and Marco Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning, 11(5-6), 2019. ISSN 1935-8245.   
+[29] Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision, 40(2):99–121, November 2000.   
+[30] Ryoma Sato, Marco Cuturi, Makoto Yamada, and Hisashi Kashima. Fast and robust comparison of probability measures in heterogeneous spaces. arXiv preprint arXiv:2002.01615, 2020.   
+[31] Meyer Scetbon and Marco Cuturi. Linear time sinkhorn divergences using positive features, 2020.   
+[32] Meyer Scetbon, Marco Cuturi, and Gabriel Peyré. Low-rank sinkhorn factorization, 2021.   
+[33] Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176 (4):928–943, 2019.   
+[34] Richard Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist., 35:876–879, 1964.   
+[35] Justin Solomon, Gabriel Peyré, Vladimir G Kim, and Suvrit Sra. Entropic metric alignment for correspondence problems. ACM Transactions on Graphics (TOG), 35(4):1–13, 2016.   
+[36] Karl-Theodor Sturm. The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces. arXiv preprint arXiv:1208.0434, 2012.   
+[37] Titouan Vayer, Laetita Chapel, Rémi Flamary, Romain Tavenard, and Nicolas Courty. Fused gromov-wasserstein distance for structured objects: theoretical foundations and mathematical properties. arXiv preprint arXiv:1811.02834, 2018.   
+[38] Titouan Vayer, Rémi Flamary, Romain Tavenard, Laetitia Chapel, and Nicolas Courty. Sliced gromov-wasserstein. arXiv preprint arXiv:1905.10124, 2019.   
+[39] Jonathan Weed and Francis Bach. Sharp asymptotic and finite-sample rates of convergence of empirical measures in wasserstein distance. Bernoulli, 25(4A):2620–2648, 2019.   
+[40] Hongteng Xu, Dixin Luo, and Lawrence Carin. Scalable gromov-wasserstein learning for graph partitioning and matching. arXiv preprint arXiv:1905.07645, 2019.   
+[41] Hongteng Xu, Dixin Luo, Hongyuan Zha, and Lawrence Carin Duke. Gromov-wasserstein learning for graph matching and node embedding. In International conference on machine learning, pages 6932–6941. PMLR, 2019.
+
+# 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] The main claim of the paper is a linear time (w.r.t. sample size, as commonly understood in OT) computation for GW. Sections $\ S 3 , 4$ and 5 build up that answer. This is experimentally validated across several experiments, both synthetic, to help the reader form intuitions, and on real data where GW was deemed useful.   
+(b) Did you describe the limitations of your work?[Yes] Because the GW problem is nonconvex, these limitations are naturally discussed in the experimental section, Section $\ S 6$ We discuss the effects of $\gamma$ and $\varepsilon$ on the method, which are not easy to parse due to the non-convexity of the method.   
+(c) Did you discuss any potential negative societal impacts of your work?[N/A] As a purely methodological paper, we do not envision potentially negative impact of this work on its own.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?[Yes] We have read these guidelines and confirm our paper does conform to them.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results?[Yes] Our theoretical results are of two nature: a bound in Proposition 1, and a guarantee on convergence in Proposition 2, which has no direct consequence on our empirical findings, yet remain useful. Theory is not our main contribution, but rather algorithms.   
+(b) Did you include complete proofs of all theoretical results?[Yes] all proofs are in the appendix. Space in the main body of the paper was prioritized to include experimental validation, which, for this non-convex problem, we believe to be equally important.
+
+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] We have included portions of the code that we have used. We pledge to make the entire code available later in the reviewing process.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?[Yes] The main contribution is compute efficiency, that we have considered across several parameter choices.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?[N/A] Our experiments are deterministic, since we use a predefined initialization.   
+(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] The experiments were run with basic computational means, a macbook pro.
+
+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] We have reused a single toolbox, and accessed data available publicly.   
+(b) Did you mention the license of the assets? [Yes] All licenses are open source, see supplementary.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Yes, code is shared in the supplementary.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+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]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]

parse/train/rHCzkRd0UK/rHCzkRd0UK_content_list.json

@@ -0,0 +1,1321 @@
+[
+    {
+        "type": "text",
+        "text": "Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs ",
+        "text_level": 1,
+        "bbox": [
+            232,
+            122,
+            766,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            578,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The ability to compare and align related datasets living in heterogeneous spaces plays an increasingly important role in machine learning. The Gromov-Wasserstein (GW) formalism can help tackle this problem. Its main goal is to seek an assignment (more generally a coupling matrix) that can register points across otherwise incomparable datasets. As a non-convex and quadratic generalization of optimal transport (OT), GW is NP-hard. Yet, heuristics are known to work reasonably well in practice, the state of the art approach being to solve a sequence of nested regularized OT problems. While popular, that heuristic remains too costly to scale, with cubic complexity in the number of samples $n$ . We show in this paper how a recent variant of the Sinkhorn algorithm can substantially speed up the resolution of GW. That variant restricts the set of admissible couplings to those admitting a low rank factorization as the product of two sub-couplings. By updating alternatively each sub-coupling, our algorithm computes a stationary point of the problem in quadratic time with respect to the number of samples. When cost matrices have themselves low rank, our algorithm has time complexity ${ \\mathcal { O } } ( n )$ . We demonstrate the efficiency of our method on simulated and real data. ",
+        "bbox": [
+            148,
+            349,
+            766,
+            570
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "17 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            598,
+            310,
+            616
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "18 The ever increasing interest for Gromov-Wasserstein... Several problems in machine learning   \n19 involve comparing families of points that live in heterogeneous spaces. This situation arises typically   \n20 when realigning two distinct sets of feature representations obtained from the similar source. Recent   \n21 applications to single-cell genomics [15] and NLP [12, 1] provide two cases in point: Thousands   \n22 of cells taken from the same tissue are split in two groups, each group is processed with a different   \n23 experimental protocol, resulting in two distinct sets of heterogeneous feature vectors; Thousands of   \n24 word embeddings for two languages are learned independently. In both cases, one expects to find   \n25 a meaningful way to register points across sets living in heteregeneous spaces, since they contain   \n26 similar overall information. That realignment is usually carried out using the Gromov-Wasserstein   \n27 (GW) machinery proposed by Mémoli [26] and Sturm [36], which seeks a relaxed assignment matrix   \n28 that is as “close” to an isometry as possible, using a quadratic score to quantify that closeness. GW   \n29 has a lot of practical appeal: It has been used in supervised learning [41], generative modeling [7],   \n30 domain adaptation [9], structured prediction [37], quantum chemistry [27] and alignment layers [17].   \n31 ... despite its cubic cost. Because it is an NP-hard problem, these applications rely on approximating   \n32 GW, typically by solving a sequence of OT problems using entropic regularization. This heuristic is   \n33 efficient yet costly, since it requires $\\mathcal { O } ( n ^ { 3 } )$ operations to register two sets of $n$ samples, a price that is   \n34 paid when re-instantiating each OT problem. Our goal is to reduce substantially that complexity by   \n35 exploiting low-factorization of both parameters (data) and variable (relaxed assignment) matrices in   \n36 the GW problem, while maintaining state of the art performance in applications.   \n37 Wasserstein: from cubic to linear complexity. A comparatively simpler problem is the registration   \n38 of two populations embedded in the same space. This corresponds to the classic optimal transport   \n39 (OT) problem, which has received considerable attention in ML [28]. OT has found applications   \n40 in computer vision [29], NLP [24], single cell tracking [33] or multi-task regression in neuro  \n41 imaging [22]. While the OT problem is originally cast as a linear program, with a ${ \\overline { { O ( n ^ { 3 } \\log ( n ) } } } )$ cost,   \n42 many of these works rely on solving instead a penalized OT problem using Sinkhorn’s algorithm [34,   \n43 13]. In its most naive implementation, the Sinkhorn has quadratic complexity [2]. Recent works   \n44 achieve $O ( n )$ complexity by targeting the matrix-vector updates in Sinkhorn’s algorithm using   \n45 low-rank approximations of the data kernel matrix [4, 3, 31]. This idea can be further improved by   \n46 imposing the low-rank constraint on the optimization variables of the original OT problem [19], to   \n47 modify Sinkorn’s steps by enforcing a low rank factorization of the coupling variable [32].   \n48 Gromov-Wasserstein: from NP-hard to linear approximations. The GW problem replaces   \n49 the linear objective function in OT by a non-convex quadratic objective. Much like OT is a re  \n50 laxation of the optimal assignment problem, GW can be seen as a relaxation of the quadratic   \n51 assignment problem (QAP). Both GW and QAP are NP-hard to solve [8]. In practice, iteratively   \n52 minimizing a linearization of that quadratic objective using Sinkhorn works surprisingly well [20, 35].   \n53 This method corresponds to a mirror-descent scheme [27],   \n54 and in the special case of Euclidean distance matrices, the   \n55 loss is concave and it can be also interpreted as a bi-linear   \n56 relaxation [23]. In the most general case, this results in an   \n57 $O ( n ^ { 4 } )$ algorithm (the objective is a quadratic function of a   \n58 $n \\times n$ relaxed assignment matrix), that is reduced to $O ( n ^ { 3 } )$   \n59 when using separable losses [27], a price that remains too   \n60 high for several ML applications. It is possible to replace   \n61 the GW distance by cheaper yet only distantly related prox  \n62 ies, such as lower bounds based on OT [26] (see also [30])   \n63 or sliced projections [38]. Whether GW can be efficiently   \n64 sped up remains an open question. We propose in this   \n65 work a novel approach that leverages, as done recently for   \n66 OT, low-rank methods. A very recent line of works attacks   \n67 this problem by quantizing first the two input spaces to   \n68 solve a GW problem of reduced size, thus effectively pro  \n69 ducing an ad-hoc low-rank coupling [11]. A nice feature   \n70 of this approach is that it maintains the triangular inequal  \n71 ity and provides a valid upper-bound on the GW distance.   \n72 Related approaches which also approximate GW distance   \n73 using clustering methods (possibly in a recursive way)   \n74 are [6] and [40]. We take in this paper a direct approach:   \n75 instead of separating clustering and GW resolution in 2   \n76 independent steps, we propose do address them simulta  \n77 neously: our method seeks the least-costly (in GW sense)   \n78 coupling with a low rank constraint, as illustrated in Fig. 1.   \n79 Contributions We introduce the low-rank-GW problem, by imposing a low rank constraint on   \n80 feasible couplings. This method works hand-in-hand with entropic regularization and leads to a   \n81 Sinkhorn-like algorithm. Because of its exclusive reliance on matrix-vector products, the method   \n82 streams well on GPUs. This method can also leverage low-rank factorizations of the input data   \n83 matrices to further reduce the complexity of each iteration to reach linear time. Numerical evaluations   \n84 on simulated and real datasets show that this low-rank approximation maintains the favorable   \n85 property of entropic-regularized GW (namely its ability to compute “good” local minima) for a linear   \n86 computational price, thus paving the way for larger scale uses of GW in ML. ",
+        "bbox": [
+            147,
+            631,
+            825,
+            810
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            816,
+            825,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            92,
+            825,
+            244
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            250,
+            825,
+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            320,
+            550,
+            678
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/75f07794a62ec3be032add0ebc59f95a5f74a2fe87bba8885a69e073ad67fc1a.jpg",
+        "image_caption": [
+            "Figure 1: Top row: we compute the GW coupling between two curves in 2D and 3D, with $n = m = 1 0 0 0 0$ points. These points are endowed with the squared L2 distance. Bottom row: coupling obtained with the SoTA entropic approach [20, 27], compared with our linear method with rank $r = 1 0$ . See Appendix D.1 for more details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            565,
+            329,
+            821,
+            527
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            684,
+            825,
+            795
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "87 2 Background on the Gromov-Wasserstein Framework ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            818,
+            648,
+            835
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "88 Comparing measured metric spaces. Let $( \\mathcal { X } , d _ { \\mathcal { X } } )$ and $( \\mathcal { V } , d _ { \\mathcal { V } } )$ be two metric spaces, and $\\mu$ and   \n89 $\\nu$ two discrete probability measures on $\\mathcal { X }$ and $\\mathcal { V }$ , respectively. We write $\\textstyle \\mu : = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } }$ and   \n90 $\\begin{array} { r } { \\nu : = \\sum _ { i = j } ^ { m } b _ { j } \\delta _ { y _ { j } } } \\end{array}$ where $n , m \\geq 1 , a , b$ are two histograms in the probability simplicies $\\Delta _ { n } , \\Delta _ { m }$ of   \n91 respective size $n$ and $m$ , and $( x _ { 1 } , \\ldots , x _ { n } )$ , $\\left( y _ { 1 } , \\ldots , y _ { m } \\right)$ are two families in $\\mathcal { X }$ and $\\mathcal { V }$ . For $q \\geq 1$ ,   \n92 let us also denote $A : = ( d _ { \\mathcal { X } } ^ { q } ( x _ { i } , x _ { i ^ { \\prime } } ) ) _ { 1 \\leq i , i ^ { \\prime } \\leq n } \\in \\mathbb { R } ^ { n \\times n }$ and $B : = ( d _ { \\mathcal { V } } ^ { q } ( x _ { j } , x _ { j ^ { \\prime } } ) ) _ { 1 \\leq i , i ^ { \\prime } \\leq m } \\in \\mathbb { R } ^ { m \\times m }$   \n93 X Ytwo pairwise cost matrices between the points in the respective supports of $\\mu$ and $\\nu$ . The Gromov  \n94 Wasserstein (GW) discrepancy between two discrete metric measure spaces $( \\mu , d _ { \\mathcal { X } } )$ and $( \\nu , d _ { 3 } )$ is   \n95 the solution of the following non-convex quadratic problem, instantiated here for simplicity as a   \n96 function of $( a , A )$ and $( b , B )$ , which contain all the information that is needed: ",
+        "bbox": [
+            147,
+            852,
+            825,
+            912
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/991251a13c341e25ad1e28131cef590f08214154c4f03e6bc278e32502a20658.jpg",
+        "text": "$$\n\\mathbf { G } \\mathbf { W } ( ( a , A ) , ( b , B ) ) = \\operatorname* { m i n } _ { P \\in \\Pi _ { a , b } } \\mathcal { E } _ { A , B } ( P ) , \\mathrm { w h e r e ~ } \\Pi _ { a , b } : = \\{ P \\in \\mathbb { R } _ { + } ^ { n \\times m } | P \\mathbf { 1 } _ { m } = a , P ^ { T } \\mathbf { 1 } _ { n } = b \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            167,
+            789,
+            194
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "97 and the energy $\\mathcal { E } _ { A , B }$ is a quadratic function parameterized by a loss $L : \\mathbb { R } \\times \\mathbb { R } \\to \\mathbb { R }$ : ",
+        "bbox": [
+            143,
+            199,
+            732,
+            215
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f5734633d6ed2d691baffd4d9b10f3558721710a9f10926f1854ae74f7b39e9.jpg",
+        "text": "$$\n\\mathcal { E } _ { A , B } ( P ) : = \\sum _ { i , j , i ^ { \\prime } , j ^ { \\prime } } L ( A _ { i , i ^ { \\prime } } , B _ { j , j ^ { \\prime } } ) P _ { i , j } P _ { i ^ { \\prime } , j ^ { \\prime } } \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            222,
+            648,
+            257
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "98 A typical choice of the loss is the $L ^ { p }$ distance $L ( a , b ) = | a - b | ^ { p }$ with $p \\geq 1$ . In that case, [26]   \n99 proves that $\\mathrm { G W } ^ { 1 / p }$ defines a distance on the space of metric measure spaces quotiented by measure  \n100 preserving isometries. When $p = 2$ , as we consider from now on, the GW objective can be evaluated   \n101 efficiently using the marginal constraints imposed on $P$ , as follows [27]: ",
+        "bbox": [
+            143,
+            268,
+            826,
+            329
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d27d509cc3df4d58dfb28fd155f93cc420834f784c437a40c4b3d434c537dfb.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { E } _ { A , B } ( P ) = \\langle A ^ { \\odot 2 } a , a \\rangle + \\langle B ^ { \\odot 2 } b , b \\rangle - 2 \\langle A P B , P \\rangle . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            333,
+            671,
+            352
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "02 Indeed, (3) can be computed efficiently in $\\mathcal { O } ( n ^ { 2 } m + n m ^ { 2 } )$ operations, using only matrix/matrix multiplications, instead of the 103 $\\mathcal { O } ( n ^ { 2 } m ^ { 2 } )$ complexity of the naive evaluation of (2). ",
+        "bbox": [
+            150,
+            358,
+            826,
+            388
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "104 Entropic Gromov-Wasserstein. The original GW problem (1) can be regularized using an entropic   \n105 term [20, 35, 27], leading to the following problem: ",
+        "bbox": [
+            148,
+            393,
+            826,
+            422
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e893c5a9770e1411ee323ce2c2a9c1eb7fe0a69b78de07a241f43b50c79d3f2c.jpg",
+        "text": "$$\n\\mathbf { G } \\mathbf { W } _ { \\varepsilon } ( ( a , A ) , ( b , B ) ) = \\operatorname* { m i n } _ { P \\in \\Pi _ { a , b } } \\mathcal { E } _ { A , B } ( P ) - \\varepsilon H ( P ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            428,
+            669,
+            454
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "106 where $\\begin{array} { r } { H ( P ) ~ : = ~ - \\sum _ { i , j } P _ { i , j } ( \\log ( P _ { i , j } ) ~ - ~ 1 ) } \\end{array}$ is the entropy of $P$ . By applying a Mirror   \n107 Descent (MD) scheme with respect to the KL divergence and by choosing the step-size to   \n108 be $\\gamma ~ = ~ 1 / \\varepsilon$ , Peyré et al. [27] provide a simple algorithm which consists in solving a se  \n109 quence of regularized OT problem as presented in Algorithm 1. Indeed, each KL pro  \n110 jection in Algorithm 1 can be computed efficiently thanks to the Sinkhorn algorithm [13].   \n112 Computational complexity. Given a cost matrix $C$ , the   \n113 KL projection of $K _ { \\varepsilon }$ onto the polytope $\\textstyle \\prod ( a , b )$ , where   \n114 ${ \\mathrm { K L } } ( { \\bar { P } } , { \\bar { Q } } ) = \\langle P , \\log ( P / Q ) - 1 \\rangle$ , is carried out in the inner   \n115 loop of Algo. 1 using the Sinkhorn algorithm, through   \n116 matrix-vector products. This quadratic complexity (in   \n117 red) is dominated by the cost of updating matrix $C$ at each   \n118 iteration in Algorithm 1, which requires $\\mathcal { O } ( n ^ { 2 } m + n m ^ { 2 } )$   \n119 algebraic operations (cubic, in violet). As noted above,   \n120 evaluating the objective $\\mathcal { E } _ { A , B } ( \\boldsymbol { P } )$ has the same order. In   \n121 the following we show that by considering a low rank   \n122 exact decomposition (or approximation) of the distance   \n123 matrices, the cubic cost of reupdating $C$ and subsequently   \n124 evaluating $\\mathcal { E } _ { A , B }$ can be brought down to quadratic. ",
+        "bbox": [
+            140,
+            459,
+            825,
+            532
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            551,
+            549,
+            732
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Entropic-GW ",
+        "text_level": 1,
+        "bbox": [
+            562,
+            565,
+            740,
+            580
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "25 3 Exploiting a Low-Rank Factorization for Cost Matrices ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            751,
+            668,
+            768
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Exact factorization of cost matrices. In this section we consider the case where the cost matrices $A$ and $B$ admit a low-rank factorization. More precisely, we make the following assumption. ",
+        "bbox": [
+            166,
+            781,
+            821,
+            811
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. Assume that $A$ and $B$ admit a low-rank factorization, that is there exists $A _ { 1 } , A _ { 2 } \\in$ $\\mathbb { R } ^ { n \\times d }$ and $B _ { 1 } , B _ { 2 } \\in \\mathbb { R } ^ { m \\times d ^ { \\prime } }$ such that $A = A _ { 1 } A _ { 2 } ^ { T }$ and $B = B _ { 1 } B _ { 2 } ^ { T }$ , where $d \\ll n , d ^ { \\prime } \\ll m$ . ",
+        "bbox": [
+            160,
+            814,
+            821,
+            845
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "130 A case in point is when both $A$ and $B$ are squared Euclidean distance matrices, with a sample size   \n131 that is larger than ambient dimension. This case is highly relevant, covering many applications of OT   \n132 to ML. The $d \\ll n$ assumption is also likely to hold for most applications, since cases where $d \\gg n$   \n133 are known to pose challenges to the estimation of OT [16, 39]. Writing $X = [ x _ { 1 } , \\ldots , x _ { n } ] \\in \\mathbb { R } ^ { d \\times n }$ , if ",
+        "bbox": [
+            143,
+            854,
+            825,
+            912
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "134 $A = \\left[ \\Vert x _ { i } - x _ { j } \\Vert _ { 2 } ^ { 2 } \\right] _ { i , j }$ , then one has, writing $z = ( X ^ { \\odot 2 } ) ^ { T } \\mathbf { 1 } _ { d } \\in \\mathbb { R } ^ { n }$ that $A = z \\mathbf { 1 } _ { n } ^ { T } + \\mathbf { 1 } _ { n } z ^ { T } - 2 X ^ { T } X$ . Therefore by denoting 135 $A _ { 1 } = [ z , \\mathbf { 1 } _ { n } , - \\sqrt { 2 } X ^ { T } ] \\in \\mathbb { R } ^ { n \\times ( d + 2 ) }$ and $A _ { 2 } = [ \\mathbf { 1 } _ { n } , z , \\sqrt { 2 } X ^ { T } ] \\in \\mathbb { R } ^ { n \\times ( d + 2 ) }$ 136 we obtain the factorization above. ",
+        "bbox": [
+            140,
+            89,
+            826,
+            138
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "137 Under Assumption 1, the complexity of Algo. 1 is downgraded to quadratic in sample size: the two   \n138 operations that make Algo. 1 cubic lie in the updates of the cost and the computation of the objective.   \n139 Observe that for any given $P \\in \\mathbb { R } ^ { n \\times m }$ , one can compute at each iteration ",
+        "bbox": [
+            142,
+            145,
+            826,
+            188
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f7ad62dce9ec791c3a08511e4612ed559e23276c62102ce7f6dfbb3546d24af.jpg",
+        "text": "$$\n\\begin{array} { r } { C = - 4 A _ { 1 } A _ { 2 } ^ { T } P B _ { 1 } B _ { 2 } ^ { T } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            190,
+            578,
+            209
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "140 in $n m ( d + d ^ { \\prime } ) + d d ^ { \\prime } ( n + m )$ algebraic operations. Moreover thanks to the reformulation of   \n141 $\\mathcal { E } _ { A , B } ( \\boldsymbol { P } )$ given in (3), one can compute it in quadratic time as well. Indeed writing $G _ { 1 } : =$   \n142 $A _ { 1 } ^ { T } P B _ { 2 }$ and $G _ { 2 } : = A _ { 2 } ^ { T } P B _ { 1 }$ , both in $\\mathbb { R } ^ { d \\times d ^ { \\prime } }$ , one has $\\langle A P B , P \\rangle = \\mathbf { 1 } _ { d } ^ { T } ( G _ { 1 } \\odot G _ { 2 } ) \\mathbf { 1 } _ { d ^ { \\prime } }$ . Com  \n143 puting $G _ { 1 } , G _ { 2 }$ given $P$ requires only $2 ( n m d + m d d ^ { \\prime } )$ , and computing their dot product adds   \n144 $d d ^ { \\prime }$ algebraic operations. The overall complexity to compute $\\mathcal { E } _ { A , B } ( { \\cal P } )$ is $\\mathcal { O } ( n m d + m d d ^ { \\prime } )$ .   \n146 General distance matrices. When the original   \n147 cost matrices $A$ , are not low-rank but describe   \n148 distances, we propose to use a recent body of   \n149 work that output their low-rank approximation   \n150 in linear time [5, 21]. These algorithms produce,   \n151 for any distance matrix $D \\in \\bar { \\mathbb { R } } ^ { n \\times m }$ and $\\gamma > 0$ ,   \n152 matrices $D _ { 1 } \\in \\mathbb { R } ^ { n \\times d }$ , $D _ { 2 } \\in \\mathbb { R } ^ { m \\times d }$ in $\\mathcal { O } ( ( m +$   \n153 $\\scriptstyle n ) \\mathtt { p o l y } ( { \\frac { d } { \\gamma } } ) )$ algebraic operations such that with   \n154 probability at least 0.99 one has ",
+        "bbox": [
+            138,
+            212,
+            826,
+            286
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            305,
+            486,
+            434
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Quadratic Entropic-GW ",
+        "text_level": 1,
+        "bbox": [
+            498,
+            308,
+            745,
+            323
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db5b384f8068bd2e2ae60e307354779afb9dcec09128a0599c89dbffa5e7b481.jpg",
+        "text": "$$\n\\| D - D _ { 1 } D _ { 2 } ^ { T } \\| _ { F } ^ { 2 } \\leq \\| D - C _ { d } \\| _ { F } ^ { 2 } + \\gamma \\| D \\| _ { F } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            436,
+            470,
+            457
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "155 where $C _ { d }$ denotes the best rank- $d$ approximation   \n156 to $D$ . We fall back on this approach to obtain a   \n157 low-rank factorization of a distance matrix in lin  \n158 ear time whenever needed, aware that this incurs   \n159 an additional approximation. See Appendix B   \n160 for more details. ",
+        "bbox": [
+            142,
+            459,
+            485,
+            542
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "end ",
+        "text_level": 1,
+        "bbox": [
+            496,
+            435,
+            526,
+            448
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2eb83bfa3d20e26add9894e25c8eb8c11237e86c2f100eb919e294bc11788e51.jpg",
+        "text": "$$\n\\begin{array} { r l } & { c _ { 1 } \\gets \\langle A ^ { \\odot 2 } a , a \\rangle + \\langle B ^ { \\odot 2 } b , b \\rangle \\quad \\mathcal { O } ( \\mathrm { n r } } \\\\ & { G _ { 2 } \\gets A _ { 2 } ^ { T } P B _ { 1 } \\quad \\mathrm { n m d ~ + ~ m d d } ^ { \\flat } , } \\\\ & { G _ { 1 } \\gets A _ { 1 } ^ { T } P B _ { 2 } \\quad \\mathrm { n m d ~ + ~ m d d } ^ { \\flat } } \\\\ & { c _ { 2 } \\gets - 2 \\mathbf { 1 } _ { d } ^ { T } ( G _ { 1 } \\odot G _ { 2 } ) \\mathbf { 1 } _ { d ^ { \\prime } } \\quad \\mathcal { O } ( \\mathrm { d d } ^ { \\flat } ) } \\\\ & { \\xi _ { A , B } ( P ) \\gets c _ { 1 } + c _ { 2 } } \\\\ & { \\mathbf { R e t u r n } \\colon \\mathcal { E } _ { A , B } ( P ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            496,
+            448,
+            738,
+            536
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "161 4 Imposing a Low Nonnegative Low-Rank for the Coupling ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            560,
+            686,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we shift our attention to a different opportunity for speed-ups, without assuming that Assumption 1 holds: we regularize the GW problem problem by decomposing the coupling as a product of two low-rank couplings, in the footsteps of [18, 32], using the following definition: ",
+        "bbox": [
+            161,
+            592,
+            826,
+            635
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "165 Definition 1. Given $M \\in \\mathbb { R } ^ { n \\times m }$ , the nonnegative (NN) rank of $M$ is the smallest number of   \n166 nonnegative rank-one matrices into which the matrix can be decomposed additively: ",
+        "bbox": [
+            147,
+            636,
+            828,
+            666
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/13f91d810128809398f56c6fedeac1befdc9558925b6f54d93f159cef66e0d34.jpg",
+        "text": "$$\n\\operatorname { r k } _ { + } ( M ) : = \\operatorname* { m i n } \\left\\{ q | M = \\sum _ { i = 1 } ^ { q } R _ { i } , \\forall i , \\operatorname { r k } ( R _ { i } ) = 1 , R _ { i } \\geq 0 \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            667,
+            697,
+            710
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "167 Following [18, 32], we propose to constrain GW, enforcing a rank $r$ on the coupling: ",
+        "bbox": [
+            138,
+            719,
+            728,
+            736
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44f91ec6f3489def2c28c9970b353aad1e4cd526fc7e537ed4fb69d7aa1ab029.jpg",
+        "text": "$$\n\\mathrm { G W } \\mathrm { L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) : = \\operatorname* { m i n } _ { P \\in \\Pi _ { a , b } ( r ) } \\xi _ { A , B } ( P ) , \\mathrm { ~ w h e r e ~ } \\Pi _ { a , b } ( r ) : = \\{ P \\in \\Pi _ { a , b } , \\mathrm { r k } _ { + } ( P ) \\leq r \\} \\ : .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            739,
+            797,
+            766
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "168 Note that the minimum is always attained as $\\Pi _ { a , b } ( r )$ is compact and the objective is continuous.   \n169 In [32], the authors show that one can parameterize any coupling in $\\Pi _ { a , b } ( r )$ as a product of two   \n170 low-rank couplings linked by a common marginal. For any $g \\in \\Delta _ { r } ^ { * }$ , the interior of $\\Delta _ { r }$ , writing ",
+        "bbox": [
+            140,
+            770,
+            826,
+            814
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f502f81af8d4c21e3c33200a462c839d721f0af858256e43e7fdcca61e94d3fe.jpg",
+        "text": "$$\n\\Pi _ { a , g , b } : = \\Bigl \\{ P \\in \\mathbb { R } _ { + } ^ { n \\times m } , P = Q \\mathrm { d i a g } ( 1 / g ) { R } ^ { T } , Q \\in \\Pi _ { a , g } , \\mathrm { a n d } R \\in \\Pi _ { b , g } \\Bigr \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            816,
+            741,
+            844
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "one has that 171 $\\begin{array} { r } { \\bigcup _ { g \\in \\Delta _ { r } ^ { * } } \\Pi _ { a , g , b } = \\Pi _ { a , b } ( r ) } \\end{array}$ . Therefore GW-LR introduced in (5) can be reformulated as 172 the following optimization problem ",
+        "bbox": [
+            137,
+            845,
+            831,
+            878
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f85869cc445e27aef7e385cd9a11aef3b3b745e30b06b9bb6035078a24b0c2a0.jpg",
+        "text": "$$\n\\mathrm { G W - L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) = \\operatorname* { m i n } _ { ( Q , R , g ) \\in { \\mathcal C } ( a , b , r ) } { \\mathcal E } _ { A , B } ( Q \\mathrm { d i a g } ( 1 / g ) R ^ { T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            881,
+            723,
+            909
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "173 where $\\mathcal { C } ( a , b , r ) : = \\mathcal { C } _ { 1 } ( a , b , r ) \\cap \\mathcal { C } _ { 2 } ( r )$ , with ",
+        "bbox": [
+            142,
+            90,
+            465,
+            107
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e2d50bd28caea1ad1d6484590e9b7aaa9d4fce217cd7b5634208b89bcde1fbaa.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { C } _ { 1 } ( a , b , r ) : = \\Big \\{ ( Q , R , g ) \\in \\mathbb { R } _ { + } ^ { n \\times r } \\times \\mathbb { R } _ { + } ^ { m \\times r } \\times \\big ( \\mathbb { R } _ { + } ^ { * } \\big ) ^ { r } \\mathrm { s . t . } Q \\mathbf { 1 } _ { r } = a , R \\mathbf { 1 } _ { r } = b \\Big \\} , } \\\\ { \\mathcal { C } _ { 2 } ( r ) : = \\Big \\{ ( Q , R , g ) \\in \\mathbb { R } _ { + } ^ { n \\times r } \\times \\mathbb { R } _ { + } ^ { m \\times r } \\times \\mathbb { R } _ { + } ^ { r } \\mathrm { s . t . } Q ^ { T } \\mathbf { 1 } _ { n } = R ^ { T } \\mathbf { 1 } _ { m } = g \\Big \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            112,
+            759,
+            169
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "174 Stabilization of the Method. [32] propose to stabilize the objective defined in (6) by adding to the   \n175 constraints a lower bound $\\alpha$ on the weight vector $g$ such that $g \\geq \\alpha$ coordinate-wise. Indeed, as   \n176 a solution of (6) must satisfies $g > 0$ coordinate-wise, then for $\\alpha$ sufficiently small, the solution   \n177 of the same problem where one adds the constraint $g \\geq \\alpha$ will remain the same. Therefore let us   \n178 introduce our new set of constraints $\\mathcal { C } ( a , b , r , \\alpha ) : = \\mathcal { C } _ { 1 } ( a , b , r , \\alpha ) \\cap \\mathcal { C } _ { 2 } ( r )$ where $\\mathcal { C } _ { 1 } ( a , b , r , \\alpha ) : =$   \n179 $\\mathcal { C } _ { 1 } ( a , b , r ) \\cap \\{ ( Q , R , g ) \\mid g \\geq \\alpha \\}$ . Another way to stabilize the method is by considering a double   \n180 regularization scheme as proposed in [32] where in addition of constraining the nonnegative rank   \n181 of the coupling, we regularize the objective by adding an entropic term in $( Q , R , g )$ , which is to be   \n182 understood as that of the values of the three respective entropies evaluated for each term. ",
+        "bbox": [
+            140,
+            171,
+            825,
+            297
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e34d13f7875dad6b2addbdf85ed48c97c2c682e9f2494402836f5b985bf9731d.jpg",
+        "text": "$$\n\\mathrm { G W } \\mathrm { L R } _ { \\varepsilon , \\alpha } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) : = \\operatorname* { m i n } _ { ( Q , R , g ) \\in { \\mathscr C } ( a , b , r , \\alpha ) } { \\mathscr E } _ { A , B } ( Q \\mathrm { d i a g } ( 1 / g ) R ^ { T } ) - \\varepsilon H ( ( Q , R , g ) ) ~ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            304,
+            785,
+            332
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "183 Mirror Descent Scheme. As in [27], we propose to use a MD scheme with respect to the $\\mathrm { K L }$   \n184 divergence to approximate $\\mathbf { G } \\mathbf { W } \\mathbf { - L R } _ { \\varepsilon , \\alpha } ^ { ( r ) }$ in (7). More precisely, for any $\\varepsilon \\geq 0$ , the MD scheme leads   \n185 for all $k \\geq 0$ to the following updates which require solving a convex barycenter problem per step: ",
+        "bbox": [
+            140,
+            338,
+            826,
+            383
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edf9708285c5c8d1999d67c204842bcc7c925ee51bd843bd2f675bdf94f193e7.jpg",
+        "text": "$$\n( Q _ { k + 1 } , R _ { k + 1 } , g _ { k + 1 } ) : = \\underset { \\zeta \\in \\mathcal { C } ( a , b , r , \\alpha ) } { \\mathrm { a r g m i n ~ } } \\mathrm { K L } ( \\zeta , K _ { k } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            388,
+            651,
+            417
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "186 where $( Q _ { 0 } , R _ { 0 } , g _ { 0 } ) \\in \\mathcal { C } ( a , b , r )$ is an initial point such that $ { Q _ { 0 } } \\ > \\ 0$ and $\\begin{array} { r l r } { R _ { 0 } } & { { } > } & { 0 } \\end{array}$ ,   \n187 $P _ { k } : = Q _ { k } \\mathrm { d i a g } ( 1 / g _ { k } ) R _ { k } ^ { T }$ $\\mathring { \\mathbf { \\ i } } _ { k } , \\ \\mathbf { K } _ { k } : = \\big ( \\boldsymbol { K } _ { k } ^ { ( 1 ) } , \\boldsymbol { K } _ { k } ^ { ( 2 ) } , \\boldsymbol { K } _ { k } ^ { ( 3 ) } \\big ) , \\ \\boldsymbol { K } _ { k } ^ { ( 1 ) } : = \\mathrm { e x p } ( 4 \\gamma A P _ { k } B R _ { k } \\operatorname { d i a g } ( 1 /$   \n188 $( \\gamma \\varepsilon \\mathrm { ~ - ~ } 1 ) \\log ( Q _ { k } ) )$ , $\\begin{array} { r c l } { K _ { k } ^ { ( z ) } } & { : = } & { \\exp ( 4 \\gamma B P _ { k } ^ { T } D Q _ { k } \\mathrm { d i a g } ( 1 / g _ { k } ) - ( \\gamma \\varepsilon - 1 ) \\log ( R _ { k } ) ) } \\end{array}$ , $K _ { k } ^ { ( 3 ) } \\ : =$   \n189 $\\exp ( - 4 \\gamma \\omega _ { k } / g _ { k } ^ { 2 } - ( \\gamma \\varepsilon - 1 ) \\log ( g _ { k } ) )$ with $[ \\omega _ { k } ] _ { i } : = [ Q _ { k } ^ { T } A P _ { k } B R _ { k } ] _ { i , i }$ for all $i \\in \\{ 1 , \\ldots , r \\}$ and   \n190 $\\gamma$ is a positive step size. Solving (8) can be done efficiently thanks to the Dykstra’s Algorithm as   \n191 showed in [32]. See Appendix $\\textrm { C }$ for more details. ",
+        "bbox": [
+            140,
+            422,
+            826,
+            518
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Initialization. To initialize our algorithm, we adapt the First Lower Bound of [26] to our case of interest. More precisely, we show the following Proposition. See appendix A for the proof. ",
+        "bbox": [
+            163,
+            523,
+            823,
+            553
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "194 Proposition 1. Let us denote $\\tilde { x } = A ^ { \\odot 2 } a \\in \\mathbb { R } ^ { n }$ , $\\tilde { y } = B ^ { \\odot 2 } b \\in \\mathbb { R } ^ { m }$ and $\\tilde { C } = ( | \\tilde { x } _ { i } - \\tilde { y } _ { j } | ^ { 2 } ) _ { i , j } \\in \\mathbb { R } ^ { n \\times m }$   \n195 Then for all $\\varepsilon \\geq 0$ and $r \\geq 1$ we have, ",
+        "bbox": [
+            145,
+            556,
+            823,
+            587
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/165163749c6dedc2569351b606f6281455c75ab37b0fcaf5ff2ce82858404e52.jpg",
+        "text": "$$\n\\mathrm { G W - L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) \\geq \\operatorname* { m i n } _ { ( Q , R , g ) \\in { \\mathcal C } ( a , b , r , \\alpha ) } \\langle \\tilde { C } , Q \\mathrm { d i a g } ( 1 / g ) R ^ { T } \\rangle - \\varepsilon H ( ( Q , R , g ) ) ~ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            592,
+            784,
+            621
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "196 Note that the RHS of the inequality (9) is exactly the problem studied in [32] for which an algorithm   \n197 was proposed. Therefore to initialize our algorithm, we propose to use their approach. Note that here   \n198 the cost $\\tilde { C }$ is the squared Euclidean distance between two families $\\{ \\tilde { x } _ { 1 } , \\ldots , \\tilde { x } _ { n } \\}$ and $\\{ \\tilde { y } _ { 1 } , \\dots , \\tilde { y } _ { m } \\}$   \n199 in 1-D which admits a low-rank factorization. Therefore we can apply the linear-time version of the   \n200 algorithm presented in [32] to compute the solution. Algorithm 3 summarizes our approach, where   \n201 $\\mathcal { D } ( \\cdot )$ denotes the operator extracting the diagonal of a square matrix.   \n202 Computational Cost. Computing the initialization goes through the computations of $\\tilde { x }$ and $\\tilde { y }$ which   \n203 requires $O ( n ^ { 2 } + m ^ { 2 } )$ algebraic operations. Moreover, applying the algorithm proposed in [32]   \n204 when the underlying cost is the squared Euclidean distances between two families in 1-D needs   \n205 only $\\mathcal { O } ( ( n + m ) r )$ algebraic operations. Solving the barycenter problem as defined in (8) can be   \n206 207 done given $( K _ { k } ^ { ( 1 ) } , \\dot { K } _ { k } ^ { ( 2 ) } , K _ { k } ^ { ( 3 ) } )$ ) Dykstra’s Algorithm. Indeed in [32, Algorithm, each iteration of their algorithm requires only $\\mathcal { O } ( ( n + m ) r )$ show thatalgebraic   \n208 operations since it involves only matrix/vector multiplications. However computing the kernel   \n209 matrices $( K _ { k } ^ { ( 1 ) } , K _ { k } ^ { ( 2 ) } , K _ { k } ^ { ( 3 ) } )$ at each iteration of Algorithm 3 requires a quadratic complexity with   \n210 respect to the number of samples. Overall the proposed algorithm, while faster than the cubic   \n211 implementation proposed in [27], still needs $\\mathcal { O } ( ( n ^ { 2 } + m ^ { 2 } ) r )$ operations per iteration. In the following   \n212 we will see that by combining both nonnegative low-rank constraints on the coupling and low-rank   \n213 approximations of the distance matrices, we can obtain a linear time algorithm with respect to the   \n214 number of samples which computes an approximation of the GW distance. ",
+        "bbox": [
+            140,
+            632,
+            825,
+            719
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            138,
+            724,
+            825,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\mathrm { A l g o r i t h m } 3 \\mathrm { L o w - R a n k } \\mathrm { G W } , \\mathrm { G W - L R } _ { \\varepsilon , \\alpha } ^ { ( r ) } ( ( a , A ) , ( b , B ) )$ ",
+        "bbox": [
+            173,
+            92,
+            545,
+            111
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for $k = 1 , \\dots$ do ",
+        "bbox": [
+            173,
+            199,
+            290,
+            212
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "end ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            303,
+            202,
+            314
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Return: ",
+        "bbox": [
+            174,
+            385,
+            246,
+            397
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "215 Convergence of the mirror descent. Even if the objective (7) is not convex in $( Q , R , g )$ , we obtain   \n216 the non-asymptotic stationary convergence of the MD algorithm in this setting. For that purpose   \n217 we consider the same convergence criterion as the one proposed in [32] to obtain non-asymptotic   \n218 stationary convergence of the MD scheme defined as ",
+        "bbox": [
+            140,
+            422,
+            825,
+            478
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/092911e0842e52a37182e38874ceb6a5626fb289aae8c445942906052764ffb4.jpg",
+        "text": "$$\n\\Delta _ { \\varepsilon , \\alpha } ( \\pmb { \\xi } , \\gamma ) : = \\frac { 1 } { \\gamma ^ { 2 } } ( \\mathrm { K L } ( \\pmb { \\xi } , \\mathcal { G } _ { \\varepsilon , \\alpha } ( \\pmb { \\xi } , \\gamma ) ) + \\mathrm { K L } ( \\mathcal { G } _ { \\varepsilon , \\alpha } ( \\pmb { \\xi } , \\gamma ) , \\pmb { \\xi } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            484,
+            694,
+            517
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where 219 $\\begin{array} { r } { \\mathcal { G } _ { \\varepsilon , \\alpha } ( \\pmb { \\xi } , \\gamma ) : = \\mathrm { a r g m i n } _ { \\zeta \\in \\mathcal { C } ( a , b , r , \\alpha ) } \\{ \\langle \\nabla \\mathcal { E } _ { A , B } ( \\pmb { \\xi } ) , \\pmb { \\zeta } \\rangle + \\frac { 1 } { \\gamma } \\mathrm { K L } ( \\zeta , \\pmb { \\xi } ) \\} } \\end{array}$ . For any $1 / r \\ge \\alpha > 0$ , we 220 show in the following proposition the non-asymptotic stationary convergence of the MD scheme 221 applied to the problem (7). See Appendix A for the proof. ",
+        "bbox": [
+            142,
+            523,
+            826,
+            570
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "222 Proposition 2. Let $\\varepsilon \\geq 0$ , $\\textstyle { \\frac { 1 } { r } } \\geq \\alpha > 0$ and $N \\geq 1$ . By denoting $L _ { \\varepsilon , \\alpha } : = 2 7 ( \\| A \\| _ { 2 } \\| B \\| _ { 2 } / \\alpha ^ { 4 } + \\varepsilon )$ and by considering a constant stepsize in the MD scheme 223 $\\begin{array} { r } { { \\bf \\Phi } ^ { \\left( 8 \\right) } \\gamma = \\frac { 1 } { 2 L _ { \\varepsilon , \\alpha } } } \\end{array}$ , we obtain that ",
+        "bbox": [
+            145,
+            571,
+            828,
+            608
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a5e2eac46bbf41dde0634344967f388d78912eb7c4c1ef6b60fb388f31df82f3.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { 1 \\leq k \\leq N } \\Delta _ { \\varepsilon , \\alpha } ( ( Q _ { k } , R _ { k } , g _ { k } ) , \\gamma ) \\leq \\frac { 4 L _ { \\varepsilon , \\alpha } D _ { 0 } } { N } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            614,
+            643,
+            647
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where 224 $D _ { 0 } : = \\mathscr { E } _ { A , B } ( Q _ { 0 } \\mathrm { d i a g } ( 1 / g _ { 0 } R _ { 0 } ^ { T } ) - \\mathbf { G } \\mathbf { W } \\mathbf { - } \\mathbf { L R } ^ { ( r ) } ( ( a , A ) , ( b , B ) )$ is the distance of the initial value 225 to the optimal one. ",
+        "bbox": [
+            148,
+            655,
+            826,
+            686
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Recall that for $\\alpha$ sufficiently small, we have $\\mathrm { { \\bf G W - L R } } _ { \\varepsilon , \\alpha } ^ { ( r ) } ( ( a , A ) , ( b , B ) ) = { \\bf G W - L R } _ { \\varepsilon } ^ { ( r ) } ( ( a , A ) , ( b , B ) )$ . Thus Proposition 2 show that our algorithm reach a stationary point of (7). In particular, if $\\varepsilon = 0$ , the proposed algorithm converges towards a stationary point of (5). ",
+        "bbox": [
+            158,
+            696,
+            826,
+            742
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "229 5 Double Low-rank Approach for Linear Time GW ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            760,
+            619,
+            779
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Almost all operations in Algorithm 3 are linear time, except for the three updates highlighted in red, involving $C _ { 1 }$ and $C _ { 2 }$ , and the computations of $\\tilde { x } = A ^ { \\odot \\tilde { 2 } } a$ and $\\tilde { y } = B ^ { \\odot 2 } b$ as they still require a quadratic number of algebraic operations. When adding Assumption 1 from $\\ S 3$ to the rank constrained approach from $\\ S 4$ , we notice that the strengths of both approaches can work hand in hand, both in easier initial evaluations of $\\tilde { x } , \\tilde { y }$ , but, most importantly, at each new recomputation of a factorized linearization of the quadratic objective: ",
+        "bbox": [
+            173,
+            791,
+            825,
+            876
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Linear time outfactorization for rms. Because . Indeed, rema $A$ admits a that for a low-rank. Therefore $A ^ { \\odot 2 }$ $\\begin{array} { r } { \\boldsymbol { x } , \\boldsymbol { y } \\in \\mathbb { R } ^ { d } , \\langle \\boldsymbol { x } , \\boldsymbol { y } \\rangle ^ { 2 } = \\sum _ { i , j = 1 } ^ { d } x _ { i } x _ { j } y _ { i } y _ { j } } \\end{array}$ ",
+        "bbox": [
+            171,
+            881,
+            823,
+            914
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "by studying the rows of matrices $A _ { 1 } : = [ a _ { 1 } ^ { ( 1 ) } ; . . . ; a _ { n } ^ { ( 1 ) } ]$ and $A _ { 2 } : = [ a _ { 1 } ^ { ( 2 ) } ; . . . ; a _ { n } ^ { ( 2 ) } ]$ , if one writes $\\psi ( \\boldsymbol { x } ) : = \\mathrm { V e c t } ( \\boldsymbol { x } \\boldsymbol { x } ^ { T } ) \\in \\mathbb { R } ^ { d ^ { 2 } }$ where $\\mathrm { V e c t } ( \\cdot )$ is the vectorization operation, we obtain that ",
+        "bbox": [
+            173,
+            88,
+            825,
+            123
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2b283825516b4de7bd182f0d11f7fdf85f0596c97ce3d4e8dca6594ac5b6fb79.jpg",
+        "text": "$$\n\\begin{array} { r } { A ^ { \\odot 2 } = \\tilde { A } _ { 1 } \\tilde { A _ { 2 } } ^ { T } \\mathrm { ~ w h e r e ~ } \\tilde { A _ { 1 } } = [ \\psi ( a _ { 1 } ^ { ( 1 ) } ) , \\dots , \\psi ( a _ { n } ^ { ( 1 ) } ) ] ^ { T } , \\tilde { A _ { 2 } } = [ \\psi ( a _ { 1 } ^ { ( 2 ) } ) , \\dots , \\psi ( a _ { n } ^ { ( 2 ) } ) ] ^ { T } \\ . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            128,
+            782,
+            151
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Algorithm 3, the line “Step 236 $( \\star ) ^ { \\dagger }$ can thus be replaced by $\\tilde { x } \\gets \\tilde { A _ { 1 } } \\tilde { A _ { 2 } } ^ { T } a$ and $\\tilde { y } \\gets \\tilde { B _ { 1 } } \\tilde { B _ { 2 } } ^ { T } b$ Note 237 that computing $\\tilde { A } _ { 1 }$ given $A _ { 1 }$ requires only $\\mathcal { O } ( n d ^ { 2 } )$ operations, so that this alternate code only takes 238 $\\mathcal { O } ( n d ^ { 2 } ) \\bar { + } \\mathcal { O } ( \\bar { m } ( d ^ { \\prime } ) ^ { 2 } )$ operations. ",
+        "bbox": [
+            140,
+            156,
+            825,
+            207
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "239 Linear time linearization of the GW objective. The linearization step, the critical step in Algo.1   \n240 that consists in updating $C$ at each iteration, consumes a substantial portion of the computational   \n241 budget of GW. Introducing the low-rank Sinkhorn approach makes this step quadratic in Algo.3; the   \n242 complexity of that step is also quadratic using the low-rank assumption on costs $A$ and $B$ , in Algo.2.   \n243 There is therefore an opportunity to marry both to speed-up that important step. We argue that this is   \n244 indeed what happens, in the sense that combining the two yields indeed linear time complexities in   \n245 sample sizes, by replacing in Algorithm 3, the lines “Step $( \\star \\star ) ^ { \\flat }$ b y ",
+        "bbox": [
+            138,
+            210,
+            825,
+            309
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5d6a415aa8c39fb9491c2d8af0ba1b07a88ead577c50bddef225657275aa9e4.jpg",
+        "text": "$$\n{ \\cal C } _ { 1 } \\gets - A _ { 1 } A _ { 2 } ^ { T } Q \\mathrm { d i a g } ( 1 / g ) \\quad \\mathrm { a n d } \\quad { \\cal C } _ { 2 } \\gets R ^ { T } B _ { 2 } B _ { 1 } ^ { T } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            314,
+            679,
+            333
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "246 Note that this speed-up would not be achieved using other approaches that output a low rank   \n247 approximation of the transport plan [4, 3, 31]. The crucial obstacle to using these methods here is that   \n248 the cost matrix $C$ in GW is “synthetic“, in the sense that it is the output of a matrix product $A P B$   \n249 involving the very last transport $P$ . This stands in stark contrast with the requirements in [4, 3, 31]   \n250 that the kernel matrix corresponding to $K _ { \\varepsilon } = e ^ { - C / \\varepsilon }$ admits favorable properties, such as being p.s.d   \n251 or admitting an explicit (random or not) finite dimensional feature approximation. Since $C$ changes   \n252 at each iteration in Algo.1, they are not directly applicable.   \n53 Combining the results in $\\ S 4$ with those from $\\ S _ { \\mathbf { B } }$ results in updates for $C _ { 1 }$ and $C _ { 2 }$ that only require   \n54 $\\mathcal { O } ( n r d )$ and $\\mathcal { O } ( m r d ^ { \\prime } )$ operations. ",
+        "bbox": [
+            140,
+            338,
+            825,
+            438
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            155,
+            443,
+            823,
+            472
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Linear time GW. Finally all the quadratic operations appearing in Algorithm (3) can be replaced by linear counterparts. The iterations that have not been modified had an overall complexity of $\\bar { \\mathcal { O } } ( m r ( r + d ^ { \\prime } ) + \\bar { n } r ( r + d ) )$ at each iteration. The initialization and linearization steps can now be performed in linear time, with respective complexity of respectively $\\mathcal { O } ( n ( r + d ^ { 2 } ) + \\mathrm { \\dot { \\ m } } ( ( d ^ { \\prime } ) ^ { 2 } + r ) )$ and $\\mathcal { O } ( ( n r ( r + d ) + m r ( r + d ^ { \\prime } ) )$ . ",
+        "bbox": [
+            161,
+            477,
+            825,
+            547
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "260 6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            565,
+            312,
+            583
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our goal in this section is to demonstrate that, for a far smaller computational budget, the GW-LR approach is competitive with the direct entropic approach on datasets that are either synthesized to exhibit local clusters, or directly validated on a real high-dimensional dataset as well. Because both approaches have different hyperparameters, our goal is to stick to a realistic evaluation that stresses both optimality of solutions as a function of computational effort, as well as performance in real life applications. We start by investigating the sensitivity of hyperparamaters $\\varepsilon$ and $\\gamma$ on our method. Since GW is not convex, these may interact in unexpected ways. Experiments were run on a personal MacBook Pro 2019 laptop. We reused code from github.com/meyerscetbon/LOT, and downloaded genomics data from github.com/rsinghlab/SCOT. ",
+        "bbox": [
+            171,
+            595,
+            825,
+            722
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Benchmarks. We consider three synthetic problems and one real world problem to evaluate timeaccuracy trade-offs, and also compare the couplings obtained by our method and that of the entropic version [27]. More precisely, we compare the quadratic approach in GW-LR computed with algorithm (3) (and its linear time counterpat, Lin GW-LR as presented in $\\ S 5$ ), with EntropicGW, the cubic implementation of [27] (as well as its quadratic counterpart, Quad Entropic-GW presented in Algo. 2). For GW-LR and Lin GW-LR, and in all experiments, we set the lower bound on entries of g to ↵ = 10\u000010 . ",
+        "bbox": [
+            158,
+            727,
+            825,
+            825
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Initialization To initialize all algorithms with a common strategy, we adapted the first lower bound of [26, Def. 6.1] to the entropic case. In all experiments showing time-accuracy tradeoffs, we choose to use number of operations to provide platform independent quantities. Accuracy is measured by evaluating the ground-truth energy $\\mathcal { E } _ { , B }$ (even in scenarios when the method uses a low rank approximation for $A , B$ at optimization time). ",
+        "bbox": [
+            171,
+            830,
+            823,
+            901
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/ced3128aec5a3e2f1ee8b3a2ad54bff8f7dd8e662233522796a537c2ee35f7a0.jpg",
+        "image_caption": [
+            "Figure 3: The number of cluster in each distribution is 10 and the number of samples is $n = m = 5 0 0 0$ . The ground cost is the Euclidean distance. As we can evaluate the distance between two arbitrary points, we can obtain in linear-time an efficient approximation of the distance matrices $A$ and $B$ as presented in 3. The rank of their factorizations is fixed to be $d = d ^ { \\prime } = 1 0 0$ . GW-LR and EntropicGW corresponds to the case where the full matrices $A$ and $B$ are considered while Lin GW-LR and Quad Entropic-GW take as inputs the low-rank approximations of the distance matrices. We plot the time-accuracy tradeoff for multiple choices of $\\gamma$ and rank $r$ defined as a fraction of $n$ . For Entropic-GW and Quad Entropic-GW, we set $\\varepsilon = 1 / \\gamma$ as proposed in [27]. Recall that for low-rank methods, we set $\\varepsilon = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            87,
+            825,
+            209
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Sensitivity to $\\gamma$ and $\\varepsilon$ Here we aim at showing the dependence in both $\\gamma$ and $\\varepsilon$ of our proposed method. In Figure 2, we compare the GW loss obtained by our algorithm when varying $\\varepsilon$ and $\\gamma$ on two mixtures. We show that when $\\varepsilon = 0$ , the proposed method manage to consistently obtain small GW loss whatever $\\gamma$ is. 8 By allowing $\\varepsilon > 0$ , the algorithm is able to reach even smaller GW loss, however, the choice of $\\varepsilon$ depends highly on $\\gamma$ . Therefore in the following experiments, we fix $\\varepsilon = 0$ for our method. We also show the dependence in $\\gamma$ and $\\varepsilon$ of our method in other settings and observe similar behaviors. See Appendix D.2 for 4 more details. ",
+        "bbox": [
+            156,
+            375,
+            517,
+            553
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. As shown in Figure 8 in Appendix D.2, allowing $\\varepsilon > 0$ may also increase the speed of convergence of the algorithm. However choosing well $\\varepsilon$ for a given $\\gamma$ must be done carefully and we prefer in the following experiments to present the performance of our method in the simplest setting where $\\varepsilon = 0$ . ",
+        "bbox": [
+            173,
+            561,
+            517,
+            645
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/58a709d7ba81bc3cb25edfaf380271a8b7d1638e8c02cdcfc50025f0b4029cf8.jpg",
+        "image_caption": [
+            "Figure 2: In this experiment, we consider two mixtures of (2 and 3) Gaussians in respectively 5-D and 10-D, sampled as discrete measures with $n = m = 5 0 0 0$ points, see more details on setup in Appendix D.2. The ground cost is the squared Euclidean distance, which provides an exact low-rank factorization of the cost as presented in $\\ S \\ O 3$ . Results on speed (in Appendix) are therefore obtained using Lin GW-LR. The nonnegative rank of the coupling is set to $r = 5 0 = n / 1 0 0$ . We plot the GW loss obtained by Lin GW-LR when varying $\\epsilon$ for multiple choices of $\\gamma$ . Both size and color have been used to quantify visually the value of the loss at that parameter pair. Occasional inversions are due to the nonconvex nature of the GW problem. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            531,
+            393,
+            803,
+            545
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "01 Synthetic low-rank problem In this experiment   \n02 we aim at comparing the time-accuracy tradeoff of   \n03 the different methods when the underlying distribu  \n04 tions has a low-rank structure. For that purpose, we   \n05 consider two distributions in respectively 10-D and   \n06 15-D, where the support of each distributions is the   \n07 concatenation of clusters of points, and where the eu  \n08 clidean distance between the centroids of the clusters   \n09 is bigger than a threshold $\\beta$ . Here we set $\\beta = 1 0$ .   \n10 Both distributions are uniform, have the same number   \n311 of clusters and the same number of points in each cluster. Some illustrations of the simulated data   \n312 is provided in Appendix D.3. In Figure 3, when the underlying cost is the (not squared) Euclidean   \n313 distance, our methods manage to consistently obtain similar accuracy that the ones obtained by   \n314 entropic methods, with very low rank $r = n / 5 0 0$ , while being orders of magnitude faster. In Figure 4,   \n315 we also compare the time-accuracy tradeoffs in the more favorable case where the underlying cost   \n316 is the squared Euclidean distance and obtain similar results. We also show more experiments for   \n317 different number of clusters in Appendix D.3, leading to similar conclusions. ",
+        "bbox": [
+            151,
+            676,
+            517,
+            814
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/205bfafdd7068632b78380a9723796c677156a099b7b76bf03b8ebfa0f53f9ad.jpg",
+        "image_caption": [
+            "Figure 4: The number of clusters in each distribution is 5 and the number of samples considered here is $n = m = 1 0 0 0 0$ . The ground cost is the squared Euclidean distance. We compare Lin GW-LR and Quad Entropic-GW as we have an exact factorization of the matrices $A$ and $B$ . We plot the time-accuracy tradeoff when varying $\\gamma$ for multiple choices of $r$ . For Quad Entropic-GW, we set $\\varepsilon = 1 / \\gamma$ and for Lin GW-LR we set $\\varepsilon = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            88,
+            823,
+            196
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/f357507bc64ee0a9d0b9ea5317bd8491b5252e6d657cf1bfa6c8e18bf3b3b8cf.jpg",
+        "image_caption": [
+            "Figure 5: We plot, for each cells of the SNAREseq dataset, the FOSCTTM ranked in the increasing order for both GW-LR and Entropic-GW. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            286,
+            825,
+            398
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/44a0b85b7dc95af7583f8947b4baaddc578e9a97e8c9a0235263eef73478c8a8.jpg",
+        "image_caption": [
+            "Figure 6: Plot of the time-accuracy tradeoff when varying $\\gamma$ for multiple choices of rank $r$ on the SNAREseq dataset. For Entropic-GW we set $\\varepsilon = 1 / \\gamma$ , for GW-LR, we set $\\varepsilon = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            445,
+            823,
+            555
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experiments on Single Cell Genomics Data. We reproduce the single-cell alignment experiment introduced in [14]. The dataset consists in single-cell multi-omics data generated by co-assays. In that setup, the ground truth one–to-one correspondence information between cells is known, and can therefore be used to benchmark GW strategies. The dataset considered is the SNAREseq [10], with $n = m = 1 0 4 7$ . We apply the exact same pre-processing steps as proposed in [14] by computing intra-domain distance matrices $A$ and $B$ with a k-NN graphs based on correlations, to compute shortest path distance matrices. Note that in that case, one cannot obtain directly in linear time a low-rank factorization of $A$ and $B$ using [5, 21], since the shortest path distances need to be computed first. Therefore we only consider the quadratic GW-LR and the cubic Entropic-GW. In Figure 6, we compare the alignment performance through the “fraction of samples closer than the true match” (FOSCTTM) introduced in [25]. We see that both algorithm obtain similar performance. However, in Figure 5, we show that whatever the $\\gamma$ chosen, GW-LR reaches better accuracy while being order of magnitude faster than Entropic-GW for a very small rank $r = 1 0$ . ",
+        "bbox": [
+            171,
+            616,
+            825,
+            796
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Conclusion. While the factorization introduced in [32] held the promise to speed up classic OT, we have shown in this work that it delivers an even larger impact when applied to the GW problem: Indeed, the combination of low-rank Sinkhorn factorization with-low rank cost matrices is the only one, to our knowledge, that ensures that the linearization step of the GW objective can be carried out with a linear complexity, throughout outer iterations. This linear complexity is comparable to that of the most recent OT solvers, yet still retains the appealing properties of the Entropic approach, such as stability and convergence to meaningful solutions. ",
+        "bbox": [
+            169,
+            803,
+            825,
+            900
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            90,
+            267,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] Jean Alaux, Edouard Grave, Marco Cuturi, and Armand Joulin. Unsupervised hyperalignment for multilingual word embeddings. arXiv preprint arXiv:1811.01124, 2018.   \n[2] Jason Altschuler, Jonathan Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. arXiv preprint arXiv:1705.09634, 2017.   \n[3] Jason Altschuler, Francis Bach, Alessandro Rudi, and Jonathan Niles-Weed. Massively scalable sinkhorn distances via the nyström method, 2018.   \n[4] Jason Altschuler, Francis Bach, Alessandro Rudi, and Jonathan Weed. Approximating the quadratic transportation metric in near-linear time. arXiv preprint arXiv:1810.10046, 2018.   \n[5] Ainesh Bakshi and David P. Woodruff. Sublinear time low-rank approximation of distance matrices, 2018.   \n[6] Andrew J Blumberg, Mathieu Carriere, Michael A Mandell, Raul Rabadan, and Soledad Villar. Mrec: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data. arXiv preprint arXiv:2001.01666, 2020.   \n[7] Charlotte Bunne, David Alvarez-Melis, Andreas Krause, and Stefanie Jegelka. Learning generative models across incomparable spaces. arXiv preprint arXiv:1905.05461, 2019.   \n[8] Rainer E Burkard, Eranda Cela, Panos M Pardalos, and Leonidas S Pitsoulis. The quadratic assignment problem. In Handbook of combinatorial optimization, pages 1713–1809. Springer, 1998.   \n[9] Laetitia Chapel, Mokhtar Alaya, and Gilles Gasso. Partial optimal transport with applications on positive-unlabeled learning. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020.   \n[10] Song Chen, Blue B Lake, and Kun Zhang. High-throughput sequencing of the transcriptome and chromatin accessibility in the same cell. Nature biotechnology, 37(12):1452–1457, 2019.   \n[11] Samir Chowdhury, David Miller, and Tom Needham. Quantized gromov-wasserstein. arXiv preprint arXiv:2104.02013, 2021.   \n[12] Alexis Conneau, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, and Hervé Jégou. Word translation without parallel data. arXiv preprint arXiv:1710.04087, 2017.   \n[13] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in neural information processing systems, pages 2292–2300, 2013.   \n[14] Pinar Demetci, Rebecca Santorella, Björn Sandstede, William Stafford Noble, and Ritambhara Singh. Gromov-wasserstein optimal transport to align single-cell multi-omics data. bioRxiv, 2020. doi: 10.1101/2020.04.28.066787.   \n[15] Pinar Demetci, Rebecca Santorella, Bjorn Sandstede, William Stafford Noble, and Ritambhara Singh. Gromov-wasserstein optimal transport to align single-cell multi-omics data. BioRxiv, 2020.   \n[16] Richard Mansfield Dudley et al. Weak convergence of probabilities on nonseparable metric spaces and empirical measures on euclidean spaces. Illinois Journal of Mathematics, 10(1): 109–126, 1966.   \n[17] Danielle Ezuz, Justin Solomon, Vladimir G Kim, and Mirela Ben-Chen. Gwcnn: A metric alignment layer for deep shape analysis. In Computer Graphics Forum, volume 36, pages 49–57. Wiley Online Library, 2017.   \n[18] Aden Forrow, Jan-Christian Hütter, Mor Nitzan, Philippe Rigollet, Geoffrey Schiebinger, and Jonathan Weed. Statistical optimal transport via factored couplings, 2018.   \n[19] Aden Forrow, Jan-Christian Hütter, Mor Nitzan, Philippe Rigollet, Geoffrey Schiebinger, and Jonathan Weed. Statistical optimal transport via factored couplings. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 2454–2465. PMLR, 2019.   \n[20] Steven Gold and Anand Rangarajan. Softassign versus softmax: Benchmarks in combinatorial optimization. Advances in neural information processing systems, pages 626–632, 1996.   \n[21] Piotr Indyk, Ali Vakilian, Tal Wagner, and David Woodruff. Sample-optimal low-rank approximation of distance matrices, 2019.   \n[22] Hicham Janati, Thomas Bazeille, Bertrand Thirion, Marco Cuturi, and Alexandre Gramfort. Multi-subject meg/eeg source imaging with sparse multi-task regression. NeuroImage, page 116847, 2020.   \n[23] Hiroshi Konno. Maximization of a convex quadratic function under linear constraints. Mathematical programming, 11(1):117–127, 1976.   \n[24] Matt Kusner, Yu Sun, Nicholas Kolkin, and Kilian Q Weinberger. From word embeddings to document distances. In Proc. of the 32nd Intern. Conf. on Machine Learning, pages 957–966, 2015.   \n[25] Jie Liu, Yuanhao Huang, Ritambhara Singh, Jean-Philippe Vert, and William Stafford Noble. Jointly embedding multiple single-cell omics measurements. BioRxiv, page 644310, 2019.   \n[26] Facundo Mémoli. Gromov–wasserstein distances and the metric approach to object matching. Foundations of computational mathematics, 11(4):417–487, 2011.   \n[27] Gabriel Peyré, Marco Cuturi, and Justin Solomon. Gromov-wasserstein averaging of kernel and distance matrices. In International Conference on Machine Learning, pages 2664–2672, 2016.   \n[28] Gabriel Peyré and Marco Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning, 11(5-6), 2019. ISSN 1935-8245.   \n[29] Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision, 40(2):99–121, November 2000.   \n[30] Ryoma Sato, Marco Cuturi, Makoto Yamada, and Hisashi Kashima. Fast and robust comparison of probability measures in heterogeneous spaces. arXiv preprint arXiv:2002.01615, 2020.   \n[31] Meyer Scetbon and Marco Cuturi. Linear time sinkhorn divergences using positive features, 2020.   \n[32] Meyer Scetbon, Marco Cuturi, and Gabriel Peyré. Low-rank sinkhorn factorization, 2021.   \n[33] Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176 (4):928–943, 2019.   \n[34] Richard Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist., 35:876–879, 1964.   \n[35] Justin Solomon, Gabriel Peyré, Vladimir G Kim, and Suvrit Sra. Entropic metric alignment for correspondence problems. ACM Transactions on Graphics (TOG), 35(4):1–13, 2016.   \n[36] Karl-Theodor Sturm. The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces. arXiv preprint arXiv:1208.0434, 2012.   \n[37] Titouan Vayer, Laetita Chapel, Rémi Flamary, Romain Tavenard, and Nicolas Courty. Fused gromov-wasserstein distance for structured objects: theoretical foundations and mathematical properties. arXiv preprint arXiv:1811.02834, 2018.   \n[38] Titouan Vayer, Rémi Flamary, Romain Tavenard, Laetitia Chapel, and Nicolas Courty. Sliced gromov-wasserstein. arXiv preprint arXiv:1905.10124, 2019.   \n[39] Jonathan Weed and Francis Bach. Sharp asymptotic and finite-sample rates of convergence of empirical measures in wasserstein distance. Bernoulli, 25(4A):2620–2648, 2019.   \n[40] Hongteng Xu, Dixin Luo, and Lawrence Carin. Scalable gromov-wasserstein learning for graph partitioning and matching. arXiv preprint arXiv:1905.07645, 2019.   \n[41] Hongteng Xu, Dixin Luo, Hongyuan Zha, and Lawrence Carin Duke. Gromov-wasserstein learning for graph matching and node embedding. In International conference on machine learning, pages 6932–6941. PMLR, 2019. ",
+        "bbox": [
+            165,
+            99,
+            830,
+            912
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            166,
+            51,
+            828,
+            848
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            868,
+            254,
+            883
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            897,
+            339,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The main claim of the paper is a linear time (w.r.t. sample size, as commonly understood in OT) computation for GW. Sections $\\ S 3 , 4$ and 5 build up that answer. This is experimentally validated across several experiments, both synthetic, to help the reader form intuitions, and on real data where GW was deemed useful.   \n(b) Did you describe the limitations of your work?[Yes] Because the GW problem is nonconvex, these limitations are naturally discussed in the experimental section, Section $\\ S 6$ We discuss the effects of $\\gamma$ and $\\varepsilon$ on the method, which are not easy to parse due to the non-convexity of the method.   \n(c) Did you discuss any potential negative societal impacts of your work?[N/A] As a purely methodological paper, we do not envision potentially negative impact of this work on its own.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?[Yes] We have read these guidelines and confirm our paper does conform to them. ",
+        "bbox": [
+            238,
+            92,
+            825,
+            320
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            325,
+            493,
+            340
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results?[Yes] Our theoretical results are of two nature: a bound in Proposition 1, and a guarantee on convergence in Proposition 2, which has no direct consequence on our empirical findings, yet remain useful. Theory is not our main contribution, but rather algorithms.   \n(b) Did you include complete proofs of all theoretical results?[Yes] all proofs are in the appendix. Space in the main body of the paper was prioritized to include experimental validation, which, for this non-convex problem, we believe to be equally important. ",
+        "bbox": [
+            238,
+            344,
+            825,
+            445
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            449,
+            393,
+            463
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(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] We have included portions of the code that we have used. We pledge to make the entire code available later in the reviewing process.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?[Yes] The main contribution is compute efficiency, that we have considered across several parameter choices.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?[N/A] Our experiments are deterministic, since we use a predefined initialization.   \n(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] The experiments were run with basic computational means, a macbook pro. ",
+        "bbox": [
+            238,
+            469,
+            825,
+            656
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            209,
+            661,
+            823,
+            675
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators?[Yes] We have reused a single toolbox, and accessed data available publicly.   \n(b) Did you mention the license of the assets? [Yes] All licenses are open source, see supplementary.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Yes, code is shared in the supplementary.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            680,
+            825,
+            829
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            833,
+            705,
+            848
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] ",
+        "bbox": [
+            238,
+            852,
+            825,
+            912
+        ],
+        "page_idx": 11
+    }
+]

parse/train/rJgYxn09Fm/rJgYxn09Fm.md

@@ -0,0 +1,289 @@
+# LEARNING IMPLICITLY RECURRENT CNNS THROUGH PARAMETER SHARING
+
+Pedro Savarese TTI-Chicago savarese@ttic.edu
+
+Michael Maire University of Chicago mmaire@uchicago.edu
+
+# ABSTRACT
+
+We introduce a parameter sharing scheme, in which different layers of a convolutional neural network (CNN) are defined by a learned linear combination of parameter tensors from a global bank of templates. Restricting the number of templates yields a flexible hybridization of traditional CNNs and recurrent networks. Compared to traditional CNNs, we demonstrate substantial parameter savings on standard image classification tasks, while maintaining accuracy.
+
+Our simple parameter sharing scheme, though defined via soft weights, in practice often yields trained networks with near strict recurrent structure; with negligible side effects, they convert into networks with actual loops. Training these networks thus implicitly involves discovery of suitable recurrent architectures. Though considering only the design aspect of recurrent links, our trained networks achieve accuracy competitive with those built using state-of-the-art neural architecture search (NAS) procedures.
+
+Our hybridization of recurrent and convolutional networks may also represent a beneficial architectural bias. Specifically, on synthetic tasks which are algorithmic in nature, our hybrid networks both train faster and extrapolate better to test examples outside the span of the training set.
+
+# 1 INTRODUCTION
+
+The architectural details of convolutional neural networks (CNNs) have undergone rapid exploration and improvement via both human hand-design (Simonyan & Zisserman, 2015; Szegedy et al., 2015; He et al., 2016; Huang et al., 2017; Zhu et al., 2018) and automated search methods (Zoph & Le, 2017; Liu et al., 2018). Yet, this vast array of work limits itself to a circuit-like view of neural networks. Here, a CNN is regarded as a fixed-depth feed-forward circuit, with a distinct parameter governing each internal connection. These circuits are often trained to perform tasks which, in a prior era, might have been (less accurately) accomplished by running a traditional computer program coded by humans. Programs, and even traditional hardware circuits, have a more reusable internal structure, including subroutines or modules, loops, and associated control flow mechanisms.
+
+We bring one aspect of such modularity into CNNs, by making it possible to learn a set of parameters that is reused across multiple layers at different depths. As the pattern of reuse is itself learned, our scheme effectively permits learning the length (iteration count) and content of multiple loops defining the resulting CNN. We view this approach as a first step towards learning neural networks with internal organization reminiscent of computer programs. Though we focus solely on loop-like structures, leaving subroutines and dynamic control flow to future work, this simple change suffices to yield substantial quantitative and qualitative benefits over the standard baseline CNN models.
+
+While recurrent neural networks (RNNs) possess a loop-like structure by definition, their loop structure is fixed a priori, rather than learned as part of training. This can actually be a disadvantage in the event that the length of the loop is mismatched to the target task. Our parameter sharing scheme for CNNs permits a mix of loops and feed-forward layers to emerge. For example, trained with our scheme, a 50-layer CNN might learn a 2-layer loop that executes 5 times between layers 10 and 20, a 3-layer loop that runs 4 times from layers 30 to 42, while leaving the remaining layers to assume independent parameter sets. Our approach generalizes both CNNs and RNNs, creating a hybrid.
+
+![](images/a2c9b5bb263d8cd0bf0d9985922b8c7f6508fe346bcd32491b1fd6dc3f69077c.jpg)  
+Figure 1: Parameter sharing scheme. Left: A CNN (possibly a variant such as a residual network), with each convolutional layer $_ { i }$ containing an individual parameter set $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ . Middle: Parameter sharing among layers, where parameter templates $\boldsymbol { \mathsf { T } } ^ { ( 1 ) }$ , ${ \boldsymbol { \mathsf { T } } } ^ { ( 2 ) }$ are shared among each layer $_ { i }$ , which now only contains a 2-dimensional parameter $\mathbf { \pmb { \alpha } } ^ { ( i ) }$ . Weights $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ (no longer parameters, illustrated with dotted boxes) used by layer $i$ are generated from $\mathbf { \pmb { \alpha } } ^ { ( i ) }$ and templates ${ \boldsymbol { \mathsf { T } } } ^ { ( 1 ) }$ , ${ \boldsymbol { \mathsf { T } } } ^ { ( 2 ) }$ . Right: If weights $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ are outputs of a linear function (as in our method), learning parameter templates can be viewed as learning layer templates, offering a new (although equivalent) perspective for the middle diagram. Non-linearities are omitted for simplicity.
+
+Figure 1 diagrams the parameter sharing scheme facilitating this hybridization. Inspired by dictionary learning, different network layers share, via weighted combination, global parameter templates. This re-parameterization is fully differentiable, allowing learning of sharing weights and template parameters. Section 3 elaborates, and also introduces tools for analyzing learned loop structures.
+
+Section 4 demonstrates advantages of our hybrid CNNs across multiple experimental settings. Taking a modern CNN design as a baseline, and re-parameterizing it according to our scheme improves:
+
+• Parameter efficiency. Here, we experiment with the standard task of image classification using modern residual networks (He et al., 2016; Zagoruyko & Komodakis, 2016). This task is a good proxy for general usefulness in computer vision, as high-performance classification architectures often serve as a backbone for many other vision tasks, such as semantic segmentation (Chen et al., 2016; Zhao et al., 2017).
+
+Our parameter sharing scheme drastically reduces the number of unique parameters required to achieve a given accuracy on CIFAR (Krizhevsky, 2009) or ImageNet (Russakovsky et al., 2015) classification tasks. Re-parameterizing a standard residual network with our scheme cuts parameters, without triggering any drop in accuracy. This suggests that standard CNNs may be overparameterized in part because, by design (and unlike RNNs), they lack capacity to learn reusable internal operations.
+
+• Extrapolation and generalization. Here, we explore whether our hybrid networks expand the class of tasks that one can expect to train neural networks to accomplish. This line of inquiry, focusing on synthetic tasks, shares motivations with work on Neural Turing Machines (Graves et al., 2014). Specifically, we would like neural networks to be capable of learning to perform tasks for which there are concise traditional solution algorithms. Graves et al. (2014) uses sorting as an example task. As we examine an extension of CNNs, our tasks take the form of queries about planar graphs encoded as image input.
+
+On these tasks, we observe improvements to both generalization ability and learning speed for our hybrid CNNs, in comparison to standard CNNs or RNNs. Our parameter sharing scheme, by virtue of providing an architectural bias towards networks with loops, appears to assist in learning to emulate traditional algorithms.
+
+An additional side effect, seen in practice in many of our experiments, is that two different learned layers often snap to the same parameter values. That is, layers $i$ and $j$ , learn coefficient vectors $\mathbf { \alpha } \alpha ^ { ( i ) }$ and $\pmb { \alpha } ^ { ( j ) }$ (see Figure 1) that converge to be the same (up to scaling). This is a form of architecture discovery, as it permits representation of the CNN as a loopy wiring diagram between repeated layers. Section 4.3 presents example results. We also draw comparisons to existing neural architecture search (NAS) techniques. By simply learning recurrent structure as byproduct of training with standard stochastic gradient descent, we achieve accuracy competitive with current NAS procedures.
+
+Before delving into the details of our method, Section 2 provides additional context in terms of prior work on recurrent models, parameter reduction techniques, and program emulation. Sections 3 and 4 describe our hybrid shared-parameter CNN, experimental setup, and results. Section 5 concludes with commentary on our results and possible future research pathways.1
+
+# 2 RELATED WORK
+
+Recurrent variants of CNNs are used extensively for visual tasks. Recently, Zamir et al. (2017) propose utilizing a convolutional LSTM (Shi et al., 2015) as a generic feedback architecture. RNN and CNN combinations have been used for scene labeling (Pinheiro & Collobert, 2014), image captioning with attention (Xu et al., 2015), and understanding video (Donahue et al., 2015), among others. These works combine CNNs and RNNs at a coarse scale, and in a fixed hand-crafted manner. In contrast, we learn the recurrence structure itself, blending it into the inner workings of a CNN.
+
+Analysis of residual networks (He et al., 2016) reveals possible connections to recurrent networks stemming from their design (Liao & Poggio, 2016). Greff et al. (2017) provide evidence that residual networks learn to iteratively refine feature representations, making an analogy between a very deep residual network and an unrolled loop. Jastrzebski et al. (2018) further explore this connection, and experiment with training residual networks in which some layers are forced to share identical parameters. This hard parameter sharing scheme again builds a predetermined recurrence structure into the network. It yields successfully trained networks, but does not exhibit the type of performance gains that Section 4 demonstrates for our soft parameter sharing scheme.
+
+Closely related to our approach is the idea of hypernetworks (Ha et al., 2016), in which one part of a neural network is parameterized by another neural network. Our shared template-based reparameterization could be viewed as one simple choice of hypernetwork implementation. Perhaps surprisingly, this class of ideas has not been well explored for the purpose of reducing the size of neural networks. Rather, prior work has achieved parameter reduction through explicit representation bottlenecks (Iandola et al., 2016), sparsifying connection structure (Prabhu et al., 2018; Huang et al., 2018; Zhu et al., 2018), and pruning trained networks (Han et al., 2016).
+
+Orthogonal to the question of efficiency, there is substantial interest in extending neural networks to tackle new kinds of tasks, including emulation of computer programs. Some approach this problem using additional supervision in the form of execution traces (Reed & de Freitas, 2016; Cai et al., 2017), while other focus on development of network architectures that can learn from input-output pairs alone (Graves et al., 2014; 2016; Zaremba et al., 2016; Trask et al., 2018). Our experiments on synthetic tasks fall into the latter camp. At the level of architectural strategy, Trask et al. (2018) benefit from changing the form of activation function to bias the network towards correctly extrapolating common mathematical formulae. We build in a different implicit bias, towards learning iterative procedures within a CNN, and obtain a boost on correctly emulating programs.
+
+# 3 SOFT PARAMETER SHARING
+
+In convolutional neural networks (CNNs) and variants such as residual CNNs (ResNets) (He et al., 2016) and DenseNets (Huang et al., 2017), each convolutional layer $i$ contains a set of parameters $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ , with no explicit relation between parameter sets of different layers. Conversely, a strict structure is imposed to layers of recurrent neural networks (RNNs), where, in standard models (Hochreiter & Schmidhuber, 1997), a single parameter set $\pmb { \mathsf { W } }$ is shared among all time steps. This leads to a program-like computational flow, where RNNs can be seen as loops with fixed length and content. While some RNN variants (Graves et al., 2013; Koutn´ık et al., 2014; Yang et al., 2018) are less strict on the length or content of loops, these are still typically fixed beforehand.
+
+As an alternative to learning hard parameter sharing schemes – which correspond to the strict structure present in RNNs – our method consists of learning soft sharing schemes through a relaxation of this structure. We accomplish this by expressing each layer’s parameters $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ as a linear combination of parameter templates $\mathbf { \overline { { I } } } ^ { ( 1 ) } , \ldots , \mathbf { \overline { { I } } } ^ { ( k ) }$ , each with the same dimensionality as $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ :
+
+![](images/30aa8071d0e5ad98b194b3c861782577cf515638614a5ad4b9997a5766ff7be7.jpg)  
+Figure 2: Connection between the LSM matrix $S$  where $\begin{array} { r } { S _ { i , j } = \frac { | \langle { \pmb { \alpha } } ^ { ( i ) } , { \pmb { \alpha } } ^ { ( j ) } \rangle | } { \| { \pmb { \alpha } } ^ { ( i ) } \| \| { \pmb { \alpha } } ^ { ( j ) } \| } \ ) } \end{array}$ and the structure of the network. White and black entries correspond to maximum and minimum similarities $( S _ { i , j } = 1$ and $S _ { i , j } = 0$ , respectively). Left: Empirically, CNNs present no similarity between parameters of different layers. Middle: Trained with our method, the layer similarity matrix (LSM) captures similarities between different layers, including pairs with close to maximum similarity. Such pairs (depicted by same-colored coefficients and weights, and by white entries in the LSM) perform similar operations on their inputs. Right: We can tie together parameters of similar layers, creating a hard parameter sharing scheme. The network can then be folded, creating self-loops and revealing an explicit recurrent computation structure.
+
+$$
+\pmb { \mathsf { W } } ^ { ( i ) } : = \sum _ { j = 1 } ^ { k } \alpha _ { j } ^ { ( i ) } \pmb { \mathsf { T } } ^ { ( j ) }
+$$
+
+where $k$ is the number of parameter templates (chosen freely as a hyperparameter) and $\mathbf { \alpha } \alpha ^ { ( i ) }$ , a $k$ - dimensional vector, is the coefficients of layer $i$ . Figure 1 (left and middle) illustrates the difference between networks trained with and without our method. This relaxation allows for coefficients and parameter templates to be (jointly) optimized with gradient-based methods, yielding negligible extra computational cost, with a single constraint that only layers with same parameter sizes can share templates. Note that constraining coefficients $\mathbf { \alpha } \alpha ^ { ( i ) }$ to be one-hot vectors leads to hard sharing schemes, at the cost of non-differentiability.
+
+Having $k$ as a free parameter decouples the number of parameters in network from its depth. Typically, $L$ convolutional layers with constant channel and kernel sizes $C , K$ have $O ( L C ^ { 2 } { \dot { K } } ^ { 2 } )$ total parameters. Our soft sharing scheme changes the total number of parameters to $O ( k L + k C ^ { 2 } K ^ { 2 } ) =$ $\scriptstyle \dot { O } ( k C ^ { 2 } K ^ { 2 } )$ . Sections 4.1 and 4.2 show that we can decrease the parameter count of standard models without significantly impacting accuracy, or simply attain higher accuracy with $k = L$ .
+
+In the next two subsections, we discuss two consequences of the linearity of Equation (1). First, it enables alternative interpretations of our method. Second, and a major advantage, as is the case in many linear relaxations of integer problems, we are able to extract hard sharing schemes in practice, and consequently detect implicit self-loops in a CNN trained with our method.
+
+# 3.1 INTERPRETATION
+
+For layers $i$ that are linear in $\boldsymbol { \mathsf { W } } ^ { ( i ) }$ (e.g. matrix multiplication, convolution), we can view our method as learning template layers which are shared among a network. More specifically, for a convolutional layer $\mathbf { U } ^ { ( i ) } ( \mathbf { X } ) = \mathbf { W } ^ { ( i ) } \ast \mathbf { X }$ , and considering Equation (1):
+
+$$
+\mathbf { U } ^ { ( i ) } ( \mathbf { X } ) = \mathbf { W } ^ { ( i ) } * \mathbf { X } = \sum _ { j = 1 } ^ { k } \alpha _ { j } ^ { ( i ) } \mathbf { T } ^ { ( j ) } * \mathbf { X }
+$$
+
+where $\pmb { \mathsf { T } } ^ { ( j ) } * \pmb { \mathsf { X } }$ , the result of a convolution with filter sets $\bar { \mathsf { T } } ^ { ( j ) }$ , can be seen as the output of a template layer with individual parameters $\boldsymbol { \mathsf { T } } ^ { ( j ) }$ . Such layers can be seen as global feature extractors,
+
+and coefficients $\mathbf { \alpha } \alpha ^ { ( i ) }$ determine which features are relevant for the $i$ ’th computation of a network.   
+This is illustrated in Figure 1 (right diagram).
+
+This view gives a clear connection between coefficients $_ { \pmb { \alpha } }$ and the network’s structure. Having $\pmb { \alpha } ^ { ( i ) } = \pmb { \alpha } ^ { ( i + 2 ) }$ yields $\begin{array} { r } { \mathbf { W } ^ { ( i ) } = \sum _ { j = 1 } ^ { k } \alpha _ { j } ^ { ( i ) } \overline { { \mathbf { T } ^ { ( j ) } } } = \sum _ { j = 1 } ^ { k } \alpha _ { j } ^ { ( i + 2 ) } \overline { { \mathbf { T } ^ { ( j ) } } } = \mathbf { W } ^ { ( i + 2 ) } } \end{array}$ α(i+2)T(j) = W(i+2), and hence layers i and $i + 2$ are functionally equivalent. Such a network can be folded to generate an equivalent model with two layers and a self-loop, an explicitly recurrent network. While this is also possible for networks without parameter sharing, a learned alignment of $C ^ { 2 } K ^ { 2 }$ parameters is required (unlikely in practice), instead of aligning only $k \leq L$ parameters.
+
+# 3.2 IMPLICIT RECURRENCES
+
+To identify which layers in a network perform approximately the same operation, we can simply check whether their coefficients are similar. We can condense this information for all pairs of layers $i , j$ in a similarity matrix $S$ , where $S _ { i , j } = s ( \pmb { \alpha } ^ { ( i ) } , \pmb { \alpha } ^ { ( j ) } )$ for some similarity measure $s$ .
+
+For networks with normalization layers, the network’s output is invariant to weight rescaling. In this setting, a natural measure is $\begin{array} { r } { s ( \pmb { \alpha } ^ { ( i ) } , \pmb { \alpha } ^ { ( j ) } ) = \frac { | \langle \pmb { \alpha } ^ { ( i ) } , \pmb { \alpha } ^ { ( j ) } \rangle | } { \| \pmb { \alpha } ^ { ( i ) } \| \| \pmb { \alpha } ^ { ( j ) } \| } } \end{array}$ (absolute value of cosine similarity), since it possess this same property.2 We call $S$ the layer similarity matrix (LSM). Figure 2 illustrates and Section 4.3 shows experimentally how it can be used to extract recurrent loops from trained CNNs.
+
+While structure might emerge naturally, having a bias towards more structured (recurrent) models might be desirable. In this case, we can add a recurrence regularizer to the training objective, pushing parameters to values which result in more structure. For example, we can add the negative of sum of elements of the LSM: $\begin{array} { r } { \mathcal { L } _ { R } = \mathcal { L } - \lambda _ { R } \sum _ { i , j } S _ { i , j } } \end{array}$ , where $\mathcal { L }$ is the original objective. The larger $\lambda _ { R }$ is, the closer the elements of $S$ will be to 1. At an extreme case, this regularizer will push all elements in $S$ to 1, resulting in a network with a single layer and a self-loop.
+
+# 4 EXPERIMENTS
+
+We begin by training variants of standard models with soft parameter sharing, observing that it can offer parameter savings with little impact on performance, or increase performance at the same parameter count. Section 4.3 demonstrates conversion of a trained model into explicitly recurrent form. We then examine synthetic tasks (Section 4.4), where parameter sharing improves generalization. Appendix B contains details on the initialization for the coefficients $_ { \pmb { \alpha } }$ .
+
+# 4.1 CLASSIFICATION ON CIFAR
+
+The CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) are composed of 60, 000 colored $3 2 \times 3 2$ images, labeled among 10 and 100 classes respectively, and split into 50, 000 and 10, 000 examples for training and testing. We pre-process the training set with channel-wise normalization, and use horizontal flips and random crops for data augmentation, following He et al. (2016).
+
+Using Wide ResNets (WRN) (Zagoruyko & Komodakis, 2016) as a base model, we train networks with the proposed soft parameter sharing method. Since convolution layers have different number of channels and kernel sizes throughout the network, we create 3 layer groups and only share templates among layers in the same group. More specifically, WRNs for CIFAR consist of 3 stages whose inputs and outputs mostly have a constant number of channels ( $C$ , $2 C$ and $_ { 4 C }$ , for some $C$ ). Each stage contains $\frac { L - 4 } { 3 }$ layers for a network with depth $L$ , hence we group layers in the same stage together, except for the first two, a residual block whose input has a different number of channels.
+
+Thus, all layers except for the first 2 in each stage perform parameter sharing (illustrated in left diagram of Figure 4). Having $k$ templates per group means that $\textstyle { \frac { L - 4 } { 3 } } - 2$ convolution layers share $k$ parameter templates. We denote by SWRN- $L$ -w- $k$ a WRN with $L$ layers, widen factor $w$ and $k$ parameter templates per group (trained with our method). Setting $k \doteq \textstyle { \frac { L - 4 } { 3 } } - 2$ L−4 − 2 means we have
+
+Table 1: Test error $( \% )$ on CIFAR-10 and CIFAR100. SWRN 28-10, the result of training a WRN 28- 10 with our method and one template per layer, significantly outperforms the base model, suggesting that our method aids optimization (both models have the same capacity). SWRN 28-10-1, with a single template per sharing group, performs close to WRN 28-10 while having significantly less parameters and capacity. \* indicates models trained with dropout $p = 0 . 3$ (Srivastava et al., 2014). Results are average of 5 runs.
+
+<table><tr><td rowspan=1 colspan=1>CIFAR</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=2>C-10+</td><td rowspan=1 colspan=1>C-10+</td></tr><tr><td rowspan=2 colspan=1>WRN 28-10WRN 28-10*</td><td rowspan=2 colspan=1>36M36M</td><td rowspan=1 colspan=2>4.0</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=2>3.89</td><td></td></tr><tr><td rowspan=2 colspan=1>SWRN28-10SWRN 28-10*SWRN 28-10-1</td><td rowspan=2 colspan=1>36M36M12M</td><td rowspan=1 colspan=2>3.74</td><td rowspan=1 colspan=1>18.78</td></tr><tr><td rowspan=1 colspan=2>3.884.01</td><td rowspan=1 colspan=1>18.4319.73</td></tr></table>
+
+Table 2: Performance of wider SWRNs. Parameter reduction $k = 2$ ) leads to lower errors for CIFAR-10, with models being competitive against newer model families that have bottleneck layers, group convolutions, or many layers. Best SWRN results are in bold, and best overall results are underlined.
+
+<table><tr><td rowspan=1 colspan=1>CIFAR</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=1>C-10+</td><td rowspan=1 colspan=1>C-100+</td></tr><tr><td rowspan=2 colspan=1>ResNeXt-2916x64DenseNet 100-24DenseNet 190-40</td><td rowspan=1 colspan=1>68M</td><td rowspan=1 colspan=1>3.58</td><td rowspan=1 colspan=1>17.31</td></tr><tr><td rowspan=1 colspan=1>27M26M</td><td rowspan=1 colspan=1>3.743.46</td><td rowspan=1 colspan=1>19.2517.18</td></tr><tr><td rowspan=4 colspan=1>SWRN 28-10*SWRN 28-10-2*SWRN 28-14*-SWRN 28-14-2*SWRN 28-18*SWRN 28-18-2*</td><td rowspan=3 colspan=1>36M17M71M33M</td><td rowspan=1 colspan=1>3.88</td><td rowspan=1 colspan=1>18.43</td></tr><tr><td rowspan=1 colspan=1>3.75</td><td rowspan=1 colspan=1>18.66</td></tr><tr><td rowspan=1 colspan=1>3.673.69</td><td rowspan=1 colspan=1>18.2518.37</td></tr><tr><td rowspan=1 colspan=1>118M55M</td><td rowspan=1 colspan=1>3.483.43</td><td rowspan=1 colspan=1>17.4317.75</td></tr></table>
+
+![](images/518464ecb2ecb59b86a82552d2161e01f30f11f898a98ee94b417a56caf2307c.jpg)  
+Figure 3: Parameter efficiency for different models. On both CIFAR-10 and CIFAR-100, SWRNs are significantly more efficient than WRNs. DN and RNX denotes DenseNet and ResNeXt, respectively, and are plotted for illustration: both models employ orthogonal efficiency techniques, such as bottleneck layers. Best viewed in color.
+
+one parameter template per layer, and hence no parameter reduction. We denote SWRN-L- $\mathbf { \nabla } \cdot w$ (thus omitting $k$ ) as a model in this setting.
+
+Following Zagoruyko & Komodakis (2016), we train each model for 200 epochs with SGD and Nesterov momentum of 0.9 and a batch size of 128. The learning rate is initially set to 0.1 and decays by a factor of 5 at epochs 60, 120 and 160. We also apply weight decay of $\mathrm { 5 \times 1 0 ^ { - 4 } }$ on all parameters except for the coefficients $_ \alpha$ .
+
+Tables 1 and 2 present results. Networks trained with our method yield superior performance in the setting with no parameter reduction: SWRN 28-10 presents $6 . 5 \%$ and $2 . 5 \%$ lower relative test errors on C-10 and C-100, compared to the base WRN 28-10 model. With fewer templates than layers, SWRN 28-10-1 (all 6 layers of each group perform the same operation), performs virtually the same as the base WRN 28-10 network, while having $\frac 1 3$ of its parameters. On CIFAR-10, parameter reduction $k = 2$ ) is beneficial to test performance: the best performance is achieved by SWRN 28-18-2 ( $3 . 4 3 \%$ test error), outperforming the ResNeXt-29 16x64 model (Xie et al., 2017), while having fewer parameters (55M against 68M) and no bottleneck layers.
+
+Figure 3 shows that our parameter sharing scheme uniformly improves accuracy-parameter efficiency; compare the WRN model family (solid red) to our SWRN models (dotted red).
+
+Table 4 presents a comparison between our method and neural architecture search (NAS) techniques (Zoph & Le, 2017; Xie et al., 2019; Liu et al., 2019; Pham et al., 2018; Real et al., 2018) on CIFAR-10 – results differ from Table 2 solely due to cutout (DeVries & Taylor, 2017), which is commonly used in NAS literature; NAS results are quoted from their respective papers. Our method outperforms architectures discovered by recent NAS algorithms, such as DARTS (Liu et al., 2019),
+
+Table 3: (below) ImageNet classification results: training WRN 50-2 with soft parameter sharing leads to better performance by itself, without any tuning on the number of templates $k$ . Top-1 and Top-5 errors $( \% )$ are computed using a single crop.   
+
+<table><tr><td rowspan=1 colspan=1>ImageNet</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=1>Top-1</td><td rowspan=1 colspan=1>Top-5</td></tr><tr><td rowspan=4 colspan=1>WRN 50-2DenseNet-264ResNet-200SWRN 50-2</td><td rowspan=1 colspan=1>69M</td><td rowspan=1 colspan=1>22.0</td><td rowspan=1 colspan=1>6.05</td></tr><tr><td rowspan=3 colspan=1>33M65M69M</td><td rowspan=1 colspan=1>22.15</td><td rowspan=1 colspan=1>6.12</td></tr><tr><td rowspan=1 colspan=1>21.66</td><td rowspan=2 colspan=1>5.795.95</td></tr><tr><td rowspan=1 colspan=1>21.74</td></tr></table>
+
+Table 4: (right) Test error $( \% )$ on CIFAR-10 of SWRNs and models found via neural architecture search (NAS) (all trained with cutout). Networks trained with soft parameter sharing provide competitive performance against NAS methods while having low computational cost.   
+
+<table><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>Params(M)</td><td rowspan=1 colspan=1>Training Time(GPU days)</td><td rowspan=1 colspan=1>Test Error(%)</td></tr><tr><td rowspan=8 colspan=1>NASNet-ANASNet-AAmoebaNet-BAmoebaNet-BAmoebaNet-BAmoebaNet-BDARTSSNASENAS</td><td rowspan=1 colspan=1>3.3</td><td rowspan=1 colspan=1>1800</td><td rowspan=1 colspan=1>2.65</td></tr><tr><td rowspan=1 colspan=1>27.6</td><td rowspan=1 colspan=1>1800</td><td rowspan=1 colspan=1>2.4</td></tr><tr><td rowspan=1 colspan=1>2.8</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.55</td></tr><tr><td rowspan=1 colspan=1>13.7</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.31</td></tr><tr><td rowspan=1 colspan=1>26.7</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.21</td></tr><tr><td rowspan=1 colspan=1>34.9</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.13</td></tr><tr><td rowspan=1 colspan=1>3.4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>2.83</td></tr><tr><td rowspan=1 colspan=1>2.84.6</td><td rowspan=1 colspan=1>1.50.45</td><td rowspan=1 colspan=1>2.852.89</td></tr><tr><td rowspan=1 colspan=1>WRN28-10(baseline with cutout)</td><td rowspan=1 colspan=1>36.4</td><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>3.08</td></tr><tr><td rowspan=4 colspan=1>SWRN 28-4-2SWRN 28-6-2SWRN 28-10SWRN 28-10-2SWRN 28-14SWRN 28-14-2</td><td rowspan=1 colspan=1>2.7</td><td rowspan=1 colspan=1>0.12</td><td rowspan=2 colspan=1>3.453.0</td></tr><tr><td rowspan=1 colspan=1>6.1</td><td rowspan=1 colspan=1>0.25</td></tr><tr><td rowspan=1 colspan=1>36.417.1</td><td rowspan=1 colspan=1>0.40.4</td><td rowspan=1 colspan=1>2.72.69</td></tr><tr><td rowspan=1 colspan=1>71.433.5</td><td rowspan=1 colspan=1>0.70.7</td><td rowspan=1 colspan=1>2.552.53</td></tr></table>
+
+SNAS (Xie et al., 2019) and ENAS (Pham et al., 2018), while having similarly low training cost. We achieve $2 . 6 9 \%$ test error after training less than 10 hours on a single NVIDIA GTX 1080 Ti. This accuracy is only bested by NAS techniques which are several orders of magnitude more expensive to train. Being based on Wide ResNets, our models do, admittedly, have more parameters.
+
+Comparison to recent NAS algorithms, such as DARTS and SNAS, is particularly interesting as our method, though motivated differently, bears some notable similarities. Specifically, all three methods are gradient-based and use an extra set of parameters (architecture parameters in DARTS and SNAS) to perform some kind of soft selection (over operations/paths in DARTS/SNAS; over templates in our method). As Section 4.3 will show, our learned template coefficients $_ { \pmb { \alpha } }$ can often be used to transform our networks into an explicitly recurrent form - a discovered CNN-RNN hybrid.
+
+To the extent that our method can be interpreted as a form of architecture search, it might be complementary to standard NAS methods. While NAS methods typically search over operations (e.g. activation functions; $3 \times 3$ or $5 \times 5$ convolutions; non-separable, separable, or grouped filters; dilation; pooling), our soft parameter sharing can be seen as a search over recurrent patterns (which layer processes the output at each step). These seem like orthogonal aspects of neural architectures, both of which may be worth examining in an expanded search space. When using SGD to drive architecture search, these aspects take on distinct forms at the implementation level: soft parameter sharing across layers (our method) vs hard parameter sharing across networks (recent NAS methods).
+
+# 4.2 CLASSIFICATION ON IMAGENET
+
+We use the ILSVRC 2012 dataset (Russakovsky et al., 2015) as a stronger test of our method. It is composed of 1.2M training and 50, 000 validation images, drawn from 1000 classes. We follow Gross & Wilber (2016), as in Zagoruyko & Komodakis (2016); Huang et al. (2017); Xie et al. (2017), and report Top-1 and Top-5 errors on the validation set using single $2 2 4 \times 2 2 4$ crops. For this experiment, we use WRN 50-2 as a base model, and train it with soft sharing and no parameter reduction. Having bottleneck blocks, this model presents less uniform number of channels of layer inputs and outputs. To apply our method, we group convolutions in 12 groups: for each of the 4 stages in a WRN 50-2, we create 3 groups, one for each type of layer in a bottleneck unit $C  B$ , $B  B$ and $B  C$ channel mappings, for bottleneck $B$ ). Without any change in hyperparameters, the network trained with our method outperforms the base model and also deeper models such as DenseNets (though using more parameters), and performs close to ResNet-200, a model with four times the number of layers and a similar parameter count. See Table 3.
+
+![](images/b732ae6be691db0bfcb9987fe694e903c55b587b5610872d701c1f589fd45a91.jpg)  
+Figure 4: Extracting implicit recurrences from a SWRN 28-10-4. Left: Illustration of the stages of a SWRN28-10-4 (residual connections omitted for clarity). The first two layers contain individual parameter sets, while the other six share four templates. All 3 stages of the network follow this structure. Middle: LSM for each stage after training on CIFAR-10, with many elements close to 1. Hard sharing schemes can be created for pairs with large similarity by tying their coefficients (or, equivalently, their effective weights). Right: Folding stages 2 and 3 leads to self-loops and a CNN with recurrent connections – LSM for stage 2 is a repetition of 2 rows/columns, and folding decreases the number of parameters.
+
+# 4.3 LEARNING IMPLICIT RECURRENCES
+
+Results on CIFAR suggest that training networks with few parameter templates $k$ in our soft sharing scheme results in performance comparable to the base models, which have significantly more parameters. The lower $k$ is, the larger we should expect the layer similarities to be: in the extreme case where $k = 1$ , all layers in a sharing scheme have similarity 1, and can be folded into a single layer with a self-loop.
+
+For the case $k > 1$ , there is no trivial way to fold the network, as layer similarities depend on the learned coefficients. We can inspect the model’s layer similarity matrix (LSM) and see if it presents implicit recurrences: a form of recurrence in the rows/columns of the LSM. Surprisingly, we observe that rich structures emerge naturally in networks trained with soft parameter sharing, even without the recurrence regularizer. Figure 4 shows the per-stage LSM for CIFAR-trained SWRN 28-10-4. Here, the six layers of its stage-2 block can be folded into a loop of two layers, leading to an error increase of only $0 . 0 2 \%$ . Appendix A contains an additional example of network folding, diversity of LSM patterns across different runs, and an epoch-wise evolution of the LSM, showing that many patterns are observable after as few as 5 epochs of training.
+
+# 4.4 EVALUATION ON NATURALLY RECURRENT TASKS
+
+While the propensity of our parameter sharing scheme to encourage learning of recurrent networks is a useful parameter reduction tool, we would also like to leverage it for qualitative advantages over standard CNNs. On tasks for which a natural recurrent algorithm exists, does training CNNs with soft parameter sharing lead to better extrapolation?
+
+To answer this, we set up a synthetic algorithmic task: computing shortest paths. Examples are $3 2 \times 3 2$ grids containing two query points and randomly (with probability 0.1) placed obstacles. The objective is to indicate which grid points belong to a shortest path between the query points.
+
+We use curriculum learning for training, allowing us to observe how well each model adapts to more difficult examples as training phases progress. Moreover, for this task curriculum learning causes faster learning and superior performance for all trained models.
+
+![](images/4c18a6433c8b4ef4f71dfc645665348dda2d2c2330d29e897fbe70e58b1e7d07.jpg)
+
+![](images/5797e1d23a50ae81bd8845709e604165aa331d04a7a0159a8e27a17f5d892741.jpg)
+
+(a) Generated example for the synthetic shortest paths task. Blue pixels indicate the query points; red pixels represent obstacles, and white pixels are points in a shortest path (in terms of Manhattan distance) between query pixels. The task consists of predicting the white pixels (shortest paths) from the blue and red ones (queries and obstacles).
+
+(b) Training curves for the shortest paths task, where difficulty of examples increases every 50 epochs. A SCNN adapts faster than a CNN to new phases and performs better, suggesting better extrapolation capacity. With a recurrence regularizer $\lambda _ { R } = 0 . 0 1$ (SCNN-R), the model makes faster progress on the first phase, but fails to adapt to harder examples.
+
+Figure 5: Shortest paths task. Best viewed in color.
+
+Training consists of 5 curriculum phases, each one containing 5000 examples. The maximum allowed distance between the two query points increases at each phase, thus increasing difficulty. In the first phase, each query point is within a $5 \times 5$ grid around the other query point, and the grid size increases by 2 on each side at each phase, yielding a final grid size of $2 1 \times 2 1$ at phase 5.
+
+We train a CNN, a CNN with soft parameter sharing and one template per layer (SCNN), and an SCNN with recurrence regularizer $\lambda _ { R } = 0 . 0 1$ . Each model trains for 50 epochs per phase with Adam (Kingma & Ba, 2015) and a fixed learning rate of 0.01. As classes are heavily unbalanced and the balance itself changes during phases, we compare $F _ { 1 }$ scores instead of classification error.
+
+Each model starts with a $1 \times 1$ convolution, mapping the 2 input channels to 32 output channels. Next, there are 20 channel-preserving $3 \times 3$ convolutions, followed by a final $1 \times 1$ convolution that maps 32 channels to 1. Each of the $2 0 3 \times 3$ convolutions is followed by batch normalization (Ioffe & Szegedy, 2015), a ReLU non-linearity (Nair & Hinton, 2010), and has a 1-skip connection.
+
+Figure 5 shows one example from our generated dataset and the training curves for the 3 trained models: the SCNN not only outperforms the CNN, but adapts better to harder examples at new curriculum phases. The SCNN is also advantaged over a more RNN-like model: with the recurrence regularizer $\lambda _ { R } = 0 . 0 1$ , all entries in the LSM quickly converge 1, as in a RNN. This leads to faster learning during the first phase, but presents issues in adapting to difficulty changes in latter phases.
+
+# 5 CONCLUSION
+
+In this work, we take a step toward more modular and compact CNNs by extracting recurrences from feed-forward models where parameters are shared among layers. Experimentally, parameter sharing yields models with lower error on CIFAR and ImageNet, and can be used for parameter reduction by training in a regime with fewer parameter templates than layers. Moreover, we observe that parameter sharing often leads to different layers being functionally equivalent after training, enabling us to collapse them into recurrent blocks. Results on an algorithmic task suggest that our shared parameter structure beneficially biases extrapolation. We gain a more flexible form of behavior typically attributed to RNNs, as our networks adapt better to out-of-domain examples. Our form of architecture discovery is also competitive with neural architecture search (NAS) algorithms, while having a smaller training cost than state-of-the-art gradient-based NAS.
+
+As the only requirement for our method is for a network to have groups of layers with matching parameter sizes, it can be applied to a plethora of CNN model families, making it a general technique with negligible computational cost. We hope to raise questions regarding the rigid definitions of
+
+CNNs and RNNs, and increase interest in models that fall between these definitions. Adapting our method for models with non-uniform layer parameter sizes (Huang et al., 2017; Zhu et al., 2018) might be of particular future interest.
+
+# REFERENCES
+
+Jonathon Cai, Richard Shin, and Dawn Song. Making neural programming architectures generalize via recursion. ICLR, 2017.
+
+Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs. arXiv:1606.00915, 2016.
+
+Terrance DeVries and Graham W. Taylor. Improved regularization of convolutional neural networks with cutout. Journal of Machine Learning Research, 2017.
+
+Jeffrey Donahue, Lisa Anne Hendricks, Sergio Guadarrama, Marcus Rohrbach, Subhashini Venugopalan, Kate Saenko, and Trevor Darrell. Long-term recurrent convolutional networks for visual recognition and description. CVPR, 2015.
+
+Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. ICASSP, 2013.
+
+Alex Graves, Greg Wayne, and Ivo Danihelka. Neural Turing Machines. arXiv:1410.5401, 2014.
+
+Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwinska, Sergio G ´ omez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, ´ Adria Puigdom \` enech Badia, Karl Moritz Hermann, Yori Zwols, Georg Ostrovski, Adam Cain, \` Helen King, Christopher Summerfield, Phil Blunsom, Koray Kavukcuoglu, and Demis Hassabis. Hybrid computing using a neural network with dynamic external memory. Nature, 2016.
+
+K. Greff, R. K. Srivastava, and J. Schmidhuber. Highway and residual networks learn unrolled iterative estimation. ICLR, 2017.
+
+Sam Gross and Martin Wilber. Training and investigating residual nets. https://github.com/ facebook/fb.resnet.torch, 2016.
+
+David Ha, Andrew Dai, and Quoc V. Le. Hypernetworks. arXiv:1609.09106, 2016.
+
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. ICLR, 2016.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CVPR, 2016.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 1997.
+
+Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. CVPR, 2017.
+
+Gao Huang, Shichen Liu, Laurens van der Maaten, and Kilian Q. Weinberger. CondenseNet: An efficient DenseNet using learned group convolutions. CVPR, 2018.
+
+Forrest N. Iandola, Matthew W. Moskewicz, Khalid Ashraf, Song Han, William J. Dally, and Kurt Keutzer. SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and ${ < } 1 \mathbf { M } \mathbf { B }$ model size. arXiv:1602.07360, 2016.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. ICML, 2015.
+
+Stanislaw Jastrzebski, Devansh Arpit, Nicolas Ballas, Vikas Verma, Tong Che, and Yoshua Bengio. Residual connections encourage iterative inference. ICLR, 2018.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. ICLR, 2015.
+
+Jan Koutn´ık, Klaus Greff, Faustino Gomez, and Jurgen Schmidhuber. A clockwork RNN. ¨ arXiv:1402.3511, 2014.   
+Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009.   
+Qianli Liao and Tomaso A. Poggio. Bridging the gaps between residual learning, recurrent neural networks and visual cortex. arXiv:1604.03640, 2016.   
+Chenxi Liu, Barret Zoph, Jonathon Shlens, Wei Hua, Li-Jia Li, Li Fei-Fei, Alan L. Yuille, Jonathan Huang, and Kevin Murphy. Progressive neural architecture search. ECCV, 2018.   
+Hanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: Differentiable architecture search. ICLR, 2019.   
+Vinod Nair and Geoffrey E. Hinton. Rectified linear units improve restricted boltzmann machines. ICML, 2010.   
+Hieu Pham, Melody Guan, Barret Zoph, Quoc Le, and Jeff Dean. Efficient neural architecture search via parameters sharing. ICML, 2018.   
+Pedro O. Pinheiro and Ronan Collobert. Recurrent convolutional neural networks for scene labeling. ICML, 2014.   
+Ameya Prabhu, Girish Varma, and Anoop M. Namboodiri. Deep expander networks: Efficient deep networks from graph theory. ECCV, 2018.   
+Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V Le. Regularized evolution for image classifier architecture search. arXiv:1802.01548, 2018.   
+Scott E. Reed and Nando de Freitas. Neural programmer-interpreters. ICLR, 2016.   
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael S. Bernstein, Alexander C. Berg, and Fei-Fei Li. Imagenet large scale visual recognition challenge. IJCV, 115(3), 2015.   
+Andrew M. Saxe, James L. McClelland, and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. arXiv:1312.6120, 2013.   
+Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. NIPS, 2015.   
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. ICLR, 2015.   
+Nitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1), 2014.   
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott E. Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. CVPR, 2015.   
+Andrew Trask, Felix Hill, Scott Reed, Jack Rae, Chris Dyer, and Phil Blunsom. Neural arithmetic logic units. arXiv:1808.00508, 2018.   
+Saining Xie, Ross B. Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He. Aggregated residual ´ transformations for deep neural networks. CVPR, 2017.   
+Sirui Xie, Hehui Zheng, Chunxiao Liu, and Liang Lin. SNAS: stochastic neural architecture search. ICLR, 2019.   
+Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua Bengio. Show, attend and tell: Neural image caption generation with visual attention. ICML, 2015.
+
+Yibo Yang, Zhisheng Zhong, Tiancheng Shen, and Zhouchen Lin. Convolutional neural networks with alternately updated clique. CVPR, 2018.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. BMVC, 2016.
+
+Amir Roshan Zamir, Te-Lin Wu, Lin Sun, William B. Shen, Jitendra Malik, and Silvio Savarese. Feedback networks. CVPR, 2017.
+
+Wojciech Zaremba, Tomas Mikolov, Armand Joulin, and Rob Fergus. Learning simple algorithms from examples. ICML, 2016.
+
+Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia. Pyramid scene parsing network. CVPR, 2017.
+
+Ligeng Zhu, Ruizhi Deng, Zhiwei Deng, Greg Mori, and Ping Tan. Sparsely aggregated convolutional networks. ECCV, 2018.
+
+Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. ICLR, 2017.
+
+# Appendix
+
+# A ADDITIONAL RESULTS FOR IMPLICIT RECURRENCES
+
+Section 4.3 presents an example of implicit recurrences and folding of a SWRN 28-10-4 trained on CIFAR-10, where, for example, the last 6 layers in the second stage of the network fold into 2 layers with a self-loop.
+
+Figure 6 presents an additional example, where non-trivial recurrences (unlike the one in Figure 4) emerge naturally, resulting in a model that is rich in structure.
+
+![](images/fc06843966b9e39486f71756cf26d0765bf6de008dc1755452226992594fc940.jpg)  
+Figure 6: SWRN 40-8-8 (8 parameter templates shared among groups of $\frac { 4 0 - 4 } { 3 } - 2 = 1 0$ layers) trained with soft parameter sharing on CIFAR-10. Each stage (originally with 12 layers – the first two do not participate in parameter sharing) can be folded to yield blocks with complex recurrences. For clarity, we use colors to indicate the computational flow: red takes precedence over green, which in turn has precedence over blue. Colored paths are only taken once per stage. Although not trivial to see, recurrences in each stage’s folded form are determined by row/column repetitions in the respective Layer Similarity Matrix. For example, for stage 2 we have $S _ { 5 , 3 } \approx S _ { 6 , 4 } \approx 1$ , meaning that layers 3, 4, 5 and 6 can be folded into layers 3 and 4 with a loop (captured by the red edge). The same holds for $S _ { 7 , 1 }$ $_ 1 , S _ { 8 , 2 } , S _ { 9 , 3 }$ and $S _ { 1 0 , 4 }$ , hence after the loop with layers 3 and 4, the flow returns to layer 1 and goes all the way to layer 4, which generates the stage’s output. Even though there is an approximation when folding the network (in this example, we are tying layers with similarity close to 0.8), the impact on the test error is less than $0 . 3 \%$ . Also note that the folded model has a total of 24 layers (20 in the stage diagrams, plus 4 which are not shown, corresponding to the first layer of the network and three $1 \times 1$ convolutions in skip-connections), instead of the original 40.
+
+![](images/fdc8c958fab5e3a2b0f2446ac5890d9310f5a57da70a5ba719a29d7cb24d7e4b.jpg)  
+Figure 7: LSMs of a SWRN 40-8-8 (composed of 3 stages, each with 10 layers sharing 8 templates) trained on CIFAR-10 for 5 runs with different random seeds. Although the LSMs differ across different runs, hard parameter sharing can be observed in all cases (off-diagonal elements close to 1, depicted by white), characterizing implicit recurrences which would enable network folding. Moreover, the underlying structure is similar across runs, with hard sharing typically happening among layers $_ { i }$ and $i + 2$ , leading to a “chessboard” pattern.
+
+![](images/5de972993320487ca663b8f5109f332ee4d92857e74173a62c2f72f4841412cd.jpg)  
+Figure 8: LSMs of a SWRN 40-8-8 (composed of 3 stages, each with 10 layers sharing 8 templates) at different epochs during training on CIFAR-10. The transition from an identity matrix to the final LSM happens mostly in the beginning of training: at epoch 50, the LSM is almost indistinguishable from the final LSM at epoch 200, and most of the final patterns are observable already at epoch 25.
+
+# B INITIALIZATION OF COEFFICIENTS
+
+During our initial experiments, we explored different initializations for the coefficients $_ { \pmb { \alpha } }$ of each layer, and observed that using an orthogonal initialization (Saxe et al., 2013) resulted in superior performance compared to uniform or normal initialization schemes.
+
+Denote $\pmb { A }$ as the $L \times k$ matrix ( $L$ is the number of layers sharing parameters and $k$ the number of templates) with each $\because$ ’th row containing the coefficient of the $i ^ { \because }$ ’th layer $\mathbf { \alpha } \alpha ^ { ( i ) }$ . We initialize it such that $A ^ { T } A = I$ , leading to $\forall _ { i }$ , $\langle \pmb { \alpha } ^ { ( i ) } , \pmb { \alpha } ^ { ( i ) } \rangle = 1$ and $\forall _ { i \neq j } , \langle { \pmb { \alpha } } ^ { ( i ) } , { \pmb { \alpha } } ^ { ( j ) } \rangle = 0$ . While our choice for this is mostly empirical, we believe that there is likely a connection with the motivation for using orthogonal initialization for RNNs.
+
+Moreover, we discovered that other initialization options for $\pmb { A }$ work similarly to the orthogonal one. More specifically, either initializing $\pmb { A }$ with the identity matrix when $L = k$ (which naturally leads to $A ^ { T } A = I $ ) or enforcing some sparsity (initialize $\pmb { A }$ with a uniform or normal distribution and randomly setting half of its entries to zero) performs similarly to the orthogonal initialization in a consistent manner. We believe the sparse initialization to be the simplest one, as each coefficient $_ { \pmb { \alpha } }$ can be initialized independently.
+
+Finally, note that having $A ^ { T } A = I$ results in the Layer Similarity Matrix also being the identity at initialization (check that Si,j = $\begin{array} { r } { S _ { i , j } = \frac { | \langle { \pmb { \alpha } } ^ { ( i ) } , { \pmb { \alpha } } ^ { ( j ) } \rangle | } { \| { \pmb { \alpha } } ^ { ( i ) } \| \| { \pmb { \alpha } } ^ { ( j ) } \| } = \frac { | ( { \pmb { A } } ^ { T } { \pmb { A } } ) _ { i , j } | } { \| { \pmb { \alpha } } ^ { ( i ) } \| \| { \pmb { \alpha } } ^ { ( j ) } \| } } \end{array}$ |(AT A)i,j |kα(i)kkα(j)k , so if (AT A)i,j = 1, then Si,j = 1, and the same holds for 0. Surprisingly, even though the orthogonal initialization leads to a LSM that has no structure in the beginning of training, the rich patterns that we observe still emerge naturally after optimization.

parse/train/rJgYxn09Fm/rJgYxn09Fm_content_list.json

@@ -0,0 +1,1490 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING IMPLICITLY RECURRENT CNNS THROUGH PARAMETER SHARING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pedro Savarese TTI-Chicago savarese@ttic.edu ",
+        "bbox": [
+            183,
+            170,
+            352,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Michael Maire University of Chicago mmaire@uchicago.edu ",
+        "bbox": [
+            496,
+            170,
+            686,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a parameter sharing scheme, in which different layers of a convolutional neural network (CNN) are defined by a learned linear combination of parameter tensors from a global bank of templates. Restricting the number of templates yields a flexible hybridization of traditional CNNs and recurrent networks. Compared to traditional CNNs, we demonstrate substantial parameter savings on standard image classification tasks, while maintaining accuracy. ",
+        "bbox": [
+            233,
+            280,
+            764,
+            363
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our simple parameter sharing scheme, though defined via soft weights, in practice often yields trained networks with near strict recurrent structure; with negligible side effects, they convert into networks with actual loops. Training these networks thus implicitly involves discovery of suitable recurrent architectures. Though considering only the design aspect of recurrent links, our trained networks achieve accuracy competitive with those built using state-of-the-art neural architecture search (NAS) procedures. ",
+        "bbox": [
+            233,
+            371,
+            764,
+            468
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our hybridization of recurrent and convolutional networks may also represent a beneficial architectural bias. Specifically, on synthetic tasks which are algorithmic in nature, our hybrid networks both train faster and extrapolate better to test examples outside the span of the training set. ",
+        "bbox": [
+            233,
+            477,
+            764,
+            534
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
+            336,
+            574
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The architectural details of convolutional neural networks (CNNs) have undergone rapid exploration and improvement via both human hand-design (Simonyan & Zisserman, 2015; Szegedy et al., 2015; He et al., 2016; Huang et al., 2017; Zhu et al., 2018) and automated search methods (Zoph & Le, 2017; Liu et al., 2018). Yet, this vast array of work limits itself to a circuit-like view of neural networks. Here, a CNN is regarded as a fixed-depth feed-forward circuit, with a distinct parameter governing each internal connection. These circuits are often trained to perform tasks which, in a prior era, might have been (less accurately) accomplished by running a traditional computer program coded by humans. Programs, and even traditional hardware circuits, have a more reusable internal structure, including subroutines or modules, loops, and associated control flow mechanisms. ",
+        "bbox": [
+            174,
+            589,
+            825,
+            714
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We bring one aspect of such modularity into CNNs, by making it possible to learn a set of parameters that is reused across multiple layers at different depths. As the pattern of reuse is itself learned, our scheme effectively permits learning the length (iteration count) and content of multiple loops defining the resulting CNN. We view this approach as a first step towards learning neural networks with internal organization reminiscent of computer programs. Though we focus solely on loop-like structures, leaving subroutines and dynamic control flow to future work, this simple change suffices to yield substantial quantitative and qualitative benefits over the standard baseline CNN models. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While recurrent neural networks (RNNs) possess a loop-like structure by definition, their loop structure is fixed a priori, rather than learned as part of training. This can actually be a disadvantage in the event that the length of the loop is mismatched to the target task. Our parameter sharing scheme for CNNs permits a mix of loops and feed-forward layers to emerge. For example, trained with our scheme, a 50-layer CNN might learn a 2-layer loop that executes 5 times between layers 10 and 20, a 3-layer loop that runs 4 times from layers 30 to 42, while leaving the remaining layers to assume independent parameter sets. Our approach generalizes both CNNs and RNNs, creating a hybrid. ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a2c9b5bb263d8cd0bf0d9985922b8c7f6508fe346bcd32491b1fd6dc3f69077c.jpg",
+        "image_caption": [
+            "Figure 1: Parameter sharing scheme. Left: A CNN (possibly a variant such as a residual network), with each convolutional layer $_ { i }$ containing an individual parameter set $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ . Middle: Parameter sharing among layers, where parameter templates $\\boldsymbol { \\mathsf { T } } ^ { ( 1 ) }$ , ${ \\boldsymbol { \\mathsf { T } } } ^ { ( 2 ) }$ are shared among each layer $_ { i }$ , which now only contains a 2-dimensional parameter $\\mathbf { \\pmb { \\alpha } } ^ { ( i ) }$ . Weights $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ (no longer parameters, illustrated with dotted boxes) used by layer $i$ are generated from $\\mathbf { \\pmb { \\alpha } } ^ { ( i ) }$ and templates ${ \\boldsymbol { \\mathsf { T } } } ^ { ( 1 ) }$ , ${ \\boldsymbol { \\mathsf { T } } } ^ { ( 2 ) }$ . Right: If weights $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ are outputs of a linear function (as in our method), learning parameter templates can be viewed as learning layer templates, offering a new (although equivalent) perspective for the middle diagram. Non-linearities are omitted for simplicity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            104,
+            776,
+            281
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 diagrams the parameter sharing scheme facilitating this hybridization. Inspired by dictionary learning, different network layers share, via weighted combination, global parameter templates. This re-parameterization is fully differentiable, allowing learning of sharing weights and template parameters. Section 3 elaborates, and also introduces tools for analyzing learned loop structures. ",
+        "bbox": [
+            173,
+            421,
+            825,
+            477
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Section 4 demonstrates advantages of our hybrid CNNs across multiple experimental settings. Taking a modern CNN design as a baseline, and re-parameterizing it according to our scheme improves: ",
+        "bbox": [
+            173,
+            484,
+            823,
+            512
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Parameter efficiency. Here, we experiment with the standard task of image classification using modern residual networks (He et al., 2016; Zagoruyko & Komodakis, 2016). This task is a good proxy for general usefulness in computer vision, as high-performance classification architectures often serve as a backbone for many other vision tasks, such as semantic segmentation (Chen et al., 2016; Zhao et al., 2017). ",
+        "bbox": [
+            217,
+            525,
+            825,
+            594
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our parameter sharing scheme drastically reduces the number of unique parameters required to achieve a given accuracy on CIFAR (Krizhevsky, 2009) or ImageNet (Russakovsky et al., 2015) classification tasks. Re-parameterizing a standard residual network with our scheme cuts parameters, without triggering any drop in accuracy. This suggests that standard CNNs may be overparameterized in part because, by design (and unlike RNNs), they lack capacity to learn reusable internal operations. ",
+        "bbox": [
+            215,
+            595,
+            825,
+            679
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Extrapolation and generalization. Here, we explore whether our hybrid networks expand the class of tasks that one can expect to train neural networks to accomplish. This line of inquiry, focusing on synthetic tasks, shares motivations with work on Neural Turing Machines (Graves et al., 2014). Specifically, we would like neural networks to be capable of learning to perform tasks for which there are concise traditional solution algorithms. Graves et al. (2014) uses sorting as an example task. As we examine an extension of CNNs, our tasks take the form of queries about planar graphs encoded as image input. ",
+        "bbox": [
+            218,
+            684,
+            825,
+            781
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "On these tasks, we observe improvements to both generalization ability and learning speed for our hybrid CNNs, in comparison to standard CNNs or RNNs. Our parameter sharing scheme, by virtue of providing an architectural bias towards networks with loops, appears to assist in learning to emulate traditional algorithms. ",
+        "bbox": [
+            230,
+            784,
+            823,
+            839
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An additional side effect, seen in practice in many of our experiments, is that two different learned layers often snap to the same parameter values. That is, layers $i$ and $j$ , learn coefficient vectors $\\mathbf { \\alpha } \\alpha ^ { ( i ) }$ and $\\pmb { \\alpha } ^ { ( j ) }$ (see Figure 1) that converge to be the same (up to scaling). This is a form of architecture discovery, as it permits representation of the CNN as a loopy wiring diagram between repeated layers. Section 4.3 presents example results. We also draw comparisons to existing neural architecture search (NAS) techniques. By simply learning recurrent structure as byproduct of training with standard stochastic gradient descent, we achieve accuracy competitive with current NAS procedures. ",
+        "bbox": [
+            174,
+            852,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Before delving into the details of our method, Section 2 provides additional context in terms of prior work on recurrent models, parameter reduction techniques, and program emulation. Sections 3 and 4 describe our hybrid shared-parameter CNN, experimental setup, and results. Section 5 concludes with commentary on our results and possible future research pathways.1 ",
+        "bbox": [
+            176,
+            138,
+            823,
+            194
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            217,
+            344,
+            233
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recurrent variants of CNNs are used extensively for visual tasks. Recently, Zamir et al. (2017) propose utilizing a convolutional LSTM (Shi et al., 2015) as a generic feedback architecture. RNN and CNN combinations have been used for scene labeling (Pinheiro & Collobert, 2014), image captioning with attention (Xu et al., 2015), and understanding video (Donahue et al., 2015), among others. These works combine CNNs and RNNs at a coarse scale, and in a fixed hand-crafted manner. In contrast, we learn the recurrence structure itself, blending it into the inner workings of a CNN. ",
+        "bbox": [
+            174,
+            251,
+            825,
+            334
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Analysis of residual networks (He et al., 2016) reveals possible connections to recurrent networks stemming from their design (Liao & Poggio, 2016). Greff et al. (2017) provide evidence that residual networks learn to iteratively refine feature representations, making an analogy between a very deep residual network and an unrolled loop. Jastrzebski et al. (2018) further explore this connection, and experiment with training residual networks in which some layers are forced to share identical parameters. This hard parameter sharing scheme again builds a predetermined recurrence structure into the network. It yields successfully trained networks, but does not exhibit the type of performance gains that Section 4 demonstrates for our soft parameter sharing scheme. ",
+        "bbox": [
+            174,
+            342,
+            825,
+            453
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Closely related to our approach is the idea of hypernetworks (Ha et al., 2016), in which one part of a neural network is parameterized by another neural network. Our shared template-based reparameterization could be viewed as one simple choice of hypernetwork implementation. Perhaps surprisingly, this class of ideas has not been well explored for the purpose of reducing the size of neural networks. Rather, prior work has achieved parameter reduction through explicit representation bottlenecks (Iandola et al., 2016), sparsifying connection structure (Prabhu et al., 2018; Huang et al., 2018; Zhu et al., 2018), and pruning trained networks (Han et al., 2016). ",
+        "bbox": [
+            174,
+            460,
+            825,
+            558
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Orthogonal to the question of efficiency, there is substantial interest in extending neural networks to tackle new kinds of tasks, including emulation of computer programs. Some approach this problem using additional supervision in the form of execution traces (Reed & de Freitas, 2016; Cai et al., 2017), while other focus on development of network architectures that can learn from input-output pairs alone (Graves et al., 2014; 2016; Zaremba et al., 2016; Trask et al., 2018). Our experiments on synthetic tasks fall into the latter camp. At the level of architectural strategy, Trask et al. (2018) benefit from changing the form of activation function to bias the network towards correctly extrapolating common mathematical formulae. We build in a different implicit bias, towards learning iterative procedures within a CNN, and obtain a boost on correctly emulating programs. ",
+        "bbox": [
+            174,
+            564,
+            825,
+            689
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 SOFT PARAMETER SHARING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            712,
+            441,
+            728
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In convolutional neural networks (CNNs) and variants such as residual CNNs (ResNets) (He et al., 2016) and DenseNets (Huang et al., 2017), each convolutional layer $i$ contains a set of parameters $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ , with no explicit relation between parameter sets of different layers. Conversely, a strict structure is imposed to layers of recurrent neural networks (RNNs), where, in standard models (Hochreiter & Schmidhuber, 1997), a single parameter set $\\pmb { \\mathsf { W } }$ is shared among all time steps. This leads to a program-like computational flow, where RNNs can be seen as loops with fixed length and content. While some RNN variants (Graves et al., 2013; Koutn´ık et al., 2014; Yang et al., 2018) are less strict on the length or content of loops, these are still typically fixed beforehand. ",
+        "bbox": [
+            174,
+            746,
+            825,
+            859
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As an alternative to learning hard parameter sharing schemes – which correspond to the strict structure present in RNNs – our method consists of learning soft sharing schemes through a relaxation of this structure. We accomplish this by expressing each layer’s parameters $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ as a linear combination of parameter templates $\\mathbf { \\overline { { I } } } ^ { ( 1 ) } , \\ldots , \\mathbf { \\overline { { I } } } ^ { ( k ) }$ , each with the same dimensionality as $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ : ",
+        "bbox": [
+            174,
+            867,
+            823,
+            895
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/30aa8071d0e5ad98b194b3c861782577cf515638614a5ad4b9997a5766ff7be7.jpg",
+        "image_caption": [
+            "Figure 2: Connection between the LSM matrix $S$ \u0000 where $\\begin{array} { r } { S _ { i , j } = \\frac { | \\langle { \\pmb { \\alpha } } ^ { ( i ) } , { \\pmb { \\alpha } } ^ { ( j ) } \\rangle | } { \\| { \\pmb { \\alpha } } ^ { ( i ) } \\| \\| { \\pmb { \\alpha } } ^ { ( j ) } \\| } \\ ) } \\end{array}$ and the structure of the network. White and black entries correspond to maximum and minimum similarities $( S _ { i , j } = 1$ and $S _ { i , j } = 0$ , respectively). Left: Empirically, CNNs present no similarity between parameters of different layers. Middle: Trained with our method, the layer similarity matrix (LSM) captures similarities between different layers, including pairs with close to maximum similarity. Such pairs (depicted by same-colored coefficients and weights, and by white entries in the LSM) perform similar operations on their inputs. Right: We can tie together parameters of similar layers, creating a hard parameter sharing scheme. The network can then be folded, creating self-loops and revealing an explicit recurrent computation structure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            103,
+            776,
+            276
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            420,
+            823,
+            453
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ed14c1826ceb32bb7c357e1df02cc95ae5df7cb58a999743483076edc0036f32.jpg",
+        "text": "$$\n\\pmb { \\mathsf { W } } ^ { ( i ) } : = \\sum _ { j = 1 } ^ { k } \\alpha _ { j } ^ { ( i ) } \\pmb { \\mathsf { T } } ^ { ( j ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            457,
+            568,
+            502
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $k$ is the number of parameter templates (chosen freely as a hyperparameter) and $\\mathbf { \\alpha } \\alpha ^ { ( i ) }$ , a $k$ - dimensional vector, is the coefficients of layer $i$ . Figure 1 (left and middle) illustrates the difference between networks trained with and without our method. This relaxation allows for coefficients and parameter templates to be (jointly) optimized with gradient-based methods, yielding negligible extra computational cost, with a single constraint that only layers with same parameter sizes can share templates. Note that constraining coefficients $\\mathbf { \\alpha } \\alpha ^ { ( i ) }$ to be one-hot vectors leads to hard sharing schemes, at the cost of non-differentiability. ",
+        "bbox": [
+            173,
+            508,
+            825,
+            608
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Having $k$ as a free parameter decouples the number of parameters in network from its depth. Typically, $L$ convolutional layers with constant channel and kernel sizes $C , K$ have $O ( L C ^ { 2 } { \\dot { K } } ^ { 2 } )$ total parameters. Our soft sharing scheme changes the total number of parameters to $O ( k L + k C ^ { 2 } K ^ { 2 } ) =$ $\\scriptstyle \\dot { O } ( k C ^ { 2 } K ^ { 2 } )$ . Sections 4.1 and 4.2 show that we can decrease the parameter count of standard models without significantly impacting accuracy, or simply attain higher accuracy with $k = L$ . ",
+        "bbox": [
+            173,
+            614,
+            825,
+            685
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the next two subsections, we discuss two consequences of the linearity of Equation (1). First, it enables alternative interpretations of our method. Second, and a major advantage, as is the case in many linear relaxations of integer problems, we are able to extract hard sharing schemes in practice, and consequently detect implicit self-loops in a CNN trained with our method. ",
+        "bbox": [
+            173,
+            690,
+            825,
+            747
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 INTERPRETATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            763,
+            334,
+            779
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For layers $i$ that are linear in $\\boldsymbol { \\mathsf { W } } ^ { ( i ) }$ (e.g. matrix multiplication, convolution), we can view our method as learning template layers which are shared among a network. More specifically, for a convolutional layer $\\mathbf { U } ^ { ( i ) } ( \\mathbf { X } ) = \\mathbf { W } ^ { ( i ) } \\ast \\mathbf { X }$ , and considering Equation (1): ",
+        "bbox": [
+            174,
+            787,
+            825,
+            835
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07513eb8445b300d96bc700c309a44f38f6d60609512da1f91bcb711332d7705.jpg",
+        "text": "$$\n\\mathbf { U } ^ { ( i ) } ( \\mathbf { X } ) = \\mathbf { W } ^ { ( i ) } * \\mathbf { X } = \\sum _ { j = 1 } ^ { k } \\alpha _ { j } ^ { ( i ) } \\mathbf { T } ^ { ( j ) } * \\mathbf { X }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            839,
+            630,
+            885
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { \\mathsf { T } } ^ { ( j ) } * \\pmb { \\mathsf { X } }$ , the result of a convolution with filter sets $\\bar { \\mathsf { T } } ^ { ( j ) }$ , can be seen as the output of a template layer with individual parameters $\\boldsymbol { \\mathsf { T } } ^ { ( j ) }$ . Such layers can be seen as global feature extractors, ",
+        "bbox": [
+            174,
+            890,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and coefficients $\\mathbf { \\alpha } \\alpha ^ { ( i ) }$ determine which features are relevant for the $i$ ’th computation of a network.   \nThis is illustrated in Figure 1 (right diagram). ",
+        "bbox": [
+            173,
+            103,
+            821,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This view gives a clear connection between coefficients $_ { \\pmb { \\alpha } }$ and the network’s structure. Having $\\pmb { \\alpha } ^ { ( i ) } = \\pmb { \\alpha } ^ { ( i + 2 ) }$ yields $\\begin{array} { r } { \\mathbf { W } ^ { ( i ) } = \\sum _ { j = 1 } ^ { k } \\alpha _ { j } ^ { ( i ) } \\overline { { \\mathbf { T } ^ { ( j ) } } } = \\sum _ { j = 1 } ^ { k } \\alpha _ { j } ^ { ( i + 2 ) } \\overline { { \\mathbf { T } ^ { ( j ) } } } = \\mathbf { W } ^ { ( i + 2 ) } } \\end{array}$ α(i+2)T(j) = W(i+2), and hence layers i and $i + 2$ are functionally equivalent. Such a network can be folded to generate an equivalent model with two layers and a self-loop, an explicitly recurrent network. While this is also possible for networks without parameter sharing, a learned alignment of $C ^ { 2 } K ^ { 2 }$ parameters is required (unlikely in practice), instead of aligning only $k \\leq L$ parameters. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            227
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 IMPLICIT RECURRENCES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            244,
+            387,
+            258
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To identify which layers in a network perform approximately the same operation, we can simply check whether their coefficients are similar. We can condense this information for all pairs of layers $i , j$ in a similarity matrix $S$ , where $S _ { i , j } = s ( \\pmb { \\alpha } ^ { ( i ) } , \\pmb { \\alpha } ^ { ( j ) } )$ for some similarity measure $s$ . ",
+        "bbox": [
+            174,
+            270,
+            823,
+            314
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For networks with normalization layers, the network’s output is invariant to weight rescaling. In this setting, a natural measure is $\\begin{array} { r } { s ( \\pmb { \\alpha } ^ { ( i ) } , \\pmb { \\alpha } ^ { ( j ) } ) = \\frac { | \\langle \\pmb { \\alpha } ^ { ( i ) } , \\pmb { \\alpha } ^ { ( j ) } \\rangle | } { \\| \\pmb { \\alpha } ^ { ( i ) } \\| \\| \\pmb { \\alpha } ^ { ( j ) } \\| } } \\end{array}$ (absolute value of cosine similarity), since it possess this same property.2 We call $S$ the layer similarity matrix (LSM). Figure 2 illustrates and Section 4.3 shows experimentally how it can be used to extract recurrent loops from trained CNNs. ",
+        "bbox": [
+            174,
+            320,
+            825,
+            387
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While structure might emerge naturally, having a bias towards more structured (recurrent) models might be desirable. In this case, we can add a recurrence regularizer to the training objective, pushing parameters to values which result in more structure. For example, we can add the negative of sum of elements of the LSM: $\\begin{array} { r } { \\mathcal { L } _ { R } = \\mathcal { L } - \\lambda _ { R } \\sum _ { i , j } S _ { i , j } } \\end{array}$ , where $\\mathcal { L }$ is the original objective. The larger $\\lambda _ { R }$ is, the closer the elements of $S$ will be to 1. At an extreme case, this regularizer will push all elements in $S$ to 1, resulting in a network with a single layer and a self-loop. ",
+        "bbox": [
+            174,
+            393,
+            825,
+            479
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            500,
+            326,
+            516
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We begin by training variants of standard models with soft parameter sharing, observing that it can offer parameter savings with little impact on performance, or increase performance at the same parameter count. Section 4.3 demonstrates conversion of a trained model into explicitly recurrent form. We then examine synthetic tasks (Section 4.4), where parameter sharing improves generalization. Appendix B contains details on the initialization for the coefficients $_ { \\pmb { \\alpha } }$ . ",
+        "bbox": [
+            174,
+            531,
+            825,
+            602
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 CLASSIFICATION ON CIFAR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            618,
+            411,
+            632
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) are composed of 60, 000 colored $3 2 \\times 3 2$ images, labeled among 10 and 100 classes respectively, and split into 50, 000 and 10, 000 examples for training and testing. We pre-process the training set with channel-wise normalization, and use horizontal flips and random crops for data augmentation, following He et al. (2016). ",
+        "bbox": [
+            174,
+            645,
+            825,
+            700
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using Wide ResNets (WRN) (Zagoruyko & Komodakis, 2016) as a base model, we train networks with the proposed soft parameter sharing method. Since convolution layers have different number of channels and kernel sizes throughout the network, we create 3 layer groups and only share templates among layers in the same group. More specifically, WRNs for CIFAR consist of 3 stages whose inputs and outputs mostly have a constant number of channels ( $C$ , $2 C$ and $_ { 4 C }$ , for some $C$ ). Each stage contains $\\frac { L - 4 } { 3 }$ layers for a network with depth $L$ , hence we group layers in the same stage together, except for the first two, a residual block whose input has a different number of channels. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            806
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, all layers except for the first 2 in each stage perform parameter sharing (illustrated in left diagram of Figure 4). Having $k$ templates per group means that $\\textstyle { \\frac { L - 4 } { 3 } } - 2$ convolution layers share $k$ parameter templates. We denote by SWRN- $L$ -w- $k$ a WRN with $L$ layers, widen factor $w$ and $k$ parameter templates per group (trained with our method). Setting $k \\doteq \\textstyle { \\frac { L - 4 } { 3 } } - 2$ L−4 − 2 means we have ",
+        "bbox": [
+            174,
+            813,
+            825,
+            869
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Test error $( \\% )$ on CIFAR-10 and CIFAR100. SWRN 28-10, the result of training a WRN 28- 10 with our method and one template per layer, significantly outperforms the base model, suggesting that our method aids optimization (both models have the same capacity). SWRN 28-10-1, with a single template per sharing group, performs close to WRN 28-10 while having significantly less parameters and capacity. \\* indicates models trained with dropout $p = 0 . 3$ (Srivastava et al., 2014). Results are average of 5 runs. ",
+        "bbox": [
+            174,
+            101,
+            493,
+            228
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/3bf58ac07021f70f3e89f70d7d80ee105e73bafa30fc6a512a91bbc253d5818a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>CIFAR</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=2>C-10+</td><td rowspan=1 colspan=1>C-10+</td></tr><tr><td rowspan=2 colspan=1>WRN 28-10WRN 28-10*</td><td rowspan=2 colspan=1>36M36M</td><td rowspan=1 colspan=2>4.0</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=2>3.89</td><td></td></tr><tr><td rowspan=2 colspan=1>SWRN28-10SWRN 28-10*SWRN 28-10-1</td><td rowspan=2 colspan=1>36M36M12M</td><td rowspan=1 colspan=2>3.74</td><td rowspan=1 colspan=1>18.78</td></tr><tr><td rowspan=1 colspan=2>3.884.01</td><td rowspan=1 colspan=1>18.4319.73</td></tr></table>",
+        "bbox": [
+            183,
+            239,
+            485,
+            319
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Performance of wider SWRNs. Parameter reduction $k = 2$ ) leads to lower errors for CIFAR-10, with models being competitive against newer model families that have bottleneck layers, group convolutions, or many layers. Best SWRN results are in bold, and best overall results are underlined. ",
+        "bbox": [
+            504,
+            101,
+            823,
+            176
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/32219caf57127ebe7fb1f0d9d1044f691bad4f7b7803dd90b72ef015c42bb956.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>CIFAR</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=1>C-10+</td><td rowspan=1 colspan=1>C-100+</td></tr><tr><td rowspan=2 colspan=1>ResNeXt-2916x64DenseNet 100-24DenseNet 190-40</td><td rowspan=1 colspan=1>68M</td><td rowspan=1 colspan=1>3.58</td><td rowspan=1 colspan=1>17.31</td></tr><tr><td rowspan=1 colspan=1>27M26M</td><td rowspan=1 colspan=1>3.743.46</td><td rowspan=1 colspan=1>19.2517.18</td></tr><tr><td rowspan=4 colspan=1>SWRN 28-10*SWRN 28-10-2*SWRN 28-14*-SWRN 28-14-2*SWRN 28-18*SWRN 28-18-2*</td><td rowspan=3 colspan=1>36M17M71M33M</td><td rowspan=1 colspan=1>3.88</td><td rowspan=1 colspan=1>18.43</td></tr><tr><td rowspan=1 colspan=1>3.75</td><td rowspan=1 colspan=1>18.66</td></tr><tr><td rowspan=1 colspan=1>3.673.69</td><td rowspan=1 colspan=1>18.2518.37</td></tr><tr><td rowspan=1 colspan=1>118M55M</td><td rowspan=1 colspan=1>3.483.43</td><td rowspan=1 colspan=1>17.4317.75</td></tr></table>",
+        "bbox": [
+            503,
+            189,
+            825,
+            319
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/518464ecb2ecb59b86a82552d2161e01f30f11f898a98ee94b417a56caf2307c.jpg",
+        "image_caption": [
+            "Figure 3: Parameter efficiency for different models. On both CIFAR-10 and CIFAR-100, SWRNs are significantly more efficient than WRNs. DN and RNX denotes DenseNet and ResNeXt, respectively, and are plotted for illustration: both models employ orthogonal efficiency techniques, such as bottleneck layers. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            333,
+            803,
+            507
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "one parameter template per layer, and hence no parameter reduction. We denote SWRN-L- $\\mathbf { \\nabla } \\cdot w$ (thus omitting $k$ ) as a model in this setting. ",
+        "bbox": [
+            173,
+            602,
+            823,
+            631
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Following Zagoruyko & Komodakis (2016), we train each model for 200 epochs with SGD and Nesterov momentum of 0.9 and a batch size of 128. The learning rate is initially set to 0.1 and decays by a factor of 5 at epochs 60, 120 and 160. We also apply weight decay of $\\mathrm { 5 \\times 1 0 ^ { - 4 } }$ on all parameters except for the coefficients $_ \\alpha$ . ",
+        "bbox": [
+            174,
+            637,
+            825,
+            693
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Tables 1 and 2 present results. Networks trained with our method yield superior performance in the setting with no parameter reduction: SWRN 28-10 presents $6 . 5 \\%$ and $2 . 5 \\%$ lower relative test errors on C-10 and C-100, compared to the base WRN 28-10 model. With fewer templates than layers, SWRN 28-10-1 (all 6 layers of each group perform the same operation), performs virtually the same as the base WRN 28-10 network, while having $\\frac 1 3$ of its parameters. On CIFAR-10, parameter reduction $k = 2$ ) is beneficial to test performance: the best performance is achieved by SWRN 28-18-2 ( $3 . 4 3 \\%$ test error), outperforming the ResNeXt-29 16x64 model (Xie et al., 2017), while having fewer parameters (55M against 68M) and no bottleneck layers. ",
+        "bbox": [
+            174,
+            700,
+            825,
+            811
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows that our parameter sharing scheme uniformly improves accuracy-parameter efficiency; compare the WRN model family (solid red) to our SWRN models (dotted red). ",
+        "bbox": [
+            174,
+            819,
+            821,
+            847
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4 presents a comparison between our method and neural architecture search (NAS) techniques (Zoph & Le, 2017; Xie et al., 2019; Liu et al., 2019; Pham et al., 2018; Real et al., 2018) on CIFAR-10 – results differ from Table 2 solely due to cutout (DeVries & Taylor, 2017), which is commonly used in NAS literature; NAS results are quoted from their respective papers. Our method outperforms architectures discovered by recent NAS algorithms, such as DARTS (Liu et al., 2019), ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/5a6c0ca6505fb42e138eaa5576fa8c85d0d174282021cbd2d6ee19cb662c206a.jpg",
+        "table_caption": [
+            "Table 3: (below) ImageNet classification results: training WRN 50-2 with soft parameter sharing leads to better performance by itself, without any tuning on the number of templates $k$ . Top-1 and Top-5 errors $( \\% )$ are computed using a single crop. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>ImageNet</td><td rowspan=1 colspan=1>Params</td><td rowspan=1 colspan=1>Top-1</td><td rowspan=1 colspan=1>Top-5</td></tr><tr><td rowspan=4 colspan=1>WRN 50-2DenseNet-264ResNet-200SWRN 50-2</td><td rowspan=1 colspan=1>69M</td><td rowspan=1 colspan=1>22.0</td><td rowspan=1 colspan=1>6.05</td></tr><tr><td rowspan=3 colspan=1>33M65M69M</td><td rowspan=1 colspan=1>22.15</td><td rowspan=1 colspan=1>6.12</td></tr><tr><td rowspan=1 colspan=1>21.66</td><td rowspan=2 colspan=1>5.795.95</td></tr><tr><td rowspan=1 colspan=1>21.74</td></tr></table>",
+        "bbox": [
+            174,
+            188,
+            454,
+            256
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b00389aec001bea95a460c2a9603cc1945686621bdd1c7f7bda6cd5b68b37916.jpg",
+        "table_caption": [
+            "Table 4: (right) Test error $( \\% )$ on CIFAR-10 of SWRNs and models found via neural architecture search (NAS) (all trained with cutout). Networks trained with soft parameter sharing provide competitive performance against NAS methods while having low computational cost. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>Params(M)</td><td rowspan=1 colspan=1>Training Time(GPU days)</td><td rowspan=1 colspan=1>Test Error(%)</td></tr><tr><td rowspan=8 colspan=1>NASNet-ANASNet-AAmoebaNet-BAmoebaNet-BAmoebaNet-BAmoebaNet-BDARTSSNASENAS</td><td rowspan=1 colspan=1>3.3</td><td rowspan=1 colspan=1>1800</td><td rowspan=1 colspan=1>2.65</td></tr><tr><td rowspan=1 colspan=1>27.6</td><td rowspan=1 colspan=1>1800</td><td rowspan=1 colspan=1>2.4</td></tr><tr><td rowspan=1 colspan=1>2.8</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.55</td></tr><tr><td rowspan=1 colspan=1>13.7</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.31</td></tr><tr><td rowspan=1 colspan=1>26.7</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.21</td></tr><tr><td rowspan=1 colspan=1>34.9</td><td rowspan=1 colspan=1>3150</td><td rowspan=1 colspan=1>2.13</td></tr><tr><td rowspan=1 colspan=1>3.4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>2.83</td></tr><tr><td rowspan=1 colspan=1>2.84.6</td><td rowspan=1 colspan=1>1.50.45</td><td rowspan=1 colspan=1>2.852.89</td></tr><tr><td rowspan=1 colspan=1>WRN28-10(baseline with cutout)</td><td rowspan=1 colspan=1>36.4</td><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>3.08</td></tr><tr><td rowspan=4 colspan=1>SWRN 28-4-2SWRN 28-6-2SWRN 28-10SWRN 28-10-2SWRN 28-14SWRN 28-14-2</td><td rowspan=1 colspan=1>2.7</td><td rowspan=1 colspan=1>0.12</td><td rowspan=2 colspan=1>3.453.0</td></tr><tr><td rowspan=1 colspan=1>6.1</td><td rowspan=1 colspan=1>0.25</td></tr><tr><td rowspan=1 colspan=1>36.417.1</td><td rowspan=1 colspan=1>0.40.4</td><td rowspan=1 colspan=1>2.72.69</td></tr><tr><td rowspan=1 colspan=1>71.433.5</td><td rowspan=1 colspan=1>0.70.7</td><td rowspan=1 colspan=1>2.552.53</td></tr></table>",
+        "bbox": [
+            472,
+            101,
+            821,
+            344
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "SNAS (Xie et al., 2019) and ENAS (Pham et al., 2018), while having similarly low training cost. We achieve $2 . 6 9 \\%$ test error after training less than 10 hours on a single NVIDIA GTX 1080 Ti. This accuracy is only bested by NAS techniques which are several orders of magnitude more expensive to train. Being based on Wide ResNets, our models do, admittedly, have more parameters. ",
+        "bbox": [
+            174,
+            409,
+            825,
+            465
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparison to recent NAS algorithms, such as DARTS and SNAS, is particularly interesting as our method, though motivated differently, bears some notable similarities. Specifically, all three methods are gradient-based and use an extra set of parameters (architecture parameters in DARTS and SNAS) to perform some kind of soft selection (over operations/paths in DARTS/SNAS; over templates in our method). As Section 4.3 will show, our learned template coefficients $_ { \\pmb { \\alpha } }$ can often be used to transform our networks into an explicitly recurrent form - a discovered CNN-RNN hybrid. ",
+        "bbox": [
+            174,
+            472,
+            825,
+            555
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To the extent that our method can be interpreted as a form of architecture search, it might be complementary to standard NAS methods. While NAS methods typically search over operations (e.g. activation functions; $3 \\times 3$ or $5 \\times 5$ convolutions; non-separable, separable, or grouped filters; dilation; pooling), our soft parameter sharing can be seen as a search over recurrent patterns (which layer processes the output at each step). These seem like orthogonal aspects of neural architectures, both of which may be worth examining in an expanded search space. When using SGD to drive architecture search, these aspects take on distinct forms at the implementation level: soft parameter sharing across layers (our method) vs hard parameter sharing across networks (recent NAS methods). ",
+        "bbox": [
+            174,
+            563,
+            825,
+            674
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 CLASSIFICATION ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            719,
+            436,
+            733
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use the ILSVRC 2012 dataset (Russakovsky et al., 2015) as a stronger test of our method. It is composed of 1.2M training and 50, 000 validation images, drawn from 1000 classes. We follow Gross & Wilber (2016), as in Zagoruyko & Komodakis (2016); Huang et al. (2017); Xie et al. (2017), and report Top-1 and Top-5 errors on the validation set using single $2 2 4 \\times 2 2 4$ crops. For this experiment, we use WRN 50-2 as a base model, and train it with soft sharing and no parameter reduction. Having bottleneck blocks, this model presents less uniform number of channels of layer inputs and outputs. To apply our method, we group convolutions in 12 groups: for each of the 4 stages in a WRN 50-2, we create 3 groups, one for each type of layer in a bottleneck unit $C  B$ , $B  B$ and $B  C$ channel mappings, for bottleneck $B$ ). Without any change in hyperparameters, the network trained with our method outperforms the base model and also deeper models such as DenseNets (though using more parameters), and performs close to ResNet-200, a model with four times the number of layers and a similar parameter count. See Table 3. ",
+        "bbox": [
+            173,
+            757,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b732ae6be691db0bfcb9987fe694e903c55b587b5610872d701c1f589fd45a91.jpg",
+        "image_caption": [
+            "Figure 4: Extracting implicit recurrences from a SWRN 28-10-4. Left: Illustration of the stages of a SWRN28-10-4 (residual connections omitted for clarity). The first two layers contain individual parameter sets, while the other six share four templates. All 3 stages of the network follow this structure. Middle: LSM for each stage after training on CIFAR-10, with many elements close to 1. Hard sharing schemes can be created for pairs with large similarity by tying their coefficients (or, equivalently, their effective weights). Right: Folding stages 2 and 3 leads to self-loops and a CNN with recurrent connections – LSM for stage 2 is a repetition of 2 rows/columns, and folding decreases the number of parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            104,
+            759,
+            375
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 LEARNING IMPLICIT RECURRENCES",
+        "text_level": 1,
+        "bbox": [
+            176,
+            500,
+            465,
+            513
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on CIFAR suggest that training networks with few parameter templates $k$ in our soft sharing scheme results in performance comparable to the base models, which have significantly more parameters. The lower $k$ is, the larger we should expect the layer similarities to be: in the extreme case where $k = 1$ , all layers in a sharing scheme have similarity 1, and can be folded into a single layer with a self-loop. ",
+        "bbox": [
+            174,
+            525,
+            825,
+            594
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For the case $k > 1$ , there is no trivial way to fold the network, as layer similarities depend on the learned coefficients. We can inspect the model’s layer similarity matrix (LSM) and see if it presents implicit recurrences: a form of recurrence in the rows/columns of the LSM. Surprisingly, we observe that rich structures emerge naturally in networks trained with soft parameter sharing, even without the recurrence regularizer. Figure 4 shows the per-stage LSM for CIFAR-trained SWRN 28-10-4. Here, the six layers of its stage-2 block can be folded into a loop of two layers, leading to an error increase of only $0 . 0 2 \\%$ . Appendix A contains an additional example of network folding, diversity of LSM patterns across different runs, and an epoch-wise evolution of the LSM, showing that many patterns are observable after as few as 5 epochs of training. ",
+        "bbox": [
+            174,
+            602,
+            825,
+            727
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 EVALUATION ON NATURALLY RECURRENT TASKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            743,
+            562,
+            757
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While the propensity of our parameter sharing scheme to encourage learning of recurrent networks is a useful parameter reduction tool, we would also like to leverage it for qualitative advantages over standard CNNs. On tasks for which a natural recurrent algorithm exists, does training CNNs with soft parameter sharing lead to better extrapolation? ",
+        "bbox": [
+            174,
+            770,
+            825,
+            825
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To answer this, we set up a synthetic algorithmic task: computing shortest paths. Examples are $3 2 \\times 3 2$ grids containing two query points and randomly (with probability 0.1) placed obstacles. The objective is to indicate which grid points belong to a shortest path between the query points. ",
+        "bbox": [
+            176,
+            833,
+            825,
+            875
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We use curriculum learning for training, allowing us to observe how well each model adapts to more difficult examples as training phases progress. Moreover, for this task curriculum learning causes faster learning and superior performance for all trained models. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/4c18a6433c8b4ef4f71dfc645665348dda2d2c2330d29e897fbe70e58b1e7d07.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            111,
+            433,
+            258
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/5797e1d23a50ae81bd8845709e604165aa331d04a7a0159a8e27a17f5d892741.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            529,
+            109,
+            785,
+            253
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(a) Generated example for the synthetic shortest paths task. Blue pixels indicate the query points; red pixels represent obstacles, and white pixels are points in a shortest path (in terms of Manhattan distance) between query pixels. The task consists of predicting the white pixels (shortest paths) from the blue and red ones (queries and obstacles). ",
+        "bbox": [
+            189,
+            265,
+            483,
+            353
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(b) Training curves for the shortest paths task, where difficulty of examples increases every 50 epochs. A SCNN adapts faster than a CNN to new phases and performs better, suggesting better extrapolation capacity. With a recurrence regularizer $\\lambda _ { R } = 0 . 0 1$ (SCNN-R), the model makes faster progress on the first phase, but fails to adapt to harder examples. ",
+        "bbox": [
+            514,
+            262,
+            808,
+            363
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 5: Shortest paths task. Best viewed in color. ",
+        "bbox": [
+            348,
+            376,
+            648,
+            388
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Training consists of 5 curriculum phases, each one containing 5000 examples. The maximum allowed distance between the two query points increases at each phase, thus increasing difficulty. In the first phase, each query point is within a $5 \\times 5$ grid around the other query point, and the grid size increases by 2 on each side at each phase, yielding a final grid size of $2 1 \\times 2 1$ at phase 5. ",
+        "bbox": [
+            174,
+            420,
+            825,
+            476
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We train a CNN, a CNN with soft parameter sharing and one template per layer (SCNN), and an SCNN with recurrence regularizer $\\lambda _ { R } = 0 . 0 1$ . Each model trains for 50 epochs per phase with Adam (Kingma & Ba, 2015) and a fixed learning rate of 0.01. As classes are heavily unbalanced and the balance itself changes during phases, we compare $F _ { 1 }$ scores instead of classification error. ",
+        "bbox": [
+            174,
+            482,
+            823,
+            537
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Each model starts with a $1 \\times 1$ convolution, mapping the 2 input channels to 32 output channels. Next, there are 20 channel-preserving $3 \\times 3$ convolutions, followed by a final $1 \\times 1$ convolution that maps 32 channels to 1. Each of the $2 0 3 \\times 3$ convolutions is followed by batch normalization (Ioffe & Szegedy, 2015), a ReLU non-linearity (Nair & Hinton, 2010), and has a 1-skip connection. ",
+        "bbox": [
+            174,
+            545,
+            825,
+            602
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows one example from our generated dataset and the training curves for the 3 trained models: the SCNN not only outperforms the CNN, but adapts better to harder examples at new curriculum phases. The SCNN is also advantaged over a more RNN-like model: with the recurrence regularizer $\\lambda _ { R } = 0 . 0 1$ , all entries in the LSM quickly converge 1, as in a RNN. This leads to faster learning during the first phase, but presents issues in adapting to difficulty changes in latter phases. ",
+        "bbox": [
+            174,
+            608,
+            825,
+            679
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            702,
+            318,
+            718
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we take a step toward more modular and compact CNNs by extracting recurrences from feed-forward models where parameters are shared among layers. Experimentally, parameter sharing yields models with lower error on CIFAR and ImageNet, and can be used for parameter reduction by training in a regime with fewer parameter templates than layers. Moreover, we observe that parameter sharing often leads to different layers being functionally equivalent after training, enabling us to collapse them into recurrent blocks. Results on an algorithmic task suggest that our shared parameter structure beneficially biases extrapolation. We gain a more flexible form of behavior typically attributed to RNNs, as our networks adapt better to out-of-domain examples. Our form of architecture discovery is also competitive with neural architecture search (NAS) algorithms, while having a smaller training cost than state-of-the-art gradient-based NAS. ",
+        "bbox": [
+            174,
+            734,
+            825,
+            875
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As the only requirement for our method is for a network to have groups of layers with matching parameter sizes, it can be applied to a plethora of CNN model families, making it a general technique with negligible computational cost. We hope to raise questions regarding the rigid definitions of ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "CNNs and RNNs, and increase interest in models that fall between these definitions. Adapting our method for models with non-uniform layer parameter sizes (Huang et al., 2017; Zhu et al., 2018) might be of particular future interest. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            166,
+            285,
+            183
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jonathon Cai, Richard Shin, and Dawn Song. Making neural programming architectures generalize via recursion. ICLR, 2017. ",
+        "bbox": [
+            173,
+            190,
+            823,
+            218
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs. arXiv:1606.00915, 2016. ",
+        "bbox": [
+            178,
+            227,
+            823,
+            270
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Terrance DeVries and Graham W. Taylor. Improved regularization of convolutional neural networks with cutout. Journal of Machine Learning Research, 2017. ",
+        "bbox": [
+            169,
+            279,
+            823,
+            308
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jeffrey Donahue, Lisa Anne Hendricks, Sergio Guadarrama, Marcus Rohrbach, Subhashini Venugopalan, Kate Saenko, and Trevor Darrell. Long-term recurrent convolutional networks for visual recognition and description. CVPR, 2015. ",
+        "bbox": [
+            174,
+            315,
+            823,
+            359
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. ICASSP, 2013. ",
+        "bbox": [
+            173,
+            367,
+            823,
+            397
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves, Greg Wayne, and Ivo Danihelka. Neural Turing Machines. arXiv:1410.5401, 2014. ",
+        "bbox": [
+            173,
+            405,
+            808,
+            421
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwinska, Sergio G ´ omez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, ´ Adria Puigdom \\` enech Badia, Karl Moritz Hermann, Yori Zwols, Georg Ostrovski, Adam Cain, \\` Helen King, Christopher Summerfield, Phil Blunsom, Koray Kavukcuoglu, and Demis Hassabis. Hybrid computing using a neural network with dynamic external memory. Nature, 2016. ",
+        "bbox": [
+            174,
+            429,
+            825,
+            500
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "K. Greff, R. K. Srivastava, and J. Schmidhuber. Highway and residual networks learn unrolled iterative estimation. ICLR, 2017. ",
+        "bbox": [
+            173,
+            508,
+            821,
+            537
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sam Gross and Martin Wilber. Training and investigating residual nets. https://github.com/ facebook/fb.resnet.torch, 2016. ",
+        "bbox": [
+            176,
+            546,
+            821,
+            575
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David Ha, Andrew Dai, and Quoc V. Le. Hypernetworks. arXiv:1609.09106, 2016. ",
+        "bbox": [
+            174,
+            583,
+            723,
+            599
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. ICLR, 2016. ",
+        "bbox": [
+            173,
+            608,
+            820,
+            637
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CVPR, 2016. ",
+        "bbox": [
+            174,
+            645,
+            821,
+            674
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 1997. ",
+        "bbox": [
+            176,
+            683,
+            807,
+            698
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. CVPR, 2017. ",
+        "bbox": [
+            176,
+            707,
+            821,
+            736
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gao Huang, Shichen Liu, Laurens van der Maaten, and Kilian Q. Weinberger. CondenseNet: An efficient DenseNet using learned group convolutions. CVPR, 2018. ",
+        "bbox": [
+            174,
+            744,
+            823,
+            773
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Forrest N. Iandola, Matthew W. Moskewicz, Khalid Ashraf, Song Han, William J. Dally, and Kurt Keutzer. SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and ${ < } 1 \\mathbf { M } \\mathbf { B }$ model size. arXiv:1602.07360, 2016. ",
+        "bbox": [
+            174,
+            781,
+            825,
+            824
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. ICML, 2015. ",
+        "bbox": [
+            174,
+            833,
+            821,
+            863
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stanislaw Jastrzebski, Devansh Arpit, Nicolas Ballas, Vikas Verma, Tong Che, and Yoshua Bengio. Residual connections encourage iterative inference. ICLR, 2018. ",
+        "bbox": [
+            174,
+            871,
+            820,
+            900
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. ICLR, 2015. ",
+        "bbox": [
+            173,
+            909,
+            794,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jan Koutn´ık, Klaus Greff, Faustino Gomez, and Jurgen Schmidhuber. A clockwork RNN. ¨ arXiv:1402.3511, 2014.   \nAlex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009.   \nQianli Liao and Tomaso A. Poggio. Bridging the gaps between residual learning, recurrent neural networks and visual cortex. arXiv:1604.03640, 2016.   \nChenxi Liu, Barret Zoph, Jonathon Shlens, Wei Hua, Li-Jia Li, Li Fei-Fei, Alan L. Yuille, Jonathan Huang, and Kevin Murphy. Progressive neural architecture search. ECCV, 2018.   \nHanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: Differentiable architecture search. ICLR, 2019.   \nVinod Nair and Geoffrey E. Hinton. Rectified linear units improve restricted boltzmann machines. ICML, 2010.   \nHieu Pham, Melody Guan, Barret Zoph, Quoc Le, and Jeff Dean. Efficient neural architecture search via parameters sharing. ICML, 2018.   \nPedro O. Pinheiro and Ronan Collobert. Recurrent convolutional neural networks for scene labeling. ICML, 2014.   \nAmeya Prabhu, Girish Varma, and Anoop M. Namboodiri. Deep expander networks: Efficient deep networks from graph theory. ECCV, 2018.   \nEsteban Real, Alok Aggarwal, Yanping Huang, and Quoc V Le. Regularized evolution for image classifier architecture search. arXiv:1802.01548, 2018.   \nScott E. Reed and Nando de Freitas. Neural programmer-interpreters. ICLR, 2016.   \nOlga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael S. Bernstein, Alexander C. Berg, and Fei-Fei Li. Imagenet large scale visual recognition challenge. IJCV, 115(3), 2015.   \nAndrew M. Saxe, James L. McClelland, and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. arXiv:1312.6120, 2013.   \nXingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. NIPS, 2015.   \nKaren Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. ICLR, 2015.   \nNitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1), 2014.   \nChristian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott E. Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. CVPR, 2015.   \nAndrew Trask, Felix Hill, Scott Reed, Jack Rae, Chris Dyer, and Phil Blunsom. Neural arithmetic logic units. arXiv:1808.00508, 2018.   \nSaining Xie, Ross B. Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He. Aggregated residual ´ transformations for deep neural networks. CVPR, 2017.   \nSirui Xie, Hehui Zheng, Chunxiao Liu, and Liang Lin. SNAS: stochastic neural architecture search. ICLR, 2019.   \nKelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua Bengio. Show, attend and tell: Neural image caption generation with visual attention. ICML, 2015. ",
+        "bbox": [
+            169,
+            89,
+            826,
+            930
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yibo Yang, Zhisheng Zhong, Tiancheng Shen, and Zhouchen Lin. Convolutional neural networks with alternately updated clique. CVPR, 2018. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. BMVC, 2016. ",
+        "bbox": [
+            174,
+            140,
+            710,
+            156
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Amir Roshan Zamir, Te-Lin Wu, Lin Sun, William B. Shen, Jitendra Malik, and Silvio Savarese. Feedback networks. CVPR, 2017. ",
+        "bbox": [
+            173,
+            165,
+            821,
+            194
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Wojciech Zaremba, Tomas Mikolov, Armand Joulin, and Rob Fergus. Learning simple algorithms from examples. ICML, 2016. ",
+        "bbox": [
+            173,
+            202,
+            825,
+            232
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia. Pyramid scene parsing network. CVPR, 2017. ",
+        "bbox": [
+            174,
+            239,
+            825,
+            270
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ligeng Zhu, Ruizhi Deng, Zhiwei Deng, Greg Mori, and Ping Tan. Sparsely aggregated convolutional networks. ECCV, 2018. ",
+        "bbox": [
+            176,
+            277,
+            823,
+            308
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. ICLR, 2017. ",
+        "bbox": [
+            171,
+            315,
+            820,
+            332
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Appendix ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            316,
+            127
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL RESULTS FOR IMPLICIT RECURRENCES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            148,
+            650,
+            166
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Section 4.3 presents an example of implicit recurrences and folding of a SWRN 28-10-4 trained on CIFAR-10, where, for example, the last 6 layers in the second stage of the network fold into 2 layers with a self-loop. ",
+        "bbox": [
+            173,
+            188,
+            825,
+            229
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 presents an additional example, where non-trivial recurrences (unlike the one in Figure 4) emerge naturally, resulting in a model that is rich in structure. ",
+        "bbox": [
+            173,
+            237,
+            823,
+            265
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/fc06843966b9e39486f71756cf26d0765bf6de008dc1755452226992594fc940.jpg",
+        "image_caption": [
+            "Figure 6: SWRN 40-8-8 (8 parameter templates shared among groups of $\\frac { 4 0 - 4 } { 3 } - 2 = 1 0$ layers) trained with soft parameter sharing on CIFAR-10. Each stage (originally with 12 layers – the first two do not participate in parameter sharing) can be folded to yield blocks with complex recurrences. For clarity, we use colors to indicate the computational flow: red takes precedence over green, which in turn has precedence over blue. Colored paths are only taken once per stage. Although not trivial to see, recurrences in each stage’s folded form are determined by row/column repetitions in the respective Layer Similarity Matrix. For example, for stage 2 we have $S _ { 5 , 3 } \\approx S _ { 6 , 4 } \\approx 1$ , meaning that layers 3, 4, 5 and 6 can be folded into layers 3 and 4 with a loop (captured by the red edge). The same holds for $S _ { 7 , 1 }$ $_ 1 , S _ { 8 , 2 } , S _ { 9 , 3 }$ and $S _ { 1 0 , 4 }$ , hence after the loop with layers 3 and 4, the flow returns to layer 1 and goes all the way to layer 4, which generates the stage’s output. Even though there is an approximation when folding the network (in this example, we are tying layers with similarity close to 0.8), the impact on the test error is less than $0 . 3 \\%$ . Also note that the folded model has a total of 24 layers (20 in the stage diagrams, plus 4 which are not shown, corresponding to the first layer of the network and three $1 \\times 1$ convolutions in skip-connections), instead of the original 40. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            285,
+            761,
+            723
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/fdc8c958fab5e3a2b0f2446ac5890d9310f5a57da70a5ba719a29d7cb24d7e4b.jpg",
+        "image_caption": [
+            "Figure 7: LSMs of a SWRN 40-8-8 (composed of 3 stages, each with 10 layers sharing 8 templates) trained on CIFAR-10 for 5 runs with different random seeds. Although the LSMs differ across different runs, hard parameter sharing can be observed in all cases (off-diagonal elements close to 1, depicted by white), characterizing implicit recurrences which would enable network folding. Moreover, the underlying structure is similar across runs, with hard sharing typically happening among layers $_ { i }$ and $i + 2$ , leading to a “chessboard” pattern. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            101,
+            823,
+            398
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/5de972993320487ca663b8f5109f332ee4d92857e74173a62c2f72f4841412cd.jpg",
+        "image_caption": [
+            "Figure 8: LSMs of a SWRN 40-8-8 (composed of 3 stages, each with 10 layers sharing 8 templates) at different epochs during training on CIFAR-10. The transition from an identity matrix to the final LSM happens mostly in the beginning of training: at epoch 50, the LSM is almost indistinguishable from the final LSM at epoch 200, and most of the final patterns are observable already at epoch 25. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            511,
+            823,
+            809
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B INITIALIZATION OF COEFFICIENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            496,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "During our initial experiments, we explored different initializations for the coefficients $_ { \\pmb { \\alpha } }$ of each layer, and observed that using an orthogonal initialization (Saxe et al., 2013) resulted in superior performance compared to uniform or normal initialization schemes. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            175
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Denote $\\pmb { A }$ as the $L \\times k$ matrix ( $L$ is the number of layers sharing parameters and $k$ the number of templates) with each $\\because$ ’th row containing the coefficient of the $i ^ { \\because }$ ’th layer $\\mathbf { \\alpha } \\alpha ^ { ( i ) }$ . We initialize it such that $A ^ { T } A = I$ , leading to $\\forall _ { i }$ , $\\langle \\pmb { \\alpha } ^ { ( i ) } , \\pmb { \\alpha } ^ { ( i ) } \\rangle = 1$ and $\\forall _ { i \\neq j } , \\langle { \\pmb { \\alpha } } ^ { ( i ) } , { \\pmb { \\alpha } } ^ { ( j ) } \\rangle = 0$ . While our choice for this is mostly empirical, we believe that there is likely a connection with the motivation for using orthogonal initialization for RNNs. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            255
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Moreover, we discovered that other initialization options for $\\pmb { A }$ work similarly to the orthogonal one. More specifically, either initializing $\\pmb { A }$ with the identity matrix when $L = k$ (which naturally leads to $A ^ { T } A = I $ ) or enforcing some sparsity (initialize $\\pmb { A }$ with a uniform or normal distribution and randomly setting half of its entries to zero) performs similarly to the orthogonal initialization in a consistent manner. We believe the sparse initialization to be the simplest one, as each coefficient $_ { \\pmb { \\alpha } }$ can be initialized independently. ",
+        "bbox": [
+            174,
+            261,
+            825,
+            345
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Finally, note that having $A ^ { T } A = I$ results in the Layer Similarity Matrix also being the identity at initialization (check that Si,j = $\\begin{array} { r } { S _ { i , j } = \\frac { | \\langle { \\pmb { \\alpha } } ^ { ( i ) } , { \\pmb { \\alpha } } ^ { ( j ) } \\rangle | } { \\| { \\pmb { \\alpha } } ^ { ( i ) } \\| \\| { \\pmb { \\alpha } } ^ { ( j ) } \\| } = \\frac { | ( { \\pmb { A } } ^ { T } { \\pmb { A } } ) _ { i , j } | } { \\| { \\pmb { \\alpha } } ^ { ( i ) } \\| \\| { \\pmb { \\alpha } } ^ { ( j ) } \\| } } \\end{array}$ |(AT A)i,j |kα(i)kkα(j)k , so if (AT A)i,j = 1, then Si,j = 1, and the same holds for 0. Surprisingly, even though the orthogonal initialization leads to a LSM that has no structure in the beginning of training, the rich patterns that we observe still emerge naturally after optimization. ",
+        "bbox": [
+            174,
+            352,
+            825,
+            430
+        ],
+        "page_idx": 14
+    }
+]

parse/train/rJxHsjRqFQ/rJxHsjRqFQ.md

@@ -0,0 +1,344 @@
+# HYPERBOLIC ATTENTION NETWORKS
+
+Caglar Gulcehre, Misha Denil, Mateusz Malinowski, Ali Razavi, Razvan Pascanu, Karl Moritz Hermann, Peter Battaglia, Victor Bapst, David Raposo, Adam Santoro, Nando de Freitas
+
+DeepMind
+
+# ABSTRACT
+
+Recent approaches have successfully demonstrated the benefits of learning the parameters of shallow networks in hyperbolic space. We extend this line of work by imposing hyperbolic geometry on the embeddings used to compute the ubiquitous attention mechanisms for different neural networks architectures. By only changing the geometry of embedding of object representations, we can use the embedding space more efficiently without increasing the number of parameters of the model. Mainly as the number of objects grows exponentially for any semantic distance from the query, hyperbolic geometry –as opposed to Euclidean geometry– can encode those objects without having any interference. Our method shows improvements in generalization on neural machine translation on WMT’14 (English to German), learning on graphs (both on synthetic and real-world graph tasks) and visual question answering (CLEVR) tasks while keeping the neural representations compact.
+
+# 1 INTRODUCTION
+
+The focus of this work is to endow neural network representations with suitable geometry to capture fundamental properties of data, including hierarchy and clustering behaviour. These properties emerge in many real-world scenarios that approximately follow power-law distributions (Newman, 2005; Clauset et al., 2009). This includes a wide range of natural phenomena in physics (Lin and Tegmark, 2017), biology (McGill et al., 2006), and even human-made structures such as metabolic-mass relationships (Borg, 1982), social networks (Krioukov et al., 2010; Papadopoulos et al., 2010), and frequencies of words (Powers, 1998; Piantadosi, 2014; Takahashi and Tanaka-Ishii, 2017).
+
+Complex networks (Krioukov et al., 2010), which connect distinguishable heterogeneous sets of elements represented as nodes, provide us an intuitive way of understanding these structures. They will also serve as our starting point for introducing hyperbolic geometry, which is by itself difficult to visualize. Nodes in complex networks are referred to as heterogeneous, in the sense that they can be divided into sub-nodes which are themselves distinguishable from each other. The scale-free structure of natural data manifests itself as a power law distribution on the node degrees of the complex network that describes it.
+
+Complex networks can be approximated with tree-like structures, such as taxonomies and dendrograms, and as lucidly presented by Krioukov et al. (2010), hyperbolic spaces can be thought of as smooth trees abstracting the hierarchical organization of complex networks. Let us begin by recalling a simple property of $n$ -ary trees that will help us understand hyperbolic space and why hyperbolic geometry is well suited to model relational data.
+
+In an $n$ -ary tree, the number of nodes at distance $r$ from the root and the number of nodes at distance no more than $r$ from the root both grow as $n ^ { r }$ . Similarly, in a two-dimensional hyperbolic space with curvature $- \zeta ^ { 2 } , \zeta > 0$ , the circumference and area of a disc of radius $r$ grows as $2 \pi { \mathrm { s i n h } } ( \zeta r )$ and $2 \pi ( \cosh ( \zeta r ) - 1 )$ , respectively, both of are exponential in $r$ (Krioukov et al., 2009; 2010). The growth of volume in hyperbolic space should be contrasted with Euclidean space where the corresponding quantities expand polynomially, circumference as $2 \pi r$ and area as $\pi r ^ { 2 }$ .
+
+In the two-dimensional example of Figure 1, the expanding rings show examples at a fixed semantic distance from the central object (“pug”). The number of concepts grows quickly with semantic distance forcing each successive ring to be more crowded in order to maintain a fixed distance to the center. In contrast, the extra volume of hyperbolic spheres (depicted by reducing the size of the examples) allows all of the examples to remain well separated from their semantic neighbours.
+
+![](images/917015691c7380fd6981c92a366234722e36b16c14b1a9c068b3a31e7d42e0b8.jpg)  
+Figure 1: An intuitive depiction of how images might be embedded in 2D. The location of the embeddings reflects the similarity between each image and that of a pug. Since the number of instances within a given semantic distance from the central object grows exponentially, the Euclidean space is not able to compactly represent such structure (left). In hyperbolic space (right) the volume grows exponentially, allowing for sufficient room to embed the images. For visualization, we have shrunk the images in this Euclidean diagram, a trick also used by Escher.
+
+Mechanically, the computed embeddings by a random network for objects at a given semantic distance might still seem epsilon distance away from each other (or crowded) as the ones obtained by using Euclidean geometry. However, enforcing hyperbolic geometry intuitively means that all operations with these embeddings take into account, the density in that particular region of the space. For example, any noise introduced in the system (e.g., in gradients) will also be corrected by the density. In contrast to working in Euclidean space, this means that the embeddings will be equally distinguishable regardless of the density.
+
+The intimate connection between hyperbolic space and scale free networks (where node degree follows a power law) is made more precise in Krioukov et al. (2010). In particular, there it is shown that the heterogeneous topology implies hyperbolic geometry, and conversely hyperbolic geometry yields heterogeneous topology. Moreover, Sarkar (2011) describes a construction that embeds trees in two-dimensional hyperbolic space with arbitrarily low distortion, which is not possible in Euclidean space of any dimension (Linial et al., 1998). Following this exciting line of research, recently the machine learning community has gained interest in learning non-Euclidean embeddings directly from data (Nickel and Kiela, 2017; Chamberlain et al., 2017; Ritter, 1999; Ontrup and Ritter, 2002; Tay et al., 2018; Bronstein et al., 2017).
+
+Fuelled by the desire of increasing the capacity of neural networks without increasing the number of trainable parameters so as to match the complexity of data, we propose hyperbolic attention networks. As opposed to previous approaches, which impose hyperbolic geometry on the parameters of shallow networks (Nickel and Kiela, 2017; Chamberlain et al., 2017), we impose hyperbolic geometry on the activations of deep networks. This allows us to exploit hyperbolic geometry to reason about embeddings produced by deep networks. We introduce efficient hyperbolic operations to express the popular, ubiquitous mechanism of attention (Bahdanau et al., 2014; Duan et al., 2017; Vaswani et al., 2017; Wang et al., 2017). Our method shows improvements in terms of generalization on neural machine translation (Vaswani et al., 2017), learning on graphs and visual question answering (Antol et al., 2015; Malinowski and Fritz, 2014; Johnson et al., 2017) tasks while keeping the representations compact. Simultaneously to our work, Cho et al. (2018) proposed a method to learn SVMs in the hyperboloid model of hyperbolic space, and Nickel and Kiela (2018) proposed a method to learn shallow embeddings of graphs in hyperbolic space by using the hyperboloid model.
+
+# 2 MODELS OF HYPERBOLIC SPACE
+
+Hyperbolic space cannot be isometrically embedded into Euclidean space (Krioukov et al., 2010); however, there are several ways to endow different subsets of Euclidean space with a hyperbolic metric, leading to different models of hyperbolic space. This leads to the well known Poincaré ball model (Iversen, 1992) and many others.
+
+The different models of hyperbolic space are all essentially the same, but different models define different coordinate systems, which offer different affordances for computation. In this paper, we primarily make use of the hyperboloid (or Lorentz) model of the hyperbolic space. Since the hyperboloid is unbounded, it a convenient target for projecting into hyperbolic space. We also make use of the Klein model, because it admits an efficient expression for the hyperbolic aggregation operation we define in Section 4.2.
+
+We briefly review the definitions of the hyperboloid and Klein models and the relationship between them, in just enough detail to support the presentation in the remainder of the paper. A more thorough treatment can be found in Iversen (1992). The geometric relationship between the Klein and hyperboloid models is diagrammed in Figure 5 of the supplementary material.
+
+Hyperboloid model: This model of $n$ dimensional hyperbolic space is a manifold in the $n + 1$ dimensional Minkowski space. The Minkowski space is $\bar { \mathbb { R } ^ { n + 1 } }$ endowed with the indefinite Minkowski bilinear form
+
+$$
+\langle \mathbf { q } , \mathbf { k } \rangle _ { M } { = } \sum _ { i = 1 } ^ { n } q _ { i } k _ { i } { - } q _ { n + 1 } k _ { n + 1 } .
+$$
+
+The hyperboloid model consists of the set
+
+$$
+\mathbb { H } ^ { n } = \{ \mathbf { x } \in \mathbb { R } ^ { n + 1 } | \langle \mathbf { x } , \mathbf { x } \rangle _ { M } = - 1 , x _ { n + 1 } > 0 \}
+$$
+
+endowed with the distance metric $d _ { \mathbb { H } } ( \mathbf { q } , \mathbf { k } ) = \operatorname { a r c c o s h } ( - \left. \mathbf { q } , \mathbf { k } \right. _ { M } )$
+
+Klein model: This model of hyperbolic space is a subset of $\mathbb { R } ^ { n }$ given by $\mathbb { K } ^ { n } = \left\{ \mathbf { x } \in \mathbb { R } ^ { n } | \| \mathbf { x } \| < 1 \right\}$ , and a point in the Klein model can be obtained from the corresponding point in the hyperboloid model by projection
+
+$$
+\pi _ { \mathbb { H }  \mathbb { K } } ( \mathbf { x } ) _ { i } = \frac { x _ { i } } { x _ { n + 1 } } ,
+$$
+
+with its inverse given by
+
+$$
+\pi _ { \mathbb { K } \to \mathbb { H } } ( \mathbf { x } ) = \frac { 1 } { \sqrt { 1 - \left\| \mathbf { x } \right\| ^ { 2 } } } ( \mathbf { x } , 1 )
+$$
+
+Distance computations in the Klein model can be inherited from the hyperboloid, in the sense that $d _ { \mathbb { K } } ( \mathbf { q } , \mathbf { k } ) = d _ { \mathbb { H } } ( \pi _ { \mathbb { K }  \mathbb { H } } ( \mathbf { k } ) , \pi _ { \mathbb { K }  \mathbb { H } } ( \mathbf { q }$ ).
+
+# 3 ATTENTION AS A BUILDING BLOCK FOR RELATIONAL REASONING
+
+Learning relations in a graph by using neural networks to model the interactions or relations has shown promising results in visual question answering (Santoro et al., 2017), modelling physical dynamics (Battaglia et al., 2016), and reasoning over graphs (Li et al., 2015; Vendrov et al., 2016; Kipf et al., 2018; Kool and Welling, 2018). Graph neural networks (Li et al., 2015; Battaglia et al., 2016) incorporate a message passing as part of the architecture in order to capture the intrinsic relations between entities. Graph convolution networks (Bruna et al., 2013; Kipf and Welling, 2016; Defferrard et al., 2016) use convolutions to efficiently learn a continuous-space representation for a graph of interest.
+
+Many of these relational reasoning models can be expressed in terms of an attentive read operation. In the following subsection, we give a general description of the attentive read, and then discuss its specific instantiations in two relational reasoning models from the literature.
+
+# 3.1 ATTENTIVE READ
+
+First introduced for translation in Bahdanau et al. (2014), attention has seen widespread use in deep learning, not only for applications in NLP but also for image processing (Wang et al., 2017) imitation
+
+learning (Duan et al., 2017) and memory (Graves et al., 2016). The core computation is the attentive read operation, which has the following form:
+
+$$
+\mathbf { r } ( \mathbf { q } _ { i } , \{ \mathbf { k } _ { j } \} _ { j } ) = \sum _ { j } \frac { f ( \mathbf { q } _ { i } , \mathbf { k } _ { j } ) } { Z } \mathbf { v } _ { i j } .
+$$
+
+Here $\mathbf { q } _ { i }$ is a vector called the query and the $\mathbf { k } _ { j }$ ’s are the keys for the memory locations being read from. The pairwise function $f ( \cdot , \cdot )$ computes a scalar matching score between a query and a key, and the vector $\mathbf { v } _ { i j }$ is a value to be read from location $j$ by query $i$ . $Z > 0$ is a normalization factor for the full sum. Both $\mathbf { v } _ { i j }$ and $Z$ are free to depend on arbitrary information, but we leave any dependencies here implicit.
+
+It will be useful in the discussion to break this operation down into two parts. The first is the matching, which computes attention weights $\alpha _ { i j } = f ( \mathbf { q } _ { i } , \mathbf { k } _ { j } )$ and the second is the aggregation, which takes a weighted average of the values using these weights,
+
+$$
+m _ { i } ( \{ \alpha _ { i j } \} _ { j } , \{ \mathbf { v } _ { i j } \} _ { j } ) = \sum _ { j } \frac { \alpha _ { i j } } { Z } \mathbf { v } _ { i j } .
+$$
+
+Instantiating a particular attentive read operation involves specifying both $f ( \cdot , \cdot )$ and $\mathbf { v } _ { i j }$ along with the normalization constant $Z$ .
+
+If one performs an attentive read for each element of the set $j$ then the resulting operation corresponds in a natural way to message passing on a graph, where each node $i$ aggregates messages $\{ \mathbf { v } _ { i j } \} _ { j }$ from its neighbours along edges of weight $f ( \mathbf { q } _ { i } , \mathbf { k } _ { j } ) / Z$ .
+
+We can express many (although not all) message passing neural network architectures (Gilmer et al., 2017) using the attentive read operation of Equation 1 as a primitive. In the following sections we do this for two architectures and then discuss how we can replace both the matching and aggregation steps with versions that leverage hyperbolic geometry.
+
+# 3.2 RELATION NETWORKS
+
+Relation Networks (RNs) (Santoro et al., 2017) are a neural network architecture designed for reasoning about the relationships between objects. An RN operates on a set of objects $O$ by applying a shared operator to each pair of objects $( \mathbf { o } _ { i } , \mathbf { o } _ { j } ) { \in } O \times O$ . The pairs can be augmented by a global information, and the result of each relational operation is passed through a further global transformation.
+
+Using the notation of the previous section, we can write the RN as
+
+$$
+R N ( O , \mathbf { c } ) = h \left( \sum _ { i } \mathbf { r } ( \mathbf { o } _ { i } , \{ \mathbf { o } _ { j } \} _ { j } ) ) \right) ,
+$$
+
+where $f ( \mathbf { o } _ { i } , \mathbf { o } _ { j } ) = 1$ , $\mathbf { v } _ { i j } = g ( \mathbf { o } _ { i } , \mathbf { o } _ { j } , \mathbf { c } )$ , $Z = 1$ . $h$ is the global transformation, $g$ is the local transformation and $\mathbf { c }$ is the global context, as described in Santoro et al. (2017). We augment the basic RN to allow $f ( \mathbf { o } _ { i } , \mathbf { o } _ { j } ) \in [ 0 , \bar { 1 } ]$ to be a general learnable function.
+
+Interpreting the RN as learned message passing on a graph over objects, the attention weights take on the semantics of edge weights, where $\alpha _ { i j }$ can be thought of as the probability of the (directed) edge $\mathbf { o } _ { j }  \mathbf { o } _ { i }$ appearing in the underlying reasoning graph.
+
+# 3.3 SCALED DOT-PRODUCT ATTENTION
+
+In the Transformer model of Vaswani et al. (2017) the authors define an all-to-all message passing operation on a set of vectors which they call scaled dot-product attention. In the language of Section 3.1 the scaled dot-product attention operation performs several attentive reads in parallel, one for each element of the input set.
+
+Vaswani et al. (2017) write scaled dot-product attention as $\scriptstyle \mathbf { R } = \operatorname { s o f t m a x } \left( { \frac { \mathbf { Q } \mathbf { K } ^ { T } } { \sqrt { d } } } \right) \mathbf { V }$ , where ${ \bf Q } , { \bf K }$ and $\mathbf { V }$ are referred to as the queries, keys, and values respectively, and $d$ is the shared dimensionality of the queries and keys. Using lowercase letters to denote rows of the corresponding matrices, we can write each row of $\mathbf { R }$ as the result of an attentive read with
+
+$$
+f ( \mathbf { q } _ { i } , \mathbf { k } _ { j } ) = \exp \left( \frac { 1 } { \sqrt { d } } \langle \mathbf { q } _ { i } , \mathbf { k } _ { j } \rangle \right) , \qquad \mathbf { v } _ { i j } = \mathbf { v } _ { j } , \qquad Z = \sum _ { j } f ( \mathbf { q } _ { i } , \mathbf { k } _ { j } ) .
+$$
+
+We experiment with both softmax and sigmoid operations for computing the attention weights in our hyperbolic models. The motivation for considering sigmoid attention weights is that in some applications (e.g. visual question answering), it makes sense for the attention weights over different entities to not compete with each other.
+
+# 4 HYPERBOLIC ATTENTION NETWORKS
+
+In this section we show how to redefine the attentive read operation of Section 3.1 as an operation on points in hyperbolic space. The key for doing this is to define new matching and aggregation functions that operate on hyperbolic points and take advantage of the metric structure of the manifold they live on. However, in order to apply these operations inside of a network we first we need a way to interpret network activations as points in hyperbolic space.
+
+We describe how to map an arbitrary point in $\mathbb { R } ^ { n }$ onto the hyperboloid, where we can interpret the result as a point in hyperbolic space. The choice of mapping is important since we must ensure that the rapid scaling behavior of hyperbolic space is maintained. Armed with an appropriate mapping we proceed to describe the hyperbolic matching and aggregation operations that operate on these points.
+
+# 4.1 HYPERBOLIC NETWORK ACTIVATIONS
+
+Mapping neural network activations into hyperbolic space requires care, since network activations might live anywhere in $\mathbb { R } ^ { n }$ , but hyperbolic structure can only be imposed on special subsets of Euclidean space (Krioukov et al., 2010). This means we need a way to map activations into an appropriate manifold. We choose to map into the hyperboloid, which is convenient since it is the only unbounded model of hyperbolic space in common use.
+
+Pseudo-polar coordinates: In polar coordinates, we express an $n$ -dimensional point as a scalar radius, and $n { - } 1$ angles. Pseudo-polar coordinates consist of a radius $r$ , as in ordinary polar coordinates, and an $n$ -dimensional vector d representing the direction of the point from the origin. In the following discussion we assume that the coordinates are normalized, i.e. that $\lVert \mathbf { d } \rVert = 1$ .
+
+If $( \mathbf { d } , r ) \in \mathbb { R } ^ { n + 1 }$ are the activations of a layer in the network, we map them onto the hyperbolid in $\mathbb { R } ^ { n + 1 }$ using $\pi ( ( \mathbf { d } , r ) ) = ( \sinh ( r ) \mathbf { d } , \cosh ( r ) )$ , which increases the scale by an exponential factor.
+
+It is easily verified that the resulting point lies in the hyperboloid, and to verify that we maintain the appropriate scaling properties we compute the distance between a point and the origin using this projection:
+
+$$
+d _ { \mathbb { H } } ( \mathbf { 0 } , ( \mathbf { d } , r ) ) { = } \mathrm { a r c c o s h } ( - \langle \pi ( \mathbf { 0 } ) , \pi ( ( \mathbf { d } , r ) ) \rangle _ { M } ) { = } r ~ ,
+$$
+
+which shows that this projection preserves exponential growth in volume for a linear increase in $r$ . Without the exponential scaling factor the effective distance of $\pi ( ( \mathbf { d } , r ) )$ from the origin grows logarithmically in hyperbolic space.1
+
+# 4.2 HYPERBOLIC ATTENTION
+
+In this section, we show how to build an attentive read operation that operates on points in hyperbolic space. We consider how to exploit hyperbolic geometry in both the matching and the aggregation steps of the attentive read operation separately.
+
+Hyperbolic matching: The most natural way to exploit hyperbolic geometry for matching pairs of points is to use the hyperbolic distance between them. Given a query $\mathbf { q } _ { i }$ and a key $\mathbf { k } _ { j }$ both lying in hyperbolic space we take,
+
+$$
+\alpha ( { \bf q } _ { i } , { \bf k } _ { j } ) = f ( - \beta d _ { \mathbb { H } } ( { \bf q } _ { i } , { \bf k } _ { j } ) - c ) ~ ,
+$$
+
+where $d _ { \mathbb { H } } ( \cdot , \cdot )$ is the hyperbolic distance, and $\beta$ and $c$ are parameters that can be set manually or learned along with the rest of the network. Having the bias parameter $c$ is useful because distances are non-negative. We take the function $f ( \cdot )$ to be either $\exp ( \cdot )$ , in which case we set the normalization appropriately to obtain a softmax, or sigmoid $( \cdot )$ .
+
+![](images/d84a53f0e9de207b75f14bab09afb67e7fbb20fbfc60af2b7bd7c6cda602effc.jpg)  
+Figure 2: The computational graph for the self-attention mechanism of the hyperbolic Transformer.
+
+Hyperbolic aggregation: The path to extend the weighted midpoint to hyperbolic space is much less obvious, but fortunately such a extension already exists as the Einstein midpoint. The Einstein midpoint is straightforward to compute by adjusting the aggregation weights appropriately (see Ungar (2005, Definition 4.21))
+
+$$
+m _ { i } ( \{ \alpha _ { i j } \} _ { j } , \{ { \bf v } _ { i j } \} _ { j } ) = \sum _ { j } \left[ \frac { \alpha _ { i j } \gamma ( { \bf v } _ { i j } ) } { \sum _ { \ell } \alpha _ { i \ell } \gamma ( { \bf v } _ { i \ell } ) } \right] { \bf v } _ { i j } ,
+$$
+
+where the $\gamma ( \mathbf { v } _ { i j } )$ are the Lorentz factors, that are given by $\begin{array} { r } { \gamma ( \mathbf { v } _ { i j } ) = \frac { 1 } { \sqrt { 1 - \| \mathbf { v } _ { i j } \| ^ { 2 } } } } \end{array}$ The norm in the denominator of the Lorentz factor is the Euclidean norm of the Klein coordinates of the point $\mathbf { v } _ { i j }$ , and the correctness of Equation 3 also relies on the points $\mathbf { v } _ { i j }$ being represented by their Klein coordinates. Fortunately the various models of hyperbolic space in common use are all isomorphic, so we can work in an arbitrary hyperbolic model and simply project to and from the Klein model to execute midpoint computations, as discussed in Section 2.
+
+The reason for using the Einstein midpoint for hyperbolic aggregation is that it obeys many of the properties that we expect from a weighted average in Euclidean space. In particular, translating the $\mathbf { v } _ { i j }$ ’s by a fixed distance in a common direction also translates the midpoint, and it is invariant to rotations of the constellation of points about the midpoint. The derivation of this operation is quite involved, and beyond the scope of this paper. We point the interested reader to Ungar (2005; 2008) for a full exposition.
+
+# 5 EXPERIMENTS
+
+We evaluate our models on synthetic and real-world tasks. Experiments where the underlying graph structure is explicitly known clearly show the benefits of using hyperbolic geometry as an inductive bias. At the same time, we show that real-world tasks within implicit graph structure such as a diagnostic visual question answering task (Johnson et al., 2017), and neural machine translation, equally benefit from relying on hyperbolic geometry. We provide experiments with feed-forward networks, the Transformer (Vaswani et al., 2017) and Relation Networks (Santoro et al., 2017) endowed with hyperbolic attention.
+
+Our results show the effectiveness of our approach on diverse tasks and architectures. The benefit of our approach is particularly prominent in relatively small models, which supports our hypothesis that hyperbolic geometry induces compact representations and is therefore better able to represent complex functions in limited space.
+
+# 5.1 MODELING SCALE-FREE GRAPHS
+
+We use the algorithm of von Looz et al. (2015) to efficiently generate large scale-free graphs, and define two predictive tasks that test our model’s ability to represent different aspects of the structure of these networks. For both tasks in this section, we train Recursive Transformer (RT) models, using hyperbolic and Euclidean attention. A Recursive Transformer is identical to the original transformer, except that the weights of each self-attention layer are tied across depth. Simultaneously to our work,
+
+![](images/d237011714960f4e38f8f8d7ea965840240a47c2ae93d9d1bbc33197f838a908.jpg)  
+Figure 3: Left: Performance of the Recursive Transformer models on the Shortest Path Length Prediction task on graphs of various sizes. The black dashed line indicates chance performance. Center: Results on Link Prediction Tasks. Right: The histogram of the radiuses for a model trained on a graph with 100 and 400 nodes.
+
+Dehghani et al. (2018) have proposed the same model as a generalization of the Transformer model and they referred to it as "Universal Transformers". We use models with 3 recursive self-attention layers, each of which has 4 heads with 4 units each for each of q, k, and v. This model has similarities to Graph Attention Networks (Velickovi ˇ c et al., 2017; Kool and Welling, 2018). ´
+
+Link prediction (LP): Link prediction is a classical graph problem, where the task is to predict if an edge exists between two nodes in the graph. We experimented with graphs of 1000 and 1200 nodes and observed that the hyperbolic RT performs better than the Euclidean RT on both tasks. We report the results in Figure 3 (middle). In general, we observed that for graphs of size 1000 and 1200 the hyperbolic transformer performs better than the Euclidean transformer given the same amount of capacity.
+
+Shortest path length prediction (SPLP): In this task, the goal is to predict the length of the shortest path between a pair of nodes in the graph. We treat this as a classification problem with a maximum pathlength of 25 which becomes naturally an unbalanced classification problem. We use rejection sampling during training to ensure the network is trained on an approximately uniform distribution of path lengths. At test time we sample paths uniformly at random, so the length distribution follows that of the underlying graphs. We report the results in Figure 3 (left). In Figure 3 (right), we visualize the distribution of the scale of the learned activations $\dot { \boldsymbol { r } }$ in the projection of Section 4.1) when training on graphs of size 100 and 400. We observe that our model tends to use larger scales for the larger graphs. As a baseline, we compare to the optimal constant predictor, which always predicts the most common expected path length. This baseline does quite well since the path length distribution on the test set is quite skewed.
+
+For both tasks, we generate training data online. Each example is a new graph in which we query the connectivity of a randomly chosen pair of nodes. To make training easier, we use a curriculum, whereby we start training on smaller graphs and gradually increase the number of vertices towards the final number. More details on the dataset generation procedure and the curriculum scheme are found in the supplementary material.
+
+# 5.2 SORT-OF-CLEVR
+
+Since we expect hyperbolic attention to be particularly well suited to relational modelling, we investigate our models on the relational variant of the Sort-of-CLEVR dataset (Santoro et al., 2017). This dataset consists of simple visual scenes allowing us to solely focus on the relational aspect of the problem. Our models extend Relation Nets (RNs) with the attention mechanism in hyperbolic space (with the Euclidean or Einstein midpoint aggregation), but otherwise we follow the standard setup-up (Santoro et al., 2017). Our best method yields accuracy of $9 9 . 2 \%$ that significantly exceeds the accuracy of the original RN $( 9 6 \% )$ .
+
+However, we are more interested in evaluating models on the low-capacity regime. Indeed, as Figure 4 (left) shows, the attention mechanism computed in the hyperbolic space improves around 20 percent points over the standard RN, where all the models use only two units of the relational MLP.
+
+# 5.3 EXPERIMENTS ON CITESEER AND CORA
+
+We use two of the standard graph transduction benchmark datasets, Citeseer and Cora (Sen et al., 2008) and used the same experimental protocol defined in Velickovi ˇ c et al. (2017). We use ´ graph attention
+
+![](images/425a0b24b227f503fb044f953043d558bef80a15982e7465eda5cd1a38dc7d38.jpg)  
+Figure 4: Left: Comparison of our models with low-capacity on the Sort-of-CLEVR dataset. The “EA” refers to the model that uses hyperbolic attention weights with Euclidean aggregation. Right: Performance of Relation Network extended by attention mechanism in either Euclidean or hyperbolic space on the CLEVR dataset.
+
+<table><tr><td colspan="3">Transductive</td></tr><tr><td>Method</td><td>Cora</td><td>Citeseer</td></tr><tr><td>GCN (Kipf and Welling, 2016)</td><td>81.5%</td><td>70.3%</td></tr><tr><td>GAT (Velickovic et al., 2017)</td><td>83.0%±0.14</td><td>72.5%± 0.14</td></tr><tr><td>H-GAT</td><td>83.5% ± 0.12</td><td>72.9% ± 0.078</td></tr></table>
+
+Table 1: Results on graph transduction tasks. We have used the same setup that is described in (Velickovi ˇ c et al., 2017). H-GAT refers to our graph attention network with hyperbolic attention ´ mechanism. Table shows the mean performance over 100 random seeds, along with $9 5 \%$ confidence intervals for this estimate.
+
+networks (GAT) as our baseline and developed a hyperbolic version of GAT (H-GAT) by replacing the original attention mechanism with the hyperbolic attention using softmax.
+
+We report our results in Table 1 and compare against the GAT with the Euclidean attention mechanism. We compute the standard deviations over 100 seeds and got improvements both on Citeseer and Cora datasets over the original GAT model. We show the visualizations of the learned hyperbolic embeddings of q and k in A.4.
+
+# 5.4 CLEVR
+
+We train our Relation Network with various attention mechanisms on the CLEVR dataset (Johnson et al., 2017). CLEVR is a synthetic visual question answering datasets consisting of 3D rendered objects like spheres, cubes, or cylinders of various size, material, or color. In contrast to other visual question answering datasets (Antol et al., 2015; Malinowski and Fritz, 2014; Zhu et al., 2016), the focus of CLEVR is on relational reasoning.
+
+In our experiments, we closely follow the procedure established in (Santoro et al., 2017), both in terms of the model architecture, capacity, or the choice of the hyperparameters, and only differ by the attention mechanism (Euclidean or hyperbolic attention), or sigmoid activations.
+
+Results are shown in Figure 4 (Right). For each model, we vary the capacity of the relational part of the network and report the resulting test accuracy. We find that hyperbolic attention with sigmoid consistently outperforms other models. Our RN with hyperbolic attention and sigmoid achieves ${ \bar { 9 } } 5 . 7 \%$ accuracy on the test set at the same capacity level as RN, whereas the latter reportedly achieves $9 5 . 5 \%$ accuracy (Santoro et al., 2017).
+
+# 5.5 NEURAL MACHINE TRANSLATION
+
+The Transformer (Vaswani et al., 2017) is a recently introduced state of the art model for neural machine translation that relies heavily on attention as its core operation. As described in Section 3.3,
+
+<table><tr><td></td><td colspan="3">WMT 2014 En-De BLEU Scores</td></tr><tr><td></td><td>Tiny</td><td>Base</td><td>Big</td></tr><tr><td>Transformer (Vaswani et al. (2017))</td><td>1</td><td>27.3</td><td>28.4</td></tr><tr><td>Transformer (Latest)</td><td>17.3</td><td>27.1</td><td>-</td></tr><tr><td>Hyperbolic Transformer (+Sigmoid)</td><td>17.5</td><td>27.4</td><td>1</td></tr><tr><td>Hyperbolic Transformer(+Softmax,+Pseudo-Polar)</td><td>17.9</td><td>27.4</td><td>=</td></tr><tr><td>Hyperbolic Transformer (+Sigmoid, +Pseudo-Polar)</td><td>18.6</td><td>27.9</td><td>28.52</td></tr></table>
+
+Table 2: Results for the WMT14 English to German translation task. Results are computed following the procedure in Vaswani et al. (2017). Citations indicate results taken from the literature. Latest is the result of training a new model using an unmodified version of the same code where we added hyperbolic attention (we have observed that the exact performance of the transformer on this task varies as the Tensor2tensor codebase evolves).
+
+we have extended the Transformer2 by replacing its scaled dot-product attention operation with its hyperbolic counterpart. We evaluate all the models on the WMT14 En-De dataset (Bojar et al., 2014).
+
+We train several versions of the Transformer model with hyperbolic attention. They use different coordinate systems (Cartesian or pseudo-polar), or different attention normalization functions (softmax or sigmoid). We consider three model sizes, referred to here as tiny, base and big. The tiny model has two layers of encoders and decoders, each with 128 units and 4 attention heads. The base model has 6 layers of encoders and decoders, each with 512 units and 8 attention heads. All hyperparameter configurations for the Euclidean versions of these models are available in the Tensor2tensor repository.
+
+Results are shown in Table 2. We observe improvements over the Euclidean model by using hyperbolic attention, in particular when coupled with the sigmoid activation function for the attention weights. The improvements are more significant when the model capacity is restricted.
+
+In addition, our best model (with sigmoid activation function and without pseudo-polar coordinates) using the big architecture from Tensor2tensor, achieves 28.52 BLEU score, whereas Vaswani et al. (2017) report 28.4 BLEU score with the original version of this model.3.
+
+# 6 CONCLUSION
+
+We have presented a novel way to impose the inductive biases from hyperbolic geometry on the activations of deep neural networks. Our proposed hyperbolic attention operation makes use of hyperbolic geometry in both the computation of the attention weights, and in the aggregation operation over values. We implemented our proposed hyperbolic attention mechanism in both Relation Networks and the Transformer and showed that we achieve improved performance on a diverse set of tasks. We have shown improved performance on link prediction and shortest path length prediction in scale free graphs, on two visual question answering datasets, real-world graph transduction tasks and finally on English to German machine translation. The gains are particularly prominent in relatively small models, which confirms our hypothesis that hyperbolic geometry induces more compact representations.
+
+Yang and Rush (2016) have proposed to imposed the activations of the neural network to lie on a Lie-group manifold in the memory. Similarly as a future work, an interesting potential future direction is to use hyperbolic geometry as an inductive bias for the activation of neural networks in the memory.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Neil Rabinowitz, Chris Dyer for constructive comments on earlier versions of this draft. We thank Yannis Asseal for helping us with the styles of the plots in this draft. We would like to thank Thomas Paine for the constructive comments.
+
+REFERENCES   
+Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C Lawrence Zitnick, and Devi Parikh. Vqa: Visual question answering. In Proceedings of the IEEE International Conference on Computer Vision, pages 2425–2433, 2015.   
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations, 2014.   
+Peter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al. Interaction networks for learning about objects, relations and physics. In Advances in neural information processing systems, pages 4502–4510, 2016.   
+Ondrej Bojar, Christian Buck, Christian Federmann, Barry Haddow, Philipp Koehn, Johannes Leveling, Christof Monz, Pavel Pecina, Matt Post, Herve Saint-Amand, Radu Soricut, Lucia Specia, and Aleš Tamchyna. Findings of the 2014 workshop on statistical machine translation. In Proceedings of the Ninth Workshop on Statistical Machine Translation, pages 12–58, Baltimore, Maryland, USA, June 2014. Association for Computational Linguistics. URL http://www.aclweb.org/anthology/W/W14/W14-3302.   
+Gunnar A Borg. Psychophysical bases of perceived exertion. Med sci sports exerc, 14(5):377–381, 1982.   
+Michael M Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18–42, 2017.   
+Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013.   
+Benjamin Paul Chamberlain, James Clough, and Marc Peter Deisenroth. Neural embeddings of graphs in hyperbolic space. arXiv preprint arXiv:1705.10359, 2017.   
+Hyunghoon Cho, Benjamin DeMeo, Jian Peng, and Bonnie Berger. Large-margin classification in hyperbolic space. arXiv preprint arXiv:1806.00437, 2018.   
+Aaron Clauset, Cosma Rohilla Shalizi, and Mark EJ Newman. Power-law distributions in empirical data. SIAM review, 51(4):661–703, 2009.   
+Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pages 3844–3852, 2016.   
+Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Łukasz Kaiser. Universal transformers. arXiv preprint arXiv:1807.03819, 2018.   
+Yan Duan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba. One-shot imitation learning. In Advances in neural information processing systems, pages 1087–1098, 2017.   
+Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. arXiv preprint arXiv:1704.01212, 2017.   
+Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka Grabska-Barwinska, ´ Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626):471, 2016.   
+Birger Iversen. Hyperbolic geometry, volume 25. Cambridge University Press, 1992.   
+Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C Lawrence Zitnick, and Ross Girshick. Clevr: A diagnostic dataset for compositional language and elementary visual reasoning. In Computer Vision and Pattern Recognition (CVPR), 2017 IEEE Conference on, pages 1988–1997. IEEE, 2017.   
+Thomas Kipf, Ethan Fetaya, Kuan-Chieh Wang, Max Welling, and Richard Zemel. Neural relational inference for interacting systems. arXiv preprint arXiv:1802.04687, 2018.   
+Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.   
+WWM Kool and M Welling. Attention solves your tsp. arXiv preprint arXiv:1803.08475, 2018.   
+Dmitri Krioukov, Fragkiskos Papadopoulos, Amin Vahdat, and Marián Boguná. Curvature and temperature of complex networks. Physical Review E, 80(3):035101, 2009.   
+Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010.   
+Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015.   
+Henry W Lin and Max Tegmark. Critical behavior in physics and probabilistic formal languages. Entropy, 19 (7):299, 2017.   
+Nathan Linial, Avner Magen, and Michael E Saks. Low distortion euclidean embeddings of trees. Israel Journal of Mathematics, 106(1):339–348, 1998.   
+Mateusz Malinowski and Mario Fritz. A multi-world approach to question answering about real-world scenes based on uncertain input. In Advances in neural information processing systems, pages 1682–1690, 2014.   
+Brian J McGill, Brian J Enquist, Evan Weiher, and Mark Westoby. Rebuilding community ecology from functional traits. Trends in ecology & evolution, 21(4):178–185, 2006.   
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015.   
+Mark EJ Newman. Power laws, pareto distributions and zipf’s law. Contemporary physics, 46(5):323–351, 2005.   
+Maximilian Nickel and Douwe Kiela. Learning continuous hierarchies in the lorentz model of hyperbolic geometry. arXiv preprint arXiv:1806.03417, 2018.   
+Maximillian Nickel and Douwe Kiela. Poincaré embeddings for learning hierarchical representations. In Advances in Neural Information Processing Systems, pages 6341–6350, 2017.   
+Jorg Ontrup and Helge Ritter. Hyperbolic self-organizing maps for semantic navigation. In Advances in neural information processing systems, pages 1417–1424, 2002.   
+Fragkiskos Papadopoulos, Dmitri Krioukov, Marián Boguñá, and Amin Vahdat. Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In INFOCOM, 2010 Proceedings IEEE, pages 1–9. IEEE, 2010.   
+Steven T Piantadosi. Zipf’s word frequency law in natural language: A critical review and future directions. Psychonomic bulletin & review, 21(5):1112–1130, 2014.   
+David MW Powers. Applications and explanations of zipf’s law. In Proceedings of the joint conferences on new methods in language processing and computational natural language learning, pages 151–160. Association for Computational Linguistics, 1998.   
+Helge Ritter. Self-organizing maps on non-euclidean spaces. In Kohonen maps, pages 97–109. Elsevier, 1999.   
+Adam Santoro, David Raposo, David G Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Tim Lillicrap. A simple neural network module for relational reasoning. In Advances in neural information processing systems, pages 4974–4983, 2017.   
+Rik Sarkar. Low distortion delaunay embedding of trees in hyperbolic plane. In International Symposium on Graph Drawing, pages 355–366. Springer, 2011.   
+Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI magazine, 29(3):93, 2008.   
+Shuntaro Takahashi and Kumiko Tanaka-Ishii. Do neural nets learn statistical laws behind natural language? PloS one, 12(12):e0189326, 2017.   
+Yi Tay, Luu Anh Tuan, and Siu Cheung Hui. Hyperbolic representation learning for fast and efficient neural question answering. In Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining, pages 583–591. ACM, 2018.   
+Abraham Albert Ungar. Analytic hyperbolic geometry: Mathematical foundations and applications. World Scientific, 2005.   
+Abraham Albert Ungar. A gyrovector space approach to hyperbolic geometry. Synthesis Lectures on Mathematics and Statistics, 1(1):1–194, 2008.   
+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, pages 6000–6010, 2017.   
+Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph ´ attention networks. arXiv preprint arXiv:1710.10903, 2017.   
+Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. In ICLR, 2016.   
+Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, pages 2692–2700, 2015.   
+Moritz von Looz, Henning Meyerhenke, and Roman Prutkin. Generating random hyperbolic graphs in subquadratic time. In International Symposium on Algorithms and Computation, pages 467–478. Springer, 2015.   
+Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. arXiv preprint arXiv:1711.07971, 2017.   
+Greg Yang and Alexander M. Rush. Lie access neural turing machine. arXiv preprint arXiv:1602.08671, 2016.   
+Yuke Zhu, Oliver Groth, Michael Bernstein, and Li Fei-Fei. Visual7w: Grounded question answering in images. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4995–5004, 2016.
+
+# A APPENDIX
+
+# A.1 MORE ON MODELS OF HYPERBOLIC SPACE
+
+In Figure 5, we illustrate the relationship between different models of hyperbolic space. There are one-to-one isometric transformations defined between each different models of the hyperbolic space. Hyperboloid model is unbounded, whereas Klein and Poincare models are bounded in a disk.
+
+![](images/41f4a49d70b83c38929ce3d8718b494f6d1c81c34f1cdb2969748fad366f5fdc.jpg)  
+Figure 5: Relationships between different representations of points used in the paper. Left: The relationship between pseudo-polar coordinates in $\mathbb { R } ^ { n }$ and the hyperboloid in $\mathbb { R } ^ { n + 1 }$ . Right: Projections relating the hyperboloid, Klein and Poincaré models of hyperbolic space.
+
+# A.2 SCALE-FREE GRAPH GENERATION
+
+We use the algorithm described by von Looz et al. (2015). In our experiments, we set the $\alpha$ to 0.95 and edge_radius_R_factor to 0.35. We will release our code both for generating and the operations in the hyperbolic space along with the camera-ready version of our paper.
+
+# A.3 SCALE-FREE GRAPH CURRICULUM
+
+Curriculum was an essential part of our training on the scale-free graph tasks. On LP and SPLP tasks, we use a curriculum where we extract the connected components from the graph by cutting the disk that the graphs generated on into slices by starting from a 30 degree angle and gradually increasing the size of the slice from the disk by increasing the angle during the training according to the number of lessons that are involved in the curriculum. This process is also visualized in Figure 6.
+
+# A.4 VISUALIZATION OF QUERY AND KEY EMBEDDINGS ON CORA
+
+In Figure 8 and 9, we visualize the embeddings of the query $( q )$ and the keys $( k )$ going into the hyperbolic matching function on the Poincare Ball model. In Figure 8, the embeddings of a model trained with dropout are bounded in a ball with smaller volume than the model trained without dropout. Also as clearly can be seen from the embedding visualizations k’s and q’s are clustered on different regions of the space.
+
+# A.5 TRAVELLING SALESMAN PROBLEM (TSP)
+
+We train an off-policy DQN-like agent (Mnih et al., 2015) with the HRT. The graphs for the TSP is generated following the procedure introduced in (Vinyals et al., 2015).
+
+On this task, as an ablation we just compared the hyperbolic networks with and The results are provided in Figure 4 (Right) with and without implicit coordinates. Overall, we found that the hyperbolic transformer networks performs better when using the implicit polar coordinates.
+
+# A.6 HYPERBOLIC RECURSIVE TRANSFORMER
+
+As shown in Figure ??, the hyperbolic RT is an extension of transformer that ties the parameters of the self-attention layers. The self-attention layer gets the representations of the nodes of the graph coming from the encoder and the decoder decodes that representation from the recursive self-attention layers for the prediction.
+
+![](images/ffa7a6033a2daacb09745bba54dd38baace27eac981747e6caed41e2406d1ee5.jpg)  
+Figure 6: We show an example of a curriculum on the hyperbolic disk. In the first lesson, we take slices from the graph only between angle 0 and $\pi / 2$ . In the second lesson we will have to take the slice from 0 to $\pi$ .
+
+![](images/2b77d45c72ea00f743e01860b6db1df360e2ea5da868bd7487ae61a8b9f8440c.jpg)  
+Figure 7: An illustration of how trees can be represented in hyperbolic (left) and Euclidean geometry (right) in a cone. In hyperbolic space, as the tree grows the angles between the edges $\mathbf { \eta } ^ { ( \theta ) }$ can be preserved from one level to the next. In Euclidean space, since the number of nodes in the tree grows faster than the rate that the volume grows, angles may not be preserved ( $\boldsymbol { \theta }$ to $\alpha$ ). Lines in the left diagram are straight in hyperbolic space, but appear curved in this Euclidean diagram.
+
+![](images/50c796e2835fe35a0d7ba5a320b7318cd3265492d5aea900d09c92c80dc78571.jpg)  
+Figure 8: Hyperbolic embedding of $q$ (red) and $k$ (blue) in a Poincare Ball on Cora dataset. Each point corresponds to a node in the graph. This visualization is obtained from a model trained with dropout. The graph on the left is the embeddings going into the attention obtained from the first layer. The Figure on the right is for the embedding of the second layer.
+
+![](images/7cca480e2e9184bdad70ba87b2d79f8801c20858f96bcacc8f3a63211a2645a5.jpg)  
+Figure 9: Hyperbolic embeddings of q (red) and k (blue) in a Poincare Ball on Cora dataset. Each point corresponds to a node in the graph. This visualization is obtained from a model trained without dropout. The figure on the left shows the embeddings going into the attention obtained from the first layer. The figure on the right shows the embedding of the second layer.
+
+![](images/bf47e1dd9f14dc878172815ae0d1d82ca15f62ff5c6e5c5671e9d8ba8cdd5112.jpg)  
+Figure 10: The comparisons between a hyperbolic recursive transformer with and without pseudo-polar (denoted as $^ +$ spherical in the legend) coordinates on the travelling salesman problem.

parse/train/rJxHsjRqFQ/rJxHsjRqFQ_content_list.json

@@ -0,0 +1,1559 @@
+[
+    {
+        "type": "text",
+        "text": "HYPERBOLIC ATTENTION NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            627,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Caglar Gulcehre, Misha Denil, Mateusz Malinowski, Ali Razavi, Razvan Pascanu, Karl Moritz Hermann, Peter Battaglia, Victor Bapst, David Raposo, Adam Santoro, Nando de Freitas ",
+        "bbox": [
+            183,
+            143,
+            669,
+            186
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "DeepMind ",
+        "bbox": [
+            184,
+            193,
+            254,
+            205
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            242,
+            544,
+            258
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent approaches have successfully demonstrated the benefits of learning the parameters of shallow networks in hyperbolic space. We extend this line of work by imposing hyperbolic geometry on the embeddings used to compute the ubiquitous attention mechanisms for different neural networks architectures. By only changing the geometry of embedding of object representations, we can use the embedding space more efficiently without increasing the number of parameters of the model. Mainly as the number of objects grows exponentially for any semantic distance from the query, hyperbolic geometry –as opposed to Euclidean geometry– can encode those objects without having any interference. Our method shows improvements in generalization on neural machine translation on WMT’14 (English to German), learning on graphs (both on synthetic and real-world graph tasks) and visual question answering (CLEVR) tasks while keeping the neural representations compact. ",
+        "bbox": [
+            233,
+            271,
+            766,
+            438
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The focus of this work is to endow neural network representations with suitable geometry to capture fundamental properties of data, including hierarchy and clustering behaviour. These properties emerge in many real-world scenarios that approximately follow power-law distributions (Newman, 2005; Clauset et al., 2009). This includes a wide range of natural phenomena in physics (Lin and Tegmark, 2017), biology (McGill et al., 2006), and even human-made structures such as metabolic-mass relationships (Borg, 1982), social networks (Krioukov et al., 2010; Papadopoulos et al., 2010), and frequencies of words (Powers, 1998; Piantadosi, 2014; Takahashi and Tanaka-Ishii, 2017). ",
+        "bbox": [
+            174,
+            492,
+            825,
+            589
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Complex networks (Krioukov et al., 2010), which connect distinguishable heterogeneous sets of elements represented as nodes, provide us an intuitive way of understanding these structures. They will also serve as our starting point for introducing hyperbolic geometry, which is by itself difficult to visualize. Nodes in complex networks are referred to as heterogeneous, in the sense that they can be divided into sub-nodes which are themselves distinguishable from each other. The scale-free structure of natural data manifests itself as a power law distribution on the node degrees of the complex network that describes it. ",
+        "bbox": [
+            174,
+            595,
+            825,
+            679
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Complex networks can be approximated with tree-like structures, such as taxonomies and dendrograms, and as lucidly presented by Krioukov et al. (2010), hyperbolic spaces can be thought of as smooth trees abstracting the hierarchical organization of complex networks. Let us begin by recalling a simple property of $n$ -ary trees that will help us understand hyperbolic space and why hyperbolic geometry is well suited to model relational data. ",
+        "bbox": [
+            174,
+            686,
+            825,
+            756
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In an $n$ -ary tree, the number of nodes at distance $r$ from the root and the number of nodes at distance no more than $r$ from the root both grow as $n ^ { r }$ . Similarly, in a two-dimensional hyperbolic space with curvature $- \\zeta ^ { 2 } , \\zeta > 0$ , the circumference and area of a disc of radius $r$ grows as $2 \\pi { \\mathrm { s i n h } } ( \\zeta r )$ and $2 \\pi ( \\cosh ( \\zeta r ) - 1 )$ , respectively, both of are exponential in $r$ (Krioukov et al., 2009; 2010). The growth of volume in hyperbolic space should be contrasted with Euclidean space where the corresponding quantities expand polynomially, circumference as $2 \\pi r$ and area as $\\pi r ^ { 2 }$ . ",
+        "bbox": [
+            174,
+            763,
+            825,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the two-dimensional example of Figure 1, the expanding rings show examples at a fixed semantic distance from the central object (“pug”). The number of concepts grows quickly with semantic distance forcing each successive ring to be more crowded in order to maintain a fixed distance to the center. In contrast, the extra volume of hyperbolic spheres (depicted by reducing the size of the examples) allows all of the examples to remain well separated from their semantic neighbours. ",
+        "bbox": [
+            176,
+            854,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/917015691c7380fd6981c92a366234722e36b16c14b1a9c068b3a31e7d42e0b8.jpg",
+        "image_caption": [
+            "Figure 1: An intuitive depiction of how images might be embedded in 2D. The location of the embeddings reflects the similarity between each image and that of a pug. Since the number of instances within a given semantic distance from the central object grows exponentially, the Euclidean space is not able to compactly represent such structure (left). In hyperbolic space (right) the volume grows exponentially, allowing for sufficient room to embed the images. For visualization, we have shrunk the images in this Euclidean diagram, a trick also used by Escher. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            102,
+            816,
+            286
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Mechanically, the computed embeddings by a random network for objects at a given semantic distance might still seem epsilon distance away from each other (or crowded) as the ones obtained by using Euclidean geometry. However, enforcing hyperbolic geometry intuitively means that all operations with these embeddings take into account, the density in that particular region of the space. For example, any noise introduced in the system (e.g., in gradients) will also be corrected by the density. In contrast to working in Euclidean space, this means that the embeddings will be equally distinguishable regardless of the density. ",
+        "bbox": [
+            174,
+            417,
+            825,
+            516
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The intimate connection between hyperbolic space and scale free networks (where node degree follows a power law) is made more precise in Krioukov et al. (2010). In particular, there it is shown that the heterogeneous topology implies hyperbolic geometry, and conversely hyperbolic geometry yields heterogeneous topology. Moreover, Sarkar (2011) describes a construction that embeds trees in two-dimensional hyperbolic space with arbitrarily low distortion, which is not possible in Euclidean space of any dimension (Linial et al., 1998). Following this exciting line of research, recently the machine learning community has gained interest in learning non-Euclidean embeddings directly from data (Nickel and Kiela, 2017; Chamberlain et al., 2017; Ritter, 1999; Ontrup and Ritter, 2002; Tay et al., 2018; Bronstein et al., 2017). ",
+        "bbox": [
+            174,
+            522,
+            825,
+            648
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fuelled by the desire of increasing the capacity of neural networks without increasing the number of trainable parameters so as to match the complexity of data, we propose hyperbolic attention networks. As opposed to previous approaches, which impose hyperbolic geometry on the parameters of shallow networks (Nickel and Kiela, 2017; Chamberlain et al., 2017), we impose hyperbolic geometry on the activations of deep networks. This allows us to exploit hyperbolic geometry to reason about embeddings produced by deep networks. We introduce efficient hyperbolic operations to express the popular, ubiquitous mechanism of attention (Bahdanau et al., 2014; Duan et al., 2017; Vaswani et al., 2017; Wang et al., 2017). Our method shows improvements in terms of generalization on neural machine translation (Vaswani et al., 2017), learning on graphs and visual question answering (Antol et al., 2015; Malinowski and Fritz, 2014; Johnson et al., 2017) tasks while keeping the representations compact. Simultaneously to our work, Cho et al. (2018) proposed a method to learn SVMs in the hyperboloid model of hyperbolic space, and Nickel and Kiela (2018) proposed a method to learn shallow embeddings of graphs in hyperbolic space by using the hyperboloid model. ",
+        "bbox": [
+            173,
+            655,
+            825,
+            835
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MODELS OF HYPERBOLIC SPACE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            861,
+            472,
+            877
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Hyperbolic space cannot be isometrically embedded into Euclidean space (Krioukov et al., 2010); however, there are several ways to endow different subsets of Euclidean space with a hyperbolic metric, leading to different models of hyperbolic space. This leads to the well known Poincaré ball model (Iversen, 1992) and many others. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The different models of hyperbolic space are all essentially the same, but different models define different coordinate systems, which offer different affordances for computation. In this paper, we primarily make use of the hyperboloid (or Lorentz) model of the hyperbolic space. Since the hyperboloid is unbounded, it a convenient target for projecting into hyperbolic space. We also make use of the Klein model, because it admits an efficient expression for the hyperbolic aggregation operation we define in Section 4.2. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            222
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We briefly review the definitions of the hyperboloid and Klein models and the relationship between them, in just enough detail to support the presentation in the remainder of the paper. A more thorough treatment can be found in Iversen (1992). The geometric relationship between the Klein and hyperboloid models is diagrammed in Figure 5 of the supplementary material. ",
+        "bbox": [
+            174,
+            229,
+            825,
+            286
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hyperboloid model: This model of $n$ dimensional hyperbolic space is a manifold in the $n + 1$ dimensional Minkowski space. The Minkowski space is $\\bar { \\mathbb { R } ^ { n + 1 } }$ endowed with the indefinite Minkowski bilinear form ",
+        "bbox": [
+            174,
+            291,
+            825,
+            334
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/830e29287a83032b45b8dce532c328d43d325323b159225f61185589a1c603cc.jpg",
+        "text": "$$\n\\langle \\mathbf { q } , \\mathbf { k } \\rangle _ { M } { = } \\sum _ { i = 1 } ^ { n } q _ { i } k _ { i } { - } q _ { n + 1 } k _ { n + 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            340,
+            602,
+            382
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The hyperboloid model consists of the set ",
+        "bbox": [
+            174,
+            388,
+            442,
+            404
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1826c194e5a12e7646a33b95d4498417d29156ec1cd91fab03d96712bacf6d42.jpg",
+        "text": "$$\n\\mathbb { H } ^ { n } = \\{ \\mathbf { x } \\in \\mathbb { R } ^ { n + 1 } | \\langle \\mathbf { x } , \\mathbf { x } \\rangle _ { M } = - 1 , x _ { n + 1 } > 0 \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            411,
+            638,
+            430
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "endowed with the distance metric $d _ { \\mathbb { H } } ( \\mathbf { q } , \\mathbf { k } ) = \\operatorname { a r c c o s h } ( - \\left. \\mathbf { q } , \\mathbf { k } \\right. _ { M } )$ ",
+        "bbox": [
+            174,
+            438,
+            606,
+            453
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Klein model: This model of hyperbolic space is a subset of $\\mathbb { R } ^ { n }$ given by $\\mathbb { K } ^ { n } = \\left\\{ \\mathbf { x } \\in \\mathbb { R } ^ { n } | \\| \\mathbf { x } \\| < 1 \\right\\}$ , and a point in the Klein model can be obtained from the corresponding point in the hyperboloid model by projection ",
+        "bbox": [
+            174,
+            459,
+            825,
+            501
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d90cf38e9e5bcacc85e0ea0ca2af4facb9048c42c027f9bb05bb5ee0c2bd839.jpg",
+        "text": "$$\n\\pi _ { \\mathbb { H }  \\mathbb { K } } ( \\mathbf { x } ) _ { i } = \\frac { x _ { i } } { x _ { n + 1 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            433,
+            508,
+            563,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with its inverse given by ",
+        "bbox": [
+            174,
+            546,
+            331,
+            561
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0ace3a2b111caf64d7c380d74ba2bda70e0f8439d77de68291cc576ec2afd45.jpg",
+        "text": "$$\n\\pi _ { \\mathbb { K } \\to \\mathbb { H } } ( \\mathbf { x } ) = \\frac { 1 } { \\sqrt { 1 - \\left\\| \\mathbf { x } \\right\\| ^ { 2 } } } ( \\mathbf { x } , 1 )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            569,
+            596,
+            613
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Distance computations in the Klein model can be inherited from the hyperboloid, in the sense that $d _ { \\mathbb { K } } ( \\mathbf { q } , \\mathbf { k } ) = d _ { \\mathbb { H } } ( \\pi _ { \\mathbb { K }  \\mathbb { H } } ( \\mathbf { k } ) , \\pi _ { \\mathbb { K }  \\mathbb { H } } ( \\mathbf { q }$ ). ",
+        "bbox": [
+            173,
+            619,
+            823,
+            650
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 ATTENTION AS A BUILDING BLOCK FOR RELATIONAL REASONING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            671,
+            743,
+            686
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning relations in a graph by using neural networks to model the interactions or relations has shown promising results in visual question answering (Santoro et al., 2017), modelling physical dynamics (Battaglia et al., 2016), and reasoning over graphs (Li et al., 2015; Vendrov et al., 2016; Kipf et al., 2018; Kool and Welling, 2018). Graph neural networks (Li et al., 2015; Battaglia et al., 2016) incorporate a message passing as part of the architecture in order to capture the intrinsic relations between entities. Graph convolution networks (Bruna et al., 2013; Kipf and Welling, 2016; Defferrard et al., 2016) use convolutions to efficiently learn a continuous-space representation for a graph of interest. ",
+        "bbox": [
+            173,
+            702,
+            826,
+            801
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Many of these relational reasoning models can be expressed in terms of an attentive read operation. In the following subsection, we give a general description of the attentive read, and then discuss its specific instantiations in two relational reasoning models from the literature. ",
+        "bbox": [
+            176,
+            808,
+            825,
+            851
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 ATTENTIVE READ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            868,
+            338,
+            882
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "First introduced for translation in Bahdanau et al. (2014), attention has seen widespread use in deep learning, not only for applications in NLP but also for image processing (Wang et al., 2017) imitation ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "learning (Duan et al., 2017) and memory (Graves et al., 2016). The core computation is the attentive read operation, which has the following form: ",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4eb6bad891ea0bd373a60684229a133f95ebc39dcd1ad0acbd277fbc32c29ce7.jpg",
+        "text": "$$\n\\mathbf { r } ( \\mathbf { q } _ { i } , \\{ \\mathbf { k } _ { j } \\} _ { j } ) = \\sum _ { j } \\frac { f ( \\mathbf { q } _ { i } , \\mathbf { k } _ { j } ) } { Z } \\mathbf { v } _ { i j } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            133,
+            607,
+            172
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here $\\mathbf { q } _ { i }$ is a vector called the query and the $\\mathbf { k } _ { j }$ ’s are the keys for the memory locations being read from. The pairwise function $f ( \\cdot , \\cdot )$ computes a scalar matching score between a query and a key, and the vector $\\mathbf { v } _ { i j }$ is a value to be read from location $j$ by query $i$ . $Z > 0$ is a normalization factor for the full sum. Both $\\mathbf { v } _ { i j }$ and $Z$ are free to depend on arbitrary information, but we leave any dependencies here implicit. ",
+        "bbox": [
+            173,
+            172,
+            826,
+            231
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It will be useful in the discussion to break this operation down into two parts. The first is the matching, which computes attention weights $\\alpha _ { i j } = f ( \\mathbf { q } _ { i } , \\mathbf { k } _ { j } )$ and the second is the aggregation, which takes a weighted average of the values using these weights, ",
+        "bbox": [
+            173,
+            237,
+            826,
+            279
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b9acc352566ba2226356ce2f4d03f6fa5fd21276a7c1014cf9f32ead7e0180d.jpg",
+        "text": "$$\nm _ { i } ( \\{ \\alpha _ { i j } \\} _ { j } , \\{ \\mathbf { v } _ { i j } \\} _ { j } ) = \\sum _ { j } \\frac { \\alpha _ { i j } } { Z } \\mathbf { v } _ { i j } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            280,
+            611,
+            315
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Instantiating a particular attentive read operation involves specifying both $f ( \\cdot , \\cdot )$ and $\\mathbf { v } _ { i j }$ along with the normalization constant $Z$ . ",
+        "bbox": [
+            174,
+            316,
+            825,
+            345
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "If one performs an attentive read for each element of the set $j$ then the resulting operation corresponds in a natural way to message passing on a graph, where each node $i$ aggregates messages $\\{ \\mathbf { v } _ { i j } \\} _ { j }$ from its neighbours along edges of weight $f ( \\mathbf { q } _ { i } , \\mathbf { k } _ { j } ) / Z$ . ",
+        "bbox": [
+            174,
+            352,
+            823,
+            396
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can express many (although not all) message passing neural network architectures (Gilmer et al., 2017) using the attentive read operation of Equation 1 as a primitive. In the following sections we do this for two architectures and then discuss how we can replace both the matching and aggregation steps with versions that leverage hyperbolic geometry. ",
+        "bbox": [
+            173,
+            401,
+            825,
+            458
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 RELATION NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            369,
+            488
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Relation Networks (RNs) (Santoro et al., 2017) are a neural network architecture designed for reasoning about the relationships between objects. An RN operates on a set of objects $O$ by applying a shared operator to each pair of objects $( \\mathbf { o } _ { i } , \\mathbf { o } _ { j } ) { \\in } O \\times O$ . The pairs can be augmented by a global information, and the result of each relational operation is passed through a further global transformation. ",
+        "bbox": [
+            173,
+            498,
+            826,
+            556
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using the notation of the previous section, we can write the RN as ",
+        "bbox": [
+            174,
+            563,
+            599,
+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a5e92b5af8db86b5926ebb69a21d30a83957cd8eaacfe690d03af7f7d1e54e63.jpg",
+        "text": "$$\nR N ( O , \\mathbf { c } ) = h \\left( \\sum _ { i } \\mathbf { r } ( \\mathbf { o } _ { i } , \\{ \\mathbf { o } _ { j } \\} _ { j } ) ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            578,
+            619,
+            621
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\mathbf { o } _ { i } , \\mathbf { o } _ { j } ) = 1$ , $\\mathbf { v } _ { i j } = g ( \\mathbf { o } _ { i } , \\mathbf { o } _ { j } , \\mathbf { c } )$ , $Z = 1$ . $h$ is the global transformation, $g$ is the local transformation and $\\mathbf { c }$ is the global context, as described in Santoro et al. (2017). We augment the basic RN to allow $f ( \\mathbf { o } _ { i } , \\mathbf { o } _ { j } ) \\in [ 0 , \\bar { 1 } ]$ to be a general learnable function. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            665
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Interpreting the RN as learned message passing on a graph over objects, the attention weights take on the semantics of edge weights, where $\\alpha _ { i j }$ can be thought of as the probability of the (directed) edge $\\mathbf { o } _ { j }  \\mathbf { o } _ { i }$ appearing in the underlying reasoning graph. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            714
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 SCALED DOT-PRODUCT ATTENTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            728,
+            459,
+            743
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the Transformer model of Vaswani et al. (2017) the authors define an all-to-all message passing operation on a set of vectors which they call scaled dot-product attention. In the language of Section 3.1 the scaled dot-product attention operation performs several attentive reads in parallel, one for each element of the input set. ",
+        "bbox": [
+            173,
+            753,
+            825,
+            811
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Vaswani et al. (2017) write scaled dot-product attention as $\\scriptstyle \\mathbf { R } = \\operatorname { s o f t m a x } \\left( { \\frac { \\mathbf { Q } \\mathbf { K } ^ { T } } { \\sqrt { d } } } \\right) \\mathbf { V }$ , where ${ \\bf Q } , { \\bf K }$ and $\\mathbf { V }$ are referred to as the queries, keys, and values respectively, and $d$ is the shared dimensionality of the queries and keys. Using lowercase letters to denote rows of the corresponding matrices, we can write each row of $\\mathbf { R }$ as the result of an attentive read with ",
+        "bbox": [
+            173,
+            818,
+            825,
+            882
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b3be467b161f001b580f42b050a83a8fb17d0b3ddd756fd128754c565836fc42.jpg",
+        "text": "$$\nf ( \\mathbf { q } _ { i } , \\mathbf { k } _ { j } ) = \\exp \\left( \\frac { 1 } { \\sqrt { d } } \\langle \\mathbf { q } _ { i } , \\mathbf { k } _ { j } \\rangle \\right) , \\qquad \\mathbf { v } _ { i j } = \\mathbf { v } _ { j } , \\qquad Z = \\sum _ { j } f ( \\mathbf { q } _ { i } , \\mathbf { k } _ { j } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            882,
+            722,
+            921
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We experiment with both softmax and sigmoid operations for computing the attention weights in our hyperbolic models. The motivation for considering sigmoid attention weights is that in some applications (e.g. visual question answering), it makes sense for the attention weights over different entities to not compete with each other. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 HYPERBOLIC ATTENTION NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            180,
+            514,
+            195
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we show how to redefine the attentive read operation of Section 3.1 as an operation on points in hyperbolic space. The key for doing this is to define new matching and aggregation functions that operate on hyperbolic points and take advantage of the metric structure of the manifold they live on. However, in order to apply these operations inside of a network we first we need a way to interpret network activations as points in hyperbolic space. ",
+        "bbox": [
+            174,
+            210,
+            825,
+            281
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We describe how to map an arbitrary point in $\\mathbb { R } ^ { n }$ onto the hyperboloid, where we can interpret the result as a point in hyperbolic space. The choice of mapping is important since we must ensure that the rapid scaling behavior of hyperbolic space is maintained. Armed with an appropriate mapping we proceed to describe the hyperbolic matching and aggregation operations that operate on these points. ",
+        "bbox": [
+            174,
+            287,
+            825,
+            344
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 HYPERBOLIC NETWORK ACTIVATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            361,
+            477,
+            375
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Mapping neural network activations into hyperbolic space requires care, since network activations might live anywhere in $\\mathbb { R } ^ { n }$ , but hyperbolic structure can only be imposed on special subsets of Euclidean space (Krioukov et al., 2010). This means we need a way to map activations into an appropriate manifold. We choose to map into the hyperboloid, which is convenient since it is the only unbounded model of hyperbolic space in common use. ",
+        "bbox": [
+            174,
+            386,
+            825,
+            457
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Pseudo-polar coordinates: In polar coordinates, we express an $n$ -dimensional point as a scalar radius, and $n { - } 1$ angles. Pseudo-polar coordinates consist of a radius $r$ , as in ordinary polar coordinates, and an $n$ -dimensional vector d representing the direction of the point from the origin. In the following discussion we assume that the coordinates are normalized, i.e. that $\\lVert \\mathbf { d } \\rVert = 1$ . ",
+        "bbox": [
+            174,
+            463,
+            825,
+            520
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If $( \\mathbf { d } , r ) \\in \\mathbb { R } ^ { n + 1 }$ are the activations of a layer in the network, we map them onto the hyperbolid in $\\mathbb { R } ^ { n + 1 }$ using $\\pi ( ( \\mathbf { d } , r ) ) = ( \\sinh ( r ) \\mathbf { d } , \\cosh ( r ) )$ , which increases the scale by an exponential factor. ",
+        "bbox": [
+            173,
+            526,
+            823,
+            555
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is easily verified that the resulting point lies in the hyperboloid, and to verify that we maintain the appropriate scaling properties we compute the distance between a point and the origin using this projection: ",
+        "bbox": [
+            174,
+            560,
+            823,
+            590
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21613ef5c9b784cab8a2118e55aec150ecdd51c2ae0d653767141eb0b8e3d3d3.jpg",
+        "text": "$$\nd _ { \\mathbb { H } } ( \\mathbf { 0 } , ( \\mathbf { d } , r ) ) { = } \\mathrm { a r c c o s h } ( - \\langle \\pi ( \\mathbf { 0 } ) , \\pi ( ( \\mathbf { d } , r ) ) \\rangle _ { M } ) { = } r ~ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            595,
+            671,
+            614
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which shows that this projection preserves exponential growth in volume for a linear increase in $r$ . Without the exponential scaling factor the effective distance of $\\pi ( ( \\mathbf { d } , r ) )$ from the origin grows logarithmically in hyperbolic space.1 ",
+        "bbox": [
+            173,
+            626,
+            826,
+            669
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 HYPERBOLIC ATTENTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            685,
+            390,
+            699
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show how to build an attentive read operation that operates on points in hyperbolic space. We consider how to exploit hyperbolic geometry in both the matching and the aggregation steps of the attentive read operation separately. ",
+        "bbox": [
+            174,
+            710,
+            825,
+            753
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Hyperbolic matching: The most natural way to exploit hyperbolic geometry for matching pairs of points is to use the hyperbolic distance between them. Given a query $\\mathbf { q } _ { i }$ and a key $\\mathbf { k } _ { j }$ both lying in hyperbolic space we take, ",
+        "bbox": [
+            174,
+            760,
+            825,
+            803
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e140aecf7aa9cfcf7dd32021dbcbf994c2c3d1f6f7552680e5aed57736c311e.jpg",
+        "text": "$$\n\\alpha ( { \\bf q } _ { i } , { \\bf k } _ { j } ) = f ( - \\beta d _ { \\mathbb { H } } ( { \\bf q } _ { i } , { \\bf k } _ { j } ) - c ) ~ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            808,
+            611,
+            825
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $d _ { \\mathbb { H } } ( \\cdot , \\cdot )$ is the hyperbolic distance, and $\\beta$ and $c$ are parameters that can be set manually or learned along with the rest of the network. Having the bias parameter $c$ is useful because distances are non-negative. We take the function $f ( \\cdot )$ to be either $\\exp ( \\cdot )$ , in which case we set the normalization appropriately to obtain a softmax, or sigmoid $( \\cdot )$ . ",
+        "bbox": [
+            174,
+            830,
+            825,
+            887
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/d84a53f0e9de207b75f14bab09afb67e7fbb20fbfc60af2b7bd7c6cda602effc.jpg",
+        "image_caption": [
+            "Figure 2: The computational graph for the self-attention mechanism of the hyperbolic Transformer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            218,
+            98,
+            779,
+            275
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Hyperbolic aggregation: The path to extend the weighted midpoint to hyperbolic space is much less obvious, but fortunately such a extension already exists as the Einstein midpoint. The Einstein midpoint is straightforward to compute by adjusting the aggregation weights appropriately (see Ungar (2005, Definition 4.21)) ",
+        "bbox": [
+            173,
+            325,
+            825,
+            382
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92e51016081d03788e10528a1af56f5395a69da044370f7f0c10cd50dd497401.jpg",
+        "text": "$$\nm _ { i } ( \\{ \\alpha _ { i j } \\} _ { j } , \\{ { \\bf v } _ { i j } \\} _ { j } ) = \\sum _ { j } \\left[ \\frac { \\alpha _ { i j } \\gamma ( { \\bf v } _ { i j } ) } { \\sum _ { \\ell } \\alpha _ { i \\ell } \\gamma ( { \\bf v } _ { i \\ell } ) } \\right] { \\bf v } _ { i j } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            386,
+            655,
+            426
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the $\\gamma ( \\mathbf { v } _ { i j } )$ are the Lorentz factors, that are given by $\\begin{array} { r } { \\gamma ( \\mathbf { v } _ { i j } ) = \\frac { 1 } { \\sqrt { 1 - \\| \\mathbf { v } _ { i j } \\| ^ { 2 } } } } \\end{array}$ The norm in the denominator of the Lorentz factor is the Euclidean norm of the Klein coordinates of the point $\\mathbf { v } _ { i j }$ , and the correctness of Equation 3 also relies on the points $\\mathbf { v } _ { i j }$ being represented by their Klein coordinates. Fortunately the various models of hyperbolic space in common use are all isomorphic, so we can work in an arbitrary hyperbolic model and simply project to and from the Klein model to execute midpoint computations, as discussed in Section 2. ",
+        "bbox": [
+            173,
+            431,
+            825,
+            523
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The reason for using the Einstein midpoint for hyperbolic aggregation is that it obeys many of the properties that we expect from a weighted average in Euclidean space. In particular, translating the $\\mathbf { v } _ { i j }$ ’s by a fixed distance in a common direction also translates the midpoint, and it is invariant to rotations of the constellation of points about the midpoint. The derivation of this operation is quite involved, and beyond the scope of this paper. We point the interested reader to Ungar (2005; 2008) for a full exposition. ",
+        "bbox": [
+            174,
+            530,
+            825,
+            601
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            619,
+            326,
+            636
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate our models on synthetic and real-world tasks. Experiments where the underlying graph structure is explicitly known clearly show the benefits of using hyperbolic geometry as an inductive bias. At the same time, we show that real-world tasks within implicit graph structure such as a diagnostic visual question answering task (Johnson et al., 2017), and neural machine translation, equally benefit from relying on hyperbolic geometry. We provide experiments with feed-forward networks, the Transformer (Vaswani et al., 2017) and Relation Networks (Santoro et al., 2017) endowed with hyperbolic attention. ",
+        "bbox": [
+            173,
+            650,
+            825,
+            748
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our results show the effectiveness of our approach on diverse tasks and architectures. The benefit of our approach is particularly prominent in relatively small models, which supports our hypothesis that hyperbolic geometry induces compact representations and is therefore better able to represent complex functions in limited space. ",
+        "bbox": [
+            174,
+            756,
+            825,
+            811
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 MODELING SCALE-FREE GRAPHS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            829,
+            442,
+            842
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use the algorithm of von Looz et al. (2015) to efficiently generate large scale-free graphs, and define two predictive tasks that test our model’s ability to represent different aspects of the structure of these networks. For both tasks in this section, we train Recursive Transformer (RT) models, using hyperbolic and Euclidean attention. A Recursive Transformer is identical to the original transformer, except that the weights of each self-attention layer are tied across depth. Simultaneously to our work, ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/d237011714960f4e38f8f8d7ea965840240a47c2ae93d9d1bbc33197f838a908.jpg",
+        "image_caption": [
+            "Figure 3: Left: Performance of the Recursive Transformer models on the Shortest Path Length Prediction task on graphs of various sizes. The black dashed line indicates chance performance. Center: Results on Link Prediction Tasks. Right: The histogram of the radiuses for a model trained on a graph with 100 and 400 nodes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            101,
+            813,
+            205
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Dehghani et al. (2018) have proposed the same model as a generalization of the Transformer model and they referred to it as \"Universal Transformers\". We use models with 3 recursive self-attention layers, each of which has 4 heads with 4 units each for each of q, k, and v. This model has similarities to Graph Attention Networks (Velickovi ˇ c et al., 2017; Kool and Welling, 2018). ´ ",
+        "bbox": [
+            174,
+            301,
+            825,
+            358
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Link prediction (LP): Link prediction is a classical graph problem, where the task is to predict if an edge exists between two nodes in the graph. We experimented with graphs of 1000 and 1200 nodes and observed that the hyperbolic RT performs better than the Euclidean RT on both tasks. We report the results in Figure 3 (middle). In general, we observed that for graphs of size 1000 and 1200 the hyperbolic transformer performs better than the Euclidean transformer given the same amount of capacity. ",
+        "bbox": [
+            174,
+            364,
+            825,
+            434
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Shortest path length prediction (SPLP): In this task, the goal is to predict the length of the shortest path between a pair of nodes in the graph. We treat this as a classification problem with a maximum pathlength of 25 which becomes naturally an unbalanced classification problem. We use rejection sampling during training to ensure the network is trained on an approximately uniform distribution of path lengths. At test time we sample paths uniformly at random, so the length distribution follows that of the underlying graphs. We report the results in Figure 3 (left). In Figure 3 (right), we visualize the distribution of the scale of the learned activations $\\dot { \\boldsymbol { r } }$ in the projection of Section 4.1) when training on graphs of size 100 and 400. We observe that our model tends to use larger scales for the larger graphs. As a baseline, we compare to the optimal constant predictor, which always predicts the most common expected path length. This baseline does quite well since the path length distribution on the test set is quite skewed. ",
+        "bbox": [
+            173,
+            441,
+            826,
+            580
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For both tasks, we generate training data online. Each example is a new graph in which we query the connectivity of a randomly chosen pair of nodes. To make training easier, we use a curriculum, whereby we start training on smaller graphs and gradually increase the number of vertices towards the final number. More details on the dataset generation procedure and the curriculum scheme are found in the supplementary material. ",
+        "bbox": [
+            174,
+            588,
+            825,
+            657
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 SORT-OF-CLEVR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            676,
+            339,
+            690
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Since we expect hyperbolic attention to be particularly well suited to relational modelling, we investigate our models on the relational variant of the Sort-of-CLEVR dataset (Santoro et al., 2017). This dataset consists of simple visual scenes allowing us to solely focus on the relational aspect of the problem. Our models extend Relation Nets (RNs) with the attention mechanism in hyperbolic space (with the Euclidean or Einstein midpoint aggregation), but otherwise we follow the standard setup-up (Santoro et al., 2017). Our best method yields accuracy of $9 9 . 2 \\%$ that significantly exceeds the accuracy of the original RN $( 9 6 \\% )$ . ",
+        "bbox": [
+            174,
+            703,
+            825,
+            801
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "However, we are more interested in evaluating models on the low-capacity regime. Indeed, as Figure 4 (left) shows, the attention mechanism computed in the hyperbolic space improves around 20 percent points over the standard RN, where all the models use only two units of the relational MLP. ",
+        "bbox": [
+            176,
+            808,
+            825,
+            849
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 EXPERIMENTS ON CITESEER AND CORA",
+        "text_level": 1,
+        "bbox": [
+            176,
+            868,
+            490,
+            882
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use two of the standard graph transduction benchmark datasets, Citeseer and Cora (Sen et al., 2008) and used the same experimental protocol defined in Velickovi ˇ c et al. (2017). We use ´ graph attention ",
+        "bbox": [
+            176,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/425a0b24b227f503fb044f953043d558bef80a15982e7465eda5cd1a38dc7d38.jpg",
+        "image_caption": [
+            "Figure 4: Left: Comparison of our models with low-capacity on the Sort-of-CLEVR dataset. The “EA” refers to the model that uses hyperbolic attention weights with Euclidean aggregation. Right: Performance of Relation Network extended by attention mechanism in either Euclidean or hyperbolic space on the CLEVR dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            102,
+            816,
+            256
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c309b4ee96f47ec461872e7284d394ee98c9d7bfb49ee69b48abd2cb67d8ccba.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Transductive</td></tr><tr><td>Method</td><td>Cora</td><td>Citeseer</td></tr><tr><td>GCN (Kipf and Welling, 2016)</td><td>81.5%</td><td>70.3%</td></tr><tr><td>GAT (Velickovic et al., 2017)</td><td>83.0%±0.14</td><td>72.5%± 0.14</td></tr><tr><td>H-GAT</td><td>83.5% ± 0.12</td><td>72.9% ± 0.078</td></tr></table>",
+        "bbox": [
+            279,
+            340,
+            717,
+            426
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Results on graph transduction tasks. We have used the same setup that is described in (Velickovi ˇ c et al., 2017). H-GAT refers to our graph attention network with hyperbolic attention ´ mechanism. Table shows the mean performance over 100 random seeds, along with $9 5 \\%$ confidence intervals for this estimate. ",
+        "bbox": [
+            176,
+            435,
+            821,
+            492
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "networks (GAT) as our baseline and developed a hyperbolic version of GAT (H-GAT) by replacing the original attention mechanism with the hyperbolic attention using softmax. ",
+        "bbox": [
+            174,
+            520,
+            821,
+            549
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We report our results in Table 1 and compare against the GAT with the Euclidean attention mechanism. We compute the standard deviations over 100 seeds and got improvements both on Citeseer and Cora datasets over the original GAT model. We show the visualizations of the learned hyperbolic embeddings of q and k in A.4. ",
+        "bbox": [
+            174,
+            555,
+            825,
+            611
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 CLEVR ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            628,
+            272,
+            643
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We train our Relation Network with various attention mechanisms on the CLEVR dataset (Johnson et al., 2017). CLEVR is a synthetic visual question answering datasets consisting of 3D rendered objects like spheres, cubes, or cylinders of various size, material, or color. In contrast to other visual question answering datasets (Antol et al., 2015; Malinowski and Fritz, 2014; Zhu et al., 2016), the focus of CLEVR is on relational reasoning. ",
+        "bbox": [
+            174,
+            655,
+            825,
+            726
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we closely follow the procedure established in (Santoro et al., 2017), both in terms of the model architecture, capacity, or the choice of the hyperparameters, and only differ by the attention mechanism (Euclidean or hyperbolic attention), or sigmoid activations. ",
+        "bbox": [
+            174,
+            732,
+            825,
+            775
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results are shown in Figure 4 (Right). For each model, we vary the capacity of the relational part of the network and report the resulting test accuracy. We find that hyperbolic attention with sigmoid consistently outperforms other models. Our RN with hyperbolic attention and sigmoid achieves ${ \\bar { 9 } } 5 . 7 \\%$ accuracy on the test set at the same capacity level as RN, whereas the latter reportedly achieves $9 5 . 5 \\%$ accuracy (Santoro et al., 2017). ",
+        "bbox": [
+            174,
+            781,
+            825,
+            852
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.5 NEURAL MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            869,
+            446,
+            883
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The Transformer (Vaswani et al., 2017) is a recently introduced state of the art model for neural machine translation that relies heavily on attention as its core operation. As described in Section 3.3, ",
+        "bbox": [
+            176,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/842e242e999ee730ea8059f36574ec928f3845fe887a305011e6f3758e8f1296.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">WMT 2014 En-De BLEU Scores</td></tr><tr><td></td><td>Tiny</td><td>Base</td><td>Big</td></tr><tr><td>Transformer (Vaswani et al. (2017))</td><td>1</td><td>27.3</td><td>28.4</td></tr><tr><td>Transformer (Latest)</td><td>17.3</td><td>27.1</td><td>-</td></tr><tr><td>Hyperbolic Transformer (+Sigmoid)</td><td>17.5</td><td>27.4</td><td>1</td></tr><tr><td>Hyperbolic Transformer(+Softmax,+Pseudo-Polar)</td><td>17.9</td><td>27.4</td><td>=</td></tr><tr><td>Hyperbolic Transformer (+Sigmoid, +Pseudo-Polar)</td><td>18.6</td><td>27.9</td><td>28.52</td></tr></table>",
+        "bbox": [
+            200,
+            101,
+            797,
+            220
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Results for the WMT14 English to German translation task. Results are computed following the procedure in Vaswani et al. (2017). Citations indicate results taken from the literature. Latest is the result of training a new model using an unmodified version of the same code where we added hyperbolic attention (we have observed that the exact performance of the transformer on this task varies as the Tensor2tensor codebase evolves). ",
+        "bbox": [
+            173,
+            229,
+            825,
+            300
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "we have extended the Transformer2 by replacing its scaled dot-product attention operation with its hyperbolic counterpart. We evaluate all the models on the WMT14 En-De dataset (Bojar et al., 2014). ",
+        "bbox": [
+            174,
+            325,
+            823,
+            353
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We train several versions of the Transformer model with hyperbolic attention. They use different coordinate systems (Cartesian or pseudo-polar), or different attention normalization functions (softmax or sigmoid). We consider three model sizes, referred to here as tiny, base and big. The tiny model has two layers of encoders and decoders, each with 128 units and 4 attention heads. The base model has 6 layers of encoders and decoders, each with 512 units and 8 attention heads. All hyperparameter configurations for the Euclidean versions of these models are available in the Tensor2tensor repository. ",
+        "bbox": [
+            174,
+            359,
+            825,
+            444
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results are shown in Table 2. We observe improvements over the Euclidean model by using hyperbolic attention, in particular when coupled with the sigmoid activation function for the attention weights. The improvements are more significant when the model capacity is restricted. ",
+        "bbox": [
+            176,
+            450,
+            821,
+            492
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In addition, our best model (with sigmoid activation function and without pseudo-polar coordinates) using the big architecture from Tensor2tensor, achieves 28.52 BLEU score, whereas Vaswani et al. (2017) report 28.4 BLEU score with the original version of this model.3. ",
+        "bbox": [
+            174,
+            500,
+            826,
+            541
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            568,
+            318,
+            583
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented a novel way to impose the inductive biases from hyperbolic geometry on the activations of deep neural networks. Our proposed hyperbolic attention operation makes use of hyperbolic geometry in both the computation of the attention weights, and in the aggregation operation over values. We implemented our proposed hyperbolic attention mechanism in both Relation Networks and the Transformer and showed that we achieve improved performance on a diverse set of tasks. We have shown improved performance on link prediction and shortest path length prediction in scale free graphs, on two visual question answering datasets, real-world graph transduction tasks and finally on English to German machine translation. The gains are particularly prominent in relatively small models, which confirms our hypothesis that hyperbolic geometry induces more compact representations. ",
+        "bbox": [
+            174,
+            602,
+            825,
+            728
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yang and Rush (2016) have proposed to imposed the activations of the neural network to lie on a Lie-group manifold in the memory. Similarly as a future work, an interesting potential future direction is to use hyperbolic geometry as an inductive bias for the activation of neural networks in the memory. ",
+        "bbox": [
+            176,
+            734,
+            825,
+            776
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            804,
+            367,
+            818
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Neil Rabinowitz, Chris Dyer for constructive comments on earlier versions of this draft. We thank Yannis Asseal for helping us with the styles of the plots in this draft. We would like to thank Thomas Paine for the constructive comments. ",
+        "bbox": [
+            176,
+            837,
+            823,
+            878
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nStanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C Lawrence Zitnick, and Devi Parikh. Vqa: Visual question answering. In Proceedings of the IEEE International Conference on Computer Vision, pages 2425–2433, 2015.   \nDzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations, 2014.   \nPeter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al. Interaction networks for learning about objects, relations and physics. In Advances in neural information processing systems, pages 4502–4510, 2016.   \nOndrej Bojar, Christian Buck, Christian Federmann, Barry Haddow, Philipp Koehn, Johannes Leveling, Christof Monz, Pavel Pecina, Matt Post, Herve Saint-Amand, Radu Soricut, Lucia Specia, and Aleš Tamchyna. Findings of the 2014 workshop on statistical machine translation. In Proceedings of the Ninth Workshop on Statistical Machine Translation, pages 12–58, Baltimore, Maryland, USA, June 2014. Association for Computational Linguistics. URL http://www.aclweb.org/anthology/W/W14/W14-3302.   \nGunnar A Borg. Psychophysical bases of perceived exertion. Med sci sports exerc, 14(5):377–381, 1982.   \nMichael M Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18–42, 2017.   \nJoan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013.   \nBenjamin Paul Chamberlain, James Clough, and Marc Peter Deisenroth. Neural embeddings of graphs in hyperbolic space. arXiv preprint arXiv:1705.10359, 2017.   \nHyunghoon Cho, Benjamin DeMeo, Jian Peng, and Bonnie Berger. Large-margin classification in hyperbolic space. arXiv preprint arXiv:1806.00437, 2018.   \nAaron Clauset, Cosma Rohilla Shalizi, and Mark EJ Newman. Power-law distributions in empirical data. SIAM review, 51(4):661–703, 2009.   \nMichaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, pages 3844–3852, 2016.   \nMostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Łukasz Kaiser. Universal transformers. arXiv preprint arXiv:1807.03819, 2018.   \nYan Duan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba. One-shot imitation learning. In Advances in neural information processing systems, pages 1087–1098, 2017.   \nJustin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. arXiv preprint arXiv:1704.01212, 2017.   \nAlex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka Grabska-Barwinska, ´ Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626):471, 2016.   \nBirger Iversen. Hyperbolic geometry, volume 25. Cambridge University Press, 1992.   \nJustin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C Lawrence Zitnick, and Ross Girshick. Clevr: A diagnostic dataset for compositional language and elementary visual reasoning. In Computer Vision and Pattern Recognition (CVPR), 2017 IEEE Conference on, pages 1988–1997. IEEE, 2017.   \nThomas Kipf, Ethan Fetaya, Kuan-Chieh Wang, Max Welling, and Richard Zemel. Neural relational inference for interacting systems. arXiv preprint arXiv:1802.04687, 2018.   \nThomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016.   \nWWM Kool and M Welling. Attention solves your tsp. arXiv preprint arXiv:1803.08475, 2018.   \nDmitri Krioukov, Fragkiskos Papadopoulos, Amin Vahdat, and Marián Boguná. Curvature and temperature of complex networks. Physical Review E, 80(3):035101, 2009.   \nDmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010.   \nYujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015.   \nHenry W Lin and Max Tegmark. Critical behavior in physics and probabilistic formal languages. Entropy, 19 (7):299, 2017.   \nNathan Linial, Avner Magen, and Michael E Saks. Low distortion euclidean embeddings of trees. Israel Journal of Mathematics, 106(1):339–348, 1998.   \nMateusz Malinowski and Mario Fritz. A multi-world approach to question answering about real-world scenes based on uncertain input. In Advances in neural information processing systems, pages 1682–1690, 2014.   \nBrian J McGill, Brian J Enquist, Evan Weiher, and Mark Westoby. Rebuilding community ecology from functional traits. Trends in ecology & evolution, 21(4):178–185, 2006.   \nVolodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015.   \nMark EJ Newman. Power laws, pareto distributions and zipf’s law. Contemporary physics, 46(5):323–351, 2005.   \nMaximilian Nickel and Douwe Kiela. Learning continuous hierarchies in the lorentz model of hyperbolic geometry. arXiv preprint arXiv:1806.03417, 2018.   \nMaximillian Nickel and Douwe Kiela. Poincaré embeddings for learning hierarchical representations. In Advances in Neural Information Processing Systems, pages 6341–6350, 2017.   \nJorg Ontrup and Helge Ritter. Hyperbolic self-organizing maps for semantic navigation. In Advances in neural information processing systems, pages 1417–1424, 2002.   \nFragkiskos Papadopoulos, Dmitri Krioukov, Marián Boguñá, and Amin Vahdat. Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In INFOCOM, 2010 Proceedings IEEE, pages 1–9. IEEE, 2010.   \nSteven T Piantadosi. Zipf’s word frequency law in natural language: A critical review and future directions. Psychonomic bulletin & review, 21(5):1112–1130, 2014.   \nDavid MW Powers. Applications and explanations of zipf’s law. In Proceedings of the joint conferences on new methods in language processing and computational natural language learning, pages 151–160. Association for Computational Linguistics, 1998.   \nHelge Ritter. Self-organizing maps on non-euclidean spaces. In Kohonen maps, pages 97–109. Elsevier, 1999.   \nAdam Santoro, David Raposo, David G Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Tim Lillicrap. A simple neural network module for relational reasoning. In Advances in neural information processing systems, pages 4974–4983, 2017.   \nRik Sarkar. Low distortion delaunay embedding of trees in hyperbolic plane. In International Symposium on Graph Drawing, pages 355–366. Springer, 2011.   \nPrithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI magazine, 29(3):93, 2008.   \nShuntaro Takahashi and Kumiko Tanaka-Ishii. Do neural nets learn statistical laws behind natural language? PloS one, 12(12):e0189326, 2017.   \nYi Tay, Luu Anh Tuan, and Siu Cheung Hui. Hyperbolic representation learning for fast and efficient neural question answering. In Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining, pages 583–591. ACM, 2018.   \nAbraham Albert Ungar. Analytic hyperbolic geometry: Mathematical foundations and applications. World Scientific, 2005.   \nAbraham Albert Ungar. A gyrovector space approach to hyperbolic geometry. Synthesis Lectures on Mathematics and Statistics, 1(1):1–194, 2008.   \nAshish 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, pages 6000–6010, 2017.   \nPetar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph ´ attention networks. arXiv preprint arXiv:1710.10903, 2017.   \nIvan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. In ICLR, 2016.   \nOriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, pages 2692–2700, 2015.   \nMoritz von Looz, Henning Meyerhenke, and Roman Prutkin. Generating random hyperbolic graphs in subquadratic time. In International Symposium on Algorithms and Computation, pages 467–478. Springer, 2015.   \nXiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. arXiv preprint arXiv:1711.07971, 2017.   \nGreg Yang and Alexander M. Rush. Lie access neural turing machine. arXiv preprint arXiv:1602.08671, 2016.   \nYuke Zhu, Oliver Groth, Michael Bernstein, and Li Fei-Fei. Visual7w: Grounded question answering in images. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4995–5004, 2016. ",
+        "bbox": [
+            171,
+            84,
+            828,
+            925
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            99,
+            828,
+            830
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 MORE ON MODELS OF HYPERBOLIC SPACE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            509,
+            147
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we illustrate the relationship between different models of hyperbolic space. There are one-to-one isometric transformations defined between each different models of the hyperbolic space. Hyperboloid model is unbounded, whereas Klein and Poincare models are bounded in a disk. ",
+        "bbox": [
+            174,
+            157,
+            823,
+            196
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/41f4a49d70b83c38929ce3d8718b494f6d1c81c34f1cdb2969748fad366f5fdc.jpg",
+        "image_caption": [
+            "Figure 5: Relationships between different representations of points used in the paper. Left: The relationship between pseudo-polar coordinates in $\\mathbb { R } ^ { n }$ and the hyperboloid in $\\mathbb { R } ^ { n + 1 }$ . Right: Projections relating the hyperboloid, Klein and Poincaré models of hyperbolic space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            208,
+            812,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 SCALE-FREE GRAPH GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            469,
+            452,
+            483
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use the algorithm described by von Looz et al. (2015). In our experiments, we set the $\\alpha$ to 0.95 and edge_radius_R_factor to 0.35. We will release our code both for generating and the operations in the hyperbolic space along with the camera-ready version of our paper. ",
+        "bbox": [
+            174,
+            494,
+            825,
+            532
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 SCALE-FREE GRAPH CURRICULUM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            549,
+            457,
+            563
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Curriculum was an essential part of our training on the scale-free graph tasks. On LP and SPLP tasks, we use a curriculum where we extract the connected components from the graph by cutting the disk that the graphs generated on into slices by starting from a 30 degree angle and gradually increasing the size of the slice from the disk by increasing the angle during the training according to the number of lessons that are involved in the curriculum. This process is also visualized in Figure 6. ",
+        "bbox": [
+            174,
+            574,
+            825,
+            637
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4 VISUALIZATION OF QUERY AND KEY EMBEDDINGS ON CORA ",
+        "text_level": 1,
+        "bbox": [
+            183,
+            654,
+            637,
+            667
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8 and 9, we visualize the embeddings of the query $( q )$ and the keys $( k )$ going into the hyperbolic matching function on the Poincare Ball model. In Figure 8, the embeddings of a model trained with dropout are bounded in a ball with smaller volume than the model trained without dropout. Also as clearly can be seen from the embedding visualizations k’s and q’s are clustered on different regions of the space. ",
+        "bbox": [
+            174,
+            679,
+            825,
+            729
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.5 TRAVELLING SALESMAN PROBLEM (TSP) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            746,
+            506,
+            760
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We train an off-policy DQN-like agent (Mnih et al., 2015) with the HRT. The graphs for the TSP is generated following the procedure introduced in (Vinyals et al., 2015). ",
+        "bbox": [
+            176,
+            771,
+            823,
+            797
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "On this task, as an ablation we just compared the hyperbolic networks with and The results are provided in Figure 4 (Right) with and without implicit coordinates. Overall, we found that the hyperbolic transformer networks performs better when using the implicit polar coordinates. ",
+        "bbox": [
+            174,
+            804,
+            823,
+            843
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.6 HYPERBOLIC RECURSIVE TRANSFORMER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            859,
+            506,
+            873
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure ??, the hyperbolic RT is an extension of transformer that ties the parameters of the self-attention layers. The self-attention layer gets the representations of the nodes of the graph coming from the encoder and the decoder decodes that representation from the recursive self-attention layers for the prediction. ",
+        "bbox": [
+            176,
+            883,
+            823,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/ffa7a6033a2daacb09745bba54dd38baace27eac981747e6caed41e2406d1ee5.jpg",
+        "image_caption": [
+            "Figure 6: We show an example of a curriculum on the hyperbolic disk. In the first lesson, we take slices from the graph only between angle 0 and $\\pi / 2$ . In the second lesson we will have to take the slice from 0 to $\\pi$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            117,
+            761,
+            588
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/2b77d45c72ea00f743e01860b6db1df360e2ea5da868bd7487ae61a8b9f8440c.jpg",
+        "image_caption": [
+            "Figure 7: An illustration of how trees can be represented in hyperbolic (left) and Euclidean geometry (right) in a cone. In hyperbolic space, as the tree grows the angles between the edges $\\mathbf { \\eta } ^ { ( \\theta ) }$ can be preserved from one level to the next. In Euclidean space, since the number of nodes in the tree grows faster than the rate that the volume grows, angles may not be preserved ( $\\boldsymbol { \\theta }$ to $\\alpha$ ). Lines in the left diagram are straight in hyperbolic space, but appear curved in this Euclidean diagram. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            724,
+            812,
+            804
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/50c796e2835fe35a0d7ba5a320b7318cd3265492d5aea900d09c92c80dc78571.jpg",
+        "image_caption": [
+            "Figure 8: Hyperbolic embedding of $q$ (red) and $k$ (blue) in a Poincare Ball on Cora dataset. Each point corresponds to a node in the graph. This visualization is obtained from a model trained with dropout. The graph on the left is the embeddings going into the attention obtained from the first layer. The Figure on the right is for the embedding of the second layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            253,
+            172,
+            748,
+            375
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/7cca480e2e9184bdad70ba87b2d79f8801c20858f96bcacc8f3a63211a2645a5.jpg",
+        "image_caption": [
+            "Figure 9: Hyperbolic embeddings of q (red) and k (blue) in a Poincare Ball on Cora dataset. Each point corresponds to a node in the graph. This visualization is obtained from a model trained without dropout. The figure on the left shows the embeddings going into the attention obtained from the first layer. The figure on the right shows the embedding of the second layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            253,
+            585,
+            746,
+            787
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/bf47e1dd9f14dc878172815ae0d1d82ca15f62ff5c6e5c5671e9d8ba8cdd5112.jpg",
+        "image_caption": [
+            "Figure 10: The comparisons between a hyperbolic recursive transformer with and without pseudo-polar (denoted as $^ +$ spherical in the legend) coordinates on the travelling salesman problem. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            367,
+            710,
+            631
+        ],
+        "page_idx": 14
+    }
+]

parse/train/xYGNO86OWDH/xYGNO86OWDH.md

@@ -0,0 +1,421 @@
+# ISOTROPY IN THE CONTEXTUAL EMBEDDING SPACE: CLUSTERS AND MANIFOLDS
+
+Xingyu Cai, Jiaji Huang, Yuchen Bian, Kenneth Church Baidu Research, 1195 Bordeaux Dr, Sunnyvale, CA 94089, USA {xingyucai,huangjiaji,yuchenbian,kennethchurch}@baidu.com
+
+# ABSTRACT
+
+The geometric properties of contextual embedding spaces for deep language models such as BERT and ERNIE, have attracted considerable attention in recent years. Investigations on the contextual embeddings demonstrate a strong anisotropic space such that most of the vectors fall within a narrow cone, leading to high cosine similarities. It is surprising that these LMs are as successful as they are, given that most of their embedding vectors are as similar to one another as they are. In this paper, we argue that the isotropy indeed exists in the space, from a different but more constructive perspective. We identify isolated clusters and low dimensional manifolds in the contextual embedding space, and introduce tools to both qualitatively and quantitatively analyze them. We hope the study in this paper could provide insights towards a better understanding of the deep language models.
+
+# 1 INTRODUCTION
+
+The polysemous English word “bank” has two common senses: 1. the money sense, a place that people save or borrow money; 2. the river sense, a slope of earth that prevents the flooding. In modern usage, the two senses are very different from one another, though interestingly, both senses share similar etymologies (and both can be traced back to the same word in Proto-Germanic). In the static embedding, multiple instances of the same word (e.g. “bank”) will be represented using the same vector. On the contrary, the contextual embedding assigns different vectors to different instances of the same word, depending on the context. Historically, static embedding models like Word2vec (Mikolov et al., 2013b) and GloVe (Pennington et al., 2014), predated contextual embedding models such as ELMo (Peters et al., 2018), GPT (Radford et al., 2018), BERT (Devlin et al., 2018) and ERNIE (Sun et al., 2019). Much of the literature on language modeling has moved to contextual embeddings recently, largely because of their superior performance on the downstreaming tasks.
+
+# 1.1 RELATED WORK
+
+The static embeddings are often found to be easier to interpret. For example, the Word2Vec and GloVe papers discuss adding and subtracting vectors, such as: $\operatorname { v e c } ( \operatorname { k i n g } ) - \operatorname { v e c } ( \operatorname* { m a n } ) + \operatorname { v e c } ( \operatorname { w o m e n } ) =$ vec(queen). Inspired by this relationship, researchers started to explore geometric properties of static embedding spaces. For example, Mu & Viswanath (2018) proposed a very counter-intuitive method that removes the top principle components (the dominating directions in the transformed embedding space), which surprisingly improved the word representations. Rather than completely discarding the principle components, Liu et al. (2019) proposed to use a technique called Conceptor Negation, to softly suppress transformed dimensions with larger variances. Both approaches, simply removing certain principle components as well as Conceptor Negation, produce significant improvements over vanilla embeddings obtained by static language models. In Huang et al. (2020), the authors studied how to effectively transform static word embeddings from one language to another.
+
+Unfortunately, the strong illustrative representation like the king-queen example above, is no longer obvious in a general contextual embedding space. Arguing that syntax structure indeed exists in the contextual embeddings, Hewitt & Manning (2019) proposed a structural probe to identify the syntax trees buried in the space, and found the evidence of implicit syntax tree in BERT and ELMo. The advantage of contextual embedding over the static counterpart, mainly come from its capability to assign different vectors to the same word, depending on the word sense in the context. Researchers in (Reif et al., 2019) found such a geometric representation of word senses in the BERT model. These papers reveal the existence of linguistic features embedded implicitly in the contextual vector spaces.
+
+The geometric properties of contextual embedding space are also investigated and compared with the static embedding space. Mimno & Thompson (2017) found anisotropy when negative sampling is used. In (Ethayarajh, 2019), the authors characterize how vectors are distributed in the contextual space. They found that most vectors occupy in a relatively narrow cone in the space. Pairs of vectors within this cone have large cosines. This phenomenon can be found in most state-of-the-art contextual embedding models. In (Gao et al., 2019), the authors named this phenomenon ”representation degeneration”, and attempted to mitigate the problem by introducing a regularization term that minimizes cosine similarities between vectors. In a very recent work, Demeter et al. (2020) suggest there is a structure weakness in the space that leads to bias when using soft-max, as is common with deep language models.
+
+# 1.2 MOTIVATION AND CONTRIBUTIONS
+
+Isotropy often makes the space more effectively utilized and more robust to perturbations (no extreme directions that lead to high condition number). It is counter-intuitive and not clear why those contextual embedding models perform remarkably well on many tasks given their anisotropic embeddings bring all the vectors close together, hard to distinguish one from another. On one hand, it is widely believed that contextual embeddings encode the relevant linguistic information (e.g. (Reif et al., 2019)), but on the other hand, it is also widely believed that the contextual space is anisotropic that representations become degenerated (e.g. (Mimno & Thompson, 2017), (Gao et al., 2019), (Ethayarajh, 2019)). These motivate us to find a reasonable understanding that bridges this gap.
+
+This paper is similar in spirit to (Mu & Viswanath, 2018), but different in three aspects. First, we generalize their work on traditional static embeddings to more modern contextual embeddings. Second, we introduce clustering methods to isolate the space, whereas they used PCA to remove dominant dimensions (that tend to dominate the variance). Finally, we identify low dimensional manifolds in the space, and introduce an alternative approach (LID) to characterize local subspaces.
+
+Key Contributions: This paper takes a deeper look into the contextual embedding spaces of popular pre-trained models. It identifies the following facts that were misunderstood or not known before: 1) We find isotropy within clusters in the contextual embedding space, in contrast to previous reports of anisotropy (caused by misleading isolated clusters). We introduce clustering and center shifting to reveal the isotropy, and show more consistent layer-wise behavior across models. 2) We find a Swiss-Roll manifold in GPT/GPT2 embeddings, but not in BERT/DistilBERT embeddings. The manifold is related to word frequency, suggesting a difference in how models evolve as they see more data. We use approximate Local Intrinsic Dimension (LID) to characterize the manifold, and find contextual embedding models, including all BERT, GPT families and ELMo, often have small LIDs. The small LIDs can be viewed as the local anisotropy of the space. The code for this paper could be found at https://github.com/TideDancer/IsotropyContxt.
+
+# 2 ANALYSIS SETTINGS
+
+# 2.1 MODELS AND DATASETS
+
+In this paper, we consider popular pre-trained contextual embedding models, including BERT, DistilBERT (Sanh et al., 2019) (or denoted as D-BERT in the rest of the paper), GPT, GPT2 (Radford et al., 2019) and ELMo. For the BERT and GPT families, we perform our evaluations on the pretrained uncased base models from Huggingface (https://huggingface.co/transformers/index.html#). The pre-trained ELMo model is from AllenNLP (https://docs.allennlp.org/v1.0.0/). BERT and DBERT are non-causal models because of their attention mechanism, where tokens can attend to any token in the input, regardless of their relative positions. In contrast, GPT and GPT2 are causal models because attention is limited to the tokens previously seen in the input.
+
+Different models achieve contextual embedding in different ways. For instance, BERT adds positional embeddings to the token embeddings, while ELMo performs vector concatenation. Most models start with an initial layer that maps token ids to vectors. This paper is not concerned with that lookup table layer, and only focuses on the layers after that. The base BERT, GPT and GPT2 models have 12 layers of interest, indexed from 0 to 11, while D-BERT has 6 layers and ELMo has two.
+
+We use Penn Tree Bank (PTB) (Marcus et al., 1993) and WikiText-2 (Merity et al., 2016) datasets. The PTB has 0.88 million words and WikiText-2 has 2 million. Both of them are the standard datasets for language models. In the rest of the paper, we report on PTB since we see similar results with both datasets. Details on WikiText-2 analysis could be found in Appendix.
+
+# 2.2 NOTATION
+
+For each position in a corpus, we have a word. Words are converted into tokens, using the appropriate tokenizer for the model. Tokenizers could split some words into subwords, therefore, the number of obtained tokens (denoted as $n$ ) could be more than number of words in the corpus. PTB, for example, contains 0.88 million words, but has $n = 1 . 2$ million tokens, when processed by BERT’s tokenizer. Let $V$ be the vocabulary, a set of distinct tokens. For any element in the vocabulary $V$ , we call it a type. For example, BERT has a vocabulary of roughly 30, 000 types. We may mix using “word” and “type” for ease of reading. We denote the $i$ -th type in $V$ as $t _ { i }$ . Let $\Phi ( t _ { i } ) = \{ \phi _ { 1 } ( t _ { i } ) , \phi _ { 2 } \bar { ( } t _ { i } ) , . . . \}$ be the set of all embedding instances of $t _ { i }$ (note that different contexts in the corpus yield different embeddings of $t _ { i }$ ). By construction, $\begin{array} { r } { \sum _ { t } | \Phi ( t ) | = n } \end{array}$ . We define the inter-type cosine similarity as
+
+$$
+S _ { \mathrm { i n t e r } } \triangleq \mathbb { E } _ { i \neq j } \left[ \cos { ( \phi ( t _ { i } ) , \phi ( t _ { j } ) ) } \right]
+$$
+
+where $\phi ( t _ { i } )$ is one random sample from $\Phi ( t _ { i } )$ , and the same for $\phi ( t _ { j } ) \in \Phi ( t _ { j } )$ . The expectation is taken over all pairs of different types. Similarly, we define the intra-type cosine similarity as
+
+$$
+S _ { \mathrm { i n t r a } } \triangleq \mathbb { E } _ { i } \left[ \mathbb { E } _ { k \neq l } \left[ \cos { ( \phi _ { k } ( t _ { i } ) , \phi _ { l } ( t _ { i } ) ) } \right] \right]
+$$
+
+where the inner expectation is over different embeddings $\phi ( t _ { i } )$ for the same type $t _ { i }$ , and the outer expectation is over all types. Both $S _ { \mathrm { i n t e r } }$ and $S _ { \mathrm { i n t r a } }$ take values between $- 1$ and 1. Note that for i.i.d. Gaussian random samples $x , y$ , the expected cosine similarity $\mathbb { E } [ \cos ( x , y ) ] = 0$ . A cosine value closer to 0 often indicates strong isotropy.
+
+Clearly, the inter-type metric describes the similarity between different types, where the intra-type one measures similarity between same type’s embedding instances. Our definitions of $S _ { \mathrm { i n t e r } }$ and $S _ { \mathrm { i n t r a } }$ are similar to the measures used in Ethayarajh (2019), but at the corpus level. Note that some types are more frequent than others, especially under a Zipfian distribution (Piantadosi, 2014), and therefore, the size of $\Phi ( t )$ varies dramatically with the frequency of type $t$ .
+
+# 2.3 AN INITIAL LOOK AT ANISOTROPY
+
+Inspired by Ethayarajh (2019), we follow their procedure and take a first look at the anisotropy identified by Mimno & Thompson (2017) and Ethayarajh (2019), in the contextual embedding space.
+
+![](images/450b687e2c311c12b27e0cf64c746ae2ed7551681cbf37cfbd957ea7f3951a75.jpg)  
+Figure 1: $S _ { \mathrm { i n t e r } }$ (left) and $S _ { \mathrm { i n t r a } }$ (right). The $S _ { \mathrm { i n t e r } }$ increases as layer goes deeper, especially for GPT2’s last layer. The $S _ { \mathrm { i n t r a } }$ are generally high. This means arbitrary vectors have high cosine similarities.
+
+Figure 1 shows strong anisotropy effects in a number of models. These findings are consistent with Ethayarajh (2019), though we use slightly different metrics. The plots show expected cosine $S _ { \mathrm { i n t e r } }$ and $S _ { \mathrm { { i n t r a . } } }$ ) as a function of layer. For efficiency, we approximate $S _ { \mathrm { i n t r a } }$ by imposing a limit of 1,000 samples for frequent types, $t$ , if $| \Phi ( t ) | > 1 0 0 0$ . From the figure we can see the following:
+
+• Both $S _ { \mathrm { i n t e r } }$ and $S _ { \mathrm { i n t r a } }$ are high $( \gg 0 )$ ) across almost all the layers and all the models. In particular, the same as reported in Ethayarajh (2019), GPT2 is relatively more anisotropic. $S _ { \mathrm { { i n t e r } } }$ tends to increase with layer, in contrast with $S _ { \mathrm { i n t r a } }$ which in general decreases but with fluctuations. This means that embeddings for different types are moving closer to one another at deeper layers, while embeddings for the same type’s instances are spreading away.
+
+• The last layer is often special. Note that the last layer has smaller cosines than the second last in most cases, with the notable exception of GPT2.
+
+In summary, we observe large cosines (across layers/models), especially for the GPT2 model. When cosines are close to 1, embeddings lie in a subspace defined by a very narrow cone (Ethayarajh, 2019). One might expect embeddings to be more effective if they took advantage of a larger subspace. Are these models missing an opportunity to have the benefits from isotropy (Mu & Viswanath, 2018)? We answer this question in the following sections.
+
+# 3 CLUSTERS IN THE EMBEDDING SPACE
+
+# 3.1 EFFECTIVE DIMENSIONS
+
+There are $m = 7 6 8$ embedding dimensions for BERT, D-BERT, GPT and GPT2, and $m = 1 0 2 4$ dimensions for ELMo. We perform PCA to reduce the number of dimensions from $m$ down to $k$ . For each layer of each model, we start with the data matrix, $M \in \mathcal { R } ^ { n \times m }$ , where $n$ is the number of input tokens $\ R = 1 . 2 M$ for PTB dataset), and $m$ is the original number of dimensions. After PCA, we end up with a smaller matrix, $\hat { M } \in \mathcal { R } ^ { n \times k }$ . Let the explained variance ratio be: $\begin{array} { r } { r _ { k } = \sum _ { i = 0 } ^ { k - 1 } \sigma _ { i } / \sum _ { i = 0 } ^ { m - 1 } \sigma _ { i } } \end{array}$ , where $\sigma _ { i }$ is the $i$ -th largest eigen value of $M$ ’s covariance matrix. In this way, we define the $\epsilon$ -effective-dimension to be: $d ( \epsilon ) \triangleq \arg \operatorname* { m i n } _ { k } r _ { k } \geq \epsilon .$ . For example, $d ( 0 . 8 ) = 2$ means that 2 dimensions capture $8 0 \%$ of the variance. There is a direct connection between $d$ and isotropy: the larger $d$ often implies more isotropy, as data spreads in multiple dimensions.
+
+Table 1: The effective dimension $d ( 0 . 8 )$   
+
+<table><tr><td>Layer</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>BERT</td><td>262</td><td>273</td><td>271</td><td>273</td><td>276</td><td>283</td><td>288</td><td>282</td><td>282</td><td>282</td><td>283</td><td>270</td></tr><tr><td>D-BERT</td><td>244</td><td>226</td><td>232</td><td>227</td><td>217</td><td>175</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GPT</td><td>265</td><td>141</td><td>65</td><td>76</td><td>173</td><td>210</td><td>205</td><td>217</td><td>221</td><td>253</td><td>269</td><td>307</td></tr><tr><td>GPT2</td><td>114</td><td>73</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>8</td><td>26</td><td>66</td><td>116</td><td>1</td></tr><tr><td>ELMo</td><td>455</td><td>367</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 1 reports $d ( 0 . 8 )$ for different layers and models. It is surprising that GPT2 has so few effective dimensions, especially, $d ( 0 . 8 ) = 1$ for layer 2 to 6. The surprisingly small effective dimensionality is another way of saying that GPT2 vectors fall in a narrow cone, and consequently, their pairwise cosines are large. If all the vectors lie on a 1-D line, all the cosines would be 1, and there would be hardly any model capacity. These observations motivate us to look deeper into the embedding space.
+
+# 3.2 ISOLATED CLUSTERS
+
+![](images/117dcd6ffe45834dcc6afef290d48b527046ad06bbf9802243931025414ad887.jpg)  
+Figure 2: Isolated clusters exist in the embedding spaces for all the models. Here we only show a few representative middle layers for each model. The full visualization can be found in supplementary.
+
+By performing PCA to project the original data into a 3-D view, we can visualize GPT2’s layer 6’s embedding space in Figure 2a. The three axes refer to the first three principle components, which account for $8 2 . 8 \%$ of the total variance. All the explained variance ratio will be reported throughout the rest of the paper. The axes values are raw coordinates after PCA. In Figure 2a, there are two disconnected islands that are far away from each other. Note that the first dimension coordinate values spans from 0 to 3000, significantly wider than the other 2 dimensions. In fact this first principle dimension dominates the total variance. The left island is bigger than the one on the right. The fact that the two islands are so well separated by the first principle component suggests that classifying points by island membership accounts for much of the variance. This two-island property is exhibited in layers 2 through 10 for GPT2. The two islands merge into a single large cluster in the last layer.
+
+We observe similar clustering behavior for all the models across all the layers, though the separations are less distinct, as illustrated in other panels of Figure 2. This is also consistent with Table 1, the less separation, the higher $d ( \epsilon )$ values. For GPT2, we had hoped to find that some types are associated with one cluster and other types are associated with the other cluster, but that is not verified in our experiments. Please refer to the supplementary for visualizations of all layers in all the models.
+
+# 3.3 CLUSTERING
+
+Previous literature estimated the space isotropy on pairs of arbitrary tokens, which could reside in two disconnected clusters. But given that the variance is dominated by distances between clusters, such estimation would be biased by the inter-cluster distances. It is more meaningful to consider a per-cluster investigation rather than a global estimate.
+
+We start by performing clustering on the embedding space. There are many methods for clustering. We chose K-Means (https://scikit-learn.org/stable/modules/classes.html#), because it is reasonably fast for large inputs $n = 1 . 2$ million vectors) in high $m \geq 7 6 8 )$ dimensions. DBSCAN algorithm (Ester et al., 1996) could be an alternative as it is density based, but only works on small dataset. We use the Silhouette method (Rousseeuw, 1987) to determine the number of clusters, $| C |$ . After running $\mathrm { K }$ -means, each point $p$ (one of the $n$ vectors in $M$ ) is assigned to one of $C$ clusters. For a data point $p$ assigned to the cluster $c \in C$ , calculate the following:
+
+$$
+a _ { p } = \frac { 1 } { | c | - 1 } \sum _ { q \in c , p \neq q } \operatorname { d i s t } ( p , q ) ; b _ { p } = \operatorname * { m i n } _ { \tilde { c } \neq c } \sum _ { q \in \tilde { c } } \operatorname { d i s t } ( p , q ) ; s _ { p } = \left\{ \frac { b _ { p } - a _ { p } } { \operatorname { m a x } ( a _ { p } , b _ { p } ) } , \mathrm { ~ i f ~ } | c | > 1 \right\} .
+$$
+
+where $a _ { p }$ is the mean distance between $p$ and other points in the same cluster; $b _ { p }$ is the minimum (min over $\tilde { c }$ ) mean distance between $p$ to points of another cluster $\tilde { c }$ ; and $s _ { p }$ is the Silhouette score for point $p \in c$ . The $s _ { p }$ takes value $\in [ - 1 , 1 ]$ . The higher $s _ { p }$ , the better assignment of $p$ to its cluster. Better choices of $| C |$ would lead to better values of $s _ { p }$ (and better clustering). We define the MaximumMean-Silhouette (MMS) score for the embedding space as: $\mathbf { M M S } { \stackrel { \Delta } { = } } \operatorname* { m a x } _ { \mathbf { d i f f e r e n t } } \left| C \right|  { \mathbb { E } } _ { p } \left[ s _ { p } \right]$ , where the maximum is over different $| C |$ values for K-Means. Since it is not feasible to evaluate all choices of $| C | \in [ 1 , n ]$ , we consider $| \dot { C } | \in [ 1 , 1 5 ]$ . The expectation $\mathbb { E } _ { p } [ s _ { p } ]$ (the mean Silhouette score), is estimated from 20, 000 sample vectors in $M$ . We select the best $| C |$ that yields MMS.
+
+The MMS values provide a systematic way to describe how the clusters are distributed in the space. If the clusters are very distinct and splitted, this yields a higher MMS. On the other hand, if clusters are overlapping, blurring together, the MMS score will be low. Note that if $\mathrm { M M S } < 0 . 1$ , we set $| C |$ to be 1, as the Silhouette score does not show significant evidence of more clusters.
+
+Table 2: Number of clusters $| C |$   
+
+<table><tr><td>Layer</td><td>BERT</td><td>D-BERT</td><td>GPT</td><td>GPT2</td><td>ELMo</td></tr><tr><td>0</td><td>6</td><td>7</td><td>1</td><td>2</td><td>2</td></tr><tr><td>１２３456７8</td><td>6</td><td>10</td><td></td><td>2</td><td>2</td></tr><tr><td></td><td>４４3</td><td>15</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>14</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>10</td><td></td><td></td><td></td></tr><tr><td></td><td>14</td><td>2</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>６２２П</td><td></td><td></td><td></td><td></td></tr><tr><td>9</td><td></td><td></td><td>２２２２22２２１</td><td></td><td></td></tr><tr><td>10</td><td>2</td><td></td><td>1</td><td>２２２２２２２２２</td><td></td></tr><tr><td>11</td><td>9</td><td></td><td>1</td><td>2</td><td></td></tr></table>
+
+![](images/af709fea2fbae351086b0ebcd74fb05e81a67ed08634b0943ae81b32cff89fb6.jpg)  
+Figure 3: The MMS for all the models. GPT2 has significantly higher MMS scores than other models from layer 1 to layer 11. This means the cluster effects are more severe in GPT2.
+
+Table 2 makes it clear that clustering plays an important role in most layers of most models. Some models (BERT and D-BERT) have more clusters, and some have fewer (GPT, GPT2, ELMo). This dichotomy of models is also reflected in Figure 2.
+
+Maximum-Mean-Silhouette scores are shown in Figure 3. There are significantly higher MMS values for GPT2, starting from the 2nd layer. Recall in Figure 2a, we showed that two far-away islands exist in the space and their distance dominates the variance. In Figure 3, the high MMS scores also verifies that. Another interesting observation is, for causal models GPT, GPT2 and ELMo, they all have higher MMS in their middle layers but lower MMS in the end. This means their initial layer and final layers’ embeddings tend to merge. On the contrary, the BERT and DistilBERT have increasing MMS in deeper layers, meaning that the clusters in their embeddings are becoming clearer in deeper layers.
+
+# 3.4 ISOTROPY IN CENTERED SPACE WITHIN CLUSTERS
+
+As suggest by Mu & Viswanath (2018), the embedding space should be measured after shifting the mean to the origin. We subtract the mean for each cluster, and calculate the adjusted $S _ { \mathrm { i n t e r } }$ . Assuming we have a total of $| C |$ clusters, let $\Phi ^ { c } ( t ) = \{ \phi _ { 1 } ^ { c } ( t ) , \phi _ { 2 } ^ { c } ( t ) , \ldots \}$ be the set of type $t$ ’s embeddings in cluster $c \in C$ , and $\phi ^ { c } ( t )$ be one random sample in $\Phi ^ { c } ( t )$ . Define the adjusted similarity:
+
+$$
+S _ { \mathrm { i n t e r } } ^ { \prime } \triangleq \mathbb { E } _ { c } \left[ \mathbb { E } _ { i \neq j } \left[ \cos \left( \bar { \phi } ^ { c } ( t _ { i } ) , \bar { \phi } ^ { c } ( t _ { j } ) \right) \right] \right] , \mathrm { w h e r e } \bar { \phi } ^ { c } ( t ) = \phi ^ { c } ( t ) - \mathbb { E } _ { \phi ^ { c } } \left[ \phi ^ { c } ( t ) \right]
+$$
+
+Here $\mathbb { E } _ { c }$ is the average over different clusters, and $\bar { \phi } ^ { c } ( t )$ is the original embedding shifted by mean (subtract the mean), where the mean is taken over the samples in cluster $c$ . Similarly we define
+
+$$
+S _ { \mathrm { i n t r a } } ^ { \prime } \triangleq \mathbb { E } _ { c } \left[ \mathbb { E } _ { i } \left[ \mathbb { E } _ { k \neq l } \left[ \cos \left( \bar { \phi } _ { k } ^ { c } ( t _ { i } ) , \bar { \phi } _ { l } ^ { c } ( t _ { i } ) \right) \right] \right] \right]
+$$
+
+![](images/7f09db5e9e8f988b0d46680cb0baf952d389782e0b41c59625673b275260dea1.jpg)  
+Figure 4: $S _ { \mathrm { i n t e r } } ^ { \prime }$ (left) and $S _ { \mathrm { i n t r a } } ^ { \prime }$ (right). The adjusted $S _ { \mathrm { i n t e r } } ^ { \prime }$ are close to zero, meaning that the space is isotropic under the adjusted measure.
+
+The Figure 4 illustrates the adjusted cosine similarities $S _ { \mathrm { i n t e r } } ^ { \prime }$ and $S _ { \mathrm { i n t r a } } ^ { \prime }$ . It reveals that:
+
+• For the adjusted inter-type cosine (the left plot), all models are having consistent near-zero $S _ { \mathrm { i n t e r } } ^ { \prime }$ last layer of GPT2 and BERT has slightly worse isotropic behavior, nevertheless, general inter-type isotropy stays across all layers. This reveals the distinguishable embedding vectors. The general decreasing trend of intra-type cosine (the right plot) shows that the multiple instances for the same type/word, is slowly spreading over the layers. This is consistent with the un-centered intra-type cosine shown in Figure 1.
+
+# 4 LOW-DIMENSIONAL MANIFOLDS
+
+# 4.1 SWISS ROLL MANIFOLD OF GPT/GPT2
+
+While BERT and D-BERT tend to distribute embeddings along more dimensions, GPT and GPT2 embed tokens in low-dimensional manifolds in their contextual embedding spaces. More specifically, we discover that most of the tokens are embedded on a spiral band, and that band gets thicker in the later layers thereafter form a Swiss Roll shaped surface.
+
+Figure 5a and 5b show the 2-D front view of the manifold in GPT and GPT2. Figure 5a zooms into the large cluster illustrated in Figure 2a (the left one), and discards the smaller one (the right one).
+
+![](images/0f374686d2f6401f149cbb706c4ff88bd2d571f02766a381e7f14d7577254e33.jpg)  
+Figure 5: The 2-D and 3-D view of low-dimensional manifold in GPT/GPT2’s embedding spaces
+
+3-D plots are shown in Figure 5c and 5d to demonstrate two manifolds, a band shaped manifold and a Swiss Roll shaped manifold. These plots were computed over PTB dataset. Similar results have been obtained from WikiText-2 in supplementary. Figure 6 tracks the progression of a narrow band into a Swiss Roll. The Swiss Roll becomes taller and taller with deeper and deeper layers.
+
+![](images/5ea1c3c6cc4901c9f8aac1cd94ebec8f2c4d49d68c0f35ede8ffcb49d15902d6.jpg)  
+Figure 6: The evolution from a narrow band into a taller and taller Swiss Roll with deeper layers.
+
+# 4.2 TOKENS IN THE SPACE
+
+To verify the manifold structure in GPT family, we study the token embeddings in the space. It is believed that similar embeddings (e.g. the embeddings for two instances of the same word) tend to stay close together in a Euclidean space, as they should have high cosine similarities. Figure 7 drills down into the embeddings for six frequent words: three punctuation symbols (“\”, “&”, “.”) and three common words (“the”, “first”, “man”). Each panel uses four colors: three colors (black, red, green) for three words of interest, plus gold color for all the other tokens.
+
+![](images/337622418101df1f4f4cd1c664f43628d5aaaa243e9bb6faffd11a34f69d72a5.jpg)  
+Figure 7: Embeddings for symbol tokens and word tokens, in layer 3 of BERT and GPT. This shows that GPT has manifold structure, such that vectors are along the spiral band. BERT’s space is closer to a Euclidean space as similar vectors are in concentrated clusters.
+
+As shown in Figure 7a 7b, the BERT model indeed group similar embeddings into small regions in the space (the red, black and green clusters). However, the GPT models are assigning similar embeddings along the manifold we observed before. In Figure 7c 7d, the embeddings for the tokens occupy a spiral band that almost cross the entire space. It does not comply with the Euclidean space geometry as points in such a spiral band would not have high cosine similarity. A Riemannian metric must exist, such that the manifold has larger distance between two spiral bands, but smaller distance on the band. Note that the 3-D plots are obtained using PCA, so there is no density-based nor non-linear reduction involved. Therefore, the manifold structures in GPT embedding spaces are verified.
+
+# 4.3 WORD FREQUENCY
+
+Another key finding is that all the models are trying to map the high frequent words/types to some specific region in the embedding spaces, rather than spreading them out to the entire space. In Figure 8, embeddings ( 8a 8c ) and corresponding word frequencies ( 8b 8d ) of GPT’s layer 8 and 9 are shown. The darker red denoted higher frequency and blue is lower frequency. The numbers at the colorbar show the number of occurrence (of a particular word / type).
+
+![](images/e89001d54c8ae5db3d86ed07710f60a19ab9b5d908681a00b0e1040a10fffb51.jpg)  
+Figure 8: Word frequency heatmap in GPT layer 8 and 9. Red is high frequency, blue is low. High frequency words are at the front end of the Swiss Roll, while low frequency words at the other end. ( 8b 8d are drawn using matplotlib.tricontourf, so the ring at 8b’s bottom should not be closed.)
+
+The Figure $\operatorname { 8 a 8 c }$ are after PCA, and selecting the two most significant dimensions. From GPT layer 8 to layer 9, as the Swiss Roll becomes taller, more variance is accounted for along the height of the Swiss Roll. Thus, the perspective switches from a front view to a side view when moving to layer 9.
+
+Figures 8b and 8d show that the most frequent words appear at the head of the Swiss Roll, followed by bands of less and less frequent words. The least frequent words appear at the far end of the Swiss Roll. This pattern suggests the model distinguishes more frequent from less frequent words. As the model finds more and more rare words, it appends them at the end of the Swiss Roll.
+
+# 4.4 MANIFOLD LOCAL INTRINSIC DIMENSION
+
+Although the original space dimension is 768 (1024 for ELMo), the manifold we observed has a lower intrinsic dimension. It means the data point on the manifold has fewer degrees of freedom to move around. For example, on a Swiss Roll in a 3-D space, any point can only have 2-D freedom thus the intrinsic dimension is only 2. A recent research on the intrinsic dimension for deep networks could be found at (Ansuini et al., 2019). In this section, we adopt the Local Intrinsic Dimension (LID) that estimates dimension locally with respect to a reference point. LID is introduced by Houle (2013), and being used in deep learning model characterization recently, e.g. (Ma et al., 2018). The LID is often derived using expansion models (Houle et al., 2012), which tries to obtain the local dimension in the vicinity of a reference point from the growth (expansion) characteristics. To illustrate this, we borrow an example from Ma et al. (2018). Let $\gamma$ be the radius of an $m$ -D ball in the Euclidean space, denote its volume as $\nu$ , then the volume’s growth rate is proportional to $\gamma ^ { m }$ , i.e. $\nu _ { 2 } / \nu _ { 1 } = ( \gamma _ { 2 } / \bar { \gamma _ { 1 } } ) ^ { m }$ , from which we can infer the local dimension $\tilde { m }$ by $\tilde { m } = \log ( \nu _ { 2 } / \nu _ { 1 } ) / \log ( \gamma _ { 2 } / \gamma _ { 1 } )$ .
+
+Accurately computing LID is a hard problem which requires a tremendous amount of data samples and enough density around the reference point. So fewer-sample estimate of LID is being studied in the past decade. One of the efficient estimation is proposed by Amsaleg et al. (2015). This technique relies on $K$ nearest neighbor search ( $K$ -NN). For a reference point $p$ , denote the set of its $K$ nearest neighbor points as $\Psi _ { p } = \{ q _ { 1 } , \dots , q _ { K } \}$ . Then the estimate of LID is computed as: $\begin{array} { r } { \mathbf { L } \mathbf { \tilde { I } } \mathbf { D } ( p ) = - \left( \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \log \frac { \mathrm { d i s t } ( p , q _ { i } ) } { \operatorname* { m a x } _ { i } ( \mathrm { d i s t } ( p , q _ { i } ) ) } \right) ^ { - 1 } } \end{array}$ , where the term inside log is the ratio of distance between $p$ to its neighbor, over the maximum distance among them. In our analysis, we use an efficient nearest neighbor computation package FAISS (Johnson et al., 2017) (https://github.com/facebookresearch/faiss) to perform the $K$ -NN. We set $K \ : = \ : 1 0 0$ , the same as in (Aumuller & Ceccarello, 2019). ¨ $\ell _ { 2 }$ distance is used, i.e. $\mathrm { d i s t } ( p , q ) = \| p - q \| _ { 2 }$ . We report the mean LID over all the samples $p$ , as $\mathbb { E } _ { p } [ \mathrm { L } \tilde { \mathrm { I D } } ( p ) ]$ , in Figure 9.
+
+![](images/05827c4a9a84e3e5c58f5e2e6d054af54762972c5d545b9fc40c214024bd9873.jpg)  
+Figure 9: The average LID using Euclidean distance. ELMo’s original embdding dimension is 1024, larger than other models’ 768.
+
+Table 3: A comparison of LIDs (using cosine similarity) among contextual and static embedding spaces.   
+
+<table><tr><td></td><td>Model</td><td>n</td><td>m</td><td>avg LID</td></tr><tr><td>Contxt Embeds</td><td>BERT D-BERT GPT GPT2</td><td>1.19 M 1.19 M 0.96 M 1.09 M</td><td>768 768 768 768</td><td>5.6 7.3 6.8 7.0</td></tr><tr><td>Static Embeds</td><td>ELMo GloVe GloVe-2M GNEWS</td><td>0.88 M 1.18 M 2.20M 3.00 M</td><td>1024 100 300 300</td><td>9.1 18.0 26.1 21.1</td></tr></table>
+
+As shown in Figure 9, the mean LIDs for all the models in all the layers are below 12. The small mean LID values reveals that the manifold’s intrinsic dimension is relatively low, especially considering that this is a 768-D (1024 for ELMo) embedding space. Since ELMo’s 1024-D is larger than other models 768-D dimension, its LID is also slightly higher than other models as shown in the figure. The existence of a low-dimensional embedding is also suggested in (Reif et al., 2019) when they study the BERT embedding geometry.
+
+In all the contextual embedding layers, there is a clear trend of increasing LID values. In Figure 9, we can also see a nearly-linear relationship between layer id and LID. With deeper and deeper layers in the net, the manifold is diffusing and slowly loses concentration. This would lead to data samples spreading, consistent with Figure 4 (recall that intra-type cosines decrease with depth). Note that as layer goes deeper, each token embedding is collecting information from context by adding their embeddings (and non-linear transforms concatenated). This could explain the spreading $/$ expanding of the local subspace, and therefore the LID increases in deeper layers.
+
+Table 3 compares LIDs for static and contextual embeddings. The table reports results for three static embeddings: GloVe / GloVe-2M (Pennington et al., 2014), and GNEWS (Mikolov et al., 2013a). Results for static embedding LIDs are based on Aumuller & Ceccarello (2019). Following Aum ¨ uller ¨ & Ceccarello (2019), we use cosine distance here: $\begin{array} { r } { \mathbf { d i s t } ^ { \prime } ( p , q ) = 1 - \cos ( p , q ) = 1 - \frac { \langle p , q \rangle } { \| p \| _ { 2 } \| q \| _ { 2 } } } \end{array}$ Note that estimates for LID using cosines are very close to the estimates using $\ell _ { 2 }$ distances. Table 3 reports averages of LIDs over each model’s layers. Even though GloVe (Pennington et al., 2014) in Table 3 has much fewer embedding dimensions (100-D compared with BERT’s 768-D), the LID is still higher than all of the contextual embedding models. From the table we can find that static embedding spaces generally have higher LID than the contextual ones. This means that the data points are more isotropic in the static embeddings, possibly due to their large vocabularies.
+
+# 5 CONCLUSIONS AND FUTURE WORK
+
+Previous works have reported the strong anisotropy in deep LMs, which is hard to explain the superior performance achieved by these models. We suggest that the anisotropy is a global view, being largely misled by distinct clusters resided in the space. Our analysis show that it is more constructive to isolate and transform the space to measure the isotropy. From this view, within the clusters, the spaces of different models all have nearly perfect isotropy that could explain the large model capacity. In addition, we investigate the space geometry for different models. Our visualization demonstrates a low-dimensional Swiss Roll manifold for GPT and GPT2 embeddings, that has not been reported before. The tokens and word frequencies are presented to qualitatively show the manifold structure. We propose to use the approximate LID to quantitatively measure the local subspace, and compared with static embedding spaces. The results show smaller LID values for the contextual embedding models, which can be seen as a local anisotropy in the space. We hope this line of research could bring a comprehensive geometric view of contextual embedding space, and gain insights on how the embeddings are affected by attention, compression, multilingualism, etc. Therefore the model performance could be further improved based on the findings.
+
+# REFERENCES
+
+Laurent Amsaleg, Oussama Chelly, Teddy Furon, Stephane Girard, Michael E Houle, Ken-ichi ´ Kawarabayashi, and Michael Nett. Estimating local intrinsic dimensionality. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 29–38, 2015.
+
+Alessio Ansuini, Alessandro Laio, Jakob H Macke, and Davide Zoccolan. Intrinsic dimension of data representations in deep neural networks. In Advances in Neural Information Processing Systems, pp. 6111–6122, 2019.
+
+Martin Aumuller and Matteo Ceccarello. The role of local intrinsic dimensionality in benchmarking ¨ nearest neighbor search. In International Conference on Similarity Search and Applications, pp. 113–127. Springer, 2019.
+
+Alexis Conneau and Guillaume Lample. Cross-lingual language model pretraining. In Advances in Neural Information Processing Systems, pp. 7059–7069, 2019.
+
+David L Davies and Donald W Bouldin. A cluster separation measure. IEEE transactions on pattern analysis and machine intelligence, (2):224–227, 1979.
+
+David Demeter, Gregory Kimmel, and Doug Downey. Stolen probability: A structural weakness of neural language models. Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, 2020.
+
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.
+
+Martin Ester, Hans-Peter Kriegel, Jorg Sander, Xiaowei Xu, et al. A density-based algorithm for ¨ discovering clusters in large spatial databases with noise. In Kdd, volume 96, pp. 226–231, 1996.
+
+Kawin Ethayarajh. How contextual are contextualized word representations? comparing the geometry of bert, elmo, and gpt-2 embeddings. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 55–65, 2019.
+
+Jun Gao, Di He, Xu Tan, Tao Qin, Liwei Wang, and Tieyan Liu. Representation degeneration problem in training natural language generation models. In International Conference on Learning Representations, 2019.
+
+John Hewitt and Christopher D Manning. A structural probe for finding syntax in word representations. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 4129–4138, 2019.
+
+Michael E Houle. Dimensionality, discriminability, density and distance distributions. In 2013 IEEE 13th International Conference on Data Mining Workshops, pp. 468–473. IEEE, 2013.
+
+Michael E Houle, Hisashi Kashima, and Michael Nett. Generalized expansion dimension. In 2012 IEEE 12th International Conference on Data Mining Workshops, pp. 587–594. IEEE, 2012.
+
+Jiaji Huang, Xingyu Cai, and Kenneth Church. Improving bilingual lexicon induction for low frequency words. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 1310–1314, 2020.
+
+Jeff Johnson, Matthijs Douze, and Herve J ´ egou. Billion-scale similarity search with gpus. ´ arXiv preprint arXiv:1702.08734, 2017.
+
+Tianlin Liu, Lyle Ungar, and Joao Sedoc. Unsupervised post-processing of word vectors via conceptor negation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 6778– 6785, 2019.
+
+Xingjun Ma, Bo Li, Yisen Wang, Sarah M Erfani, Sudanthi Wijewickrema, Grant Schoenebeck, Dawn Song, Michael E Houle, and James Bailey. Characterizing adversarial subspaces using local intrinsic dimensionality. In International Conference on Learning Representations, 2018.
+
+Mitchell Marcus, Beatrice Santorini, and Mary Ann Marcinkiewicz. Building a large annotated corpus of english: The penn treebank. 1993.
+
+Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. Pointer sentinel mixture models. arXiv preprint arXiv:1609.07843, 2016.
+
+Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013a.
+
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013b.
+
+David Mimno and Laure Thompson. The strange geometry of skip-gram with negative sampling. In Empirical Methods in Natural Language Processing, 2017.
+
+Jiaqi Mu and Pramod Viswanath. All-but-the-top: Simple and effective post-processing for word representations. In 6th International Conference on Learning Representations, ICLR 2018, 2018.
+
+Jeffrey Pennington, Richard Socher, and Christopher D Manning. Glove: Global vectors for word representation. In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), pp. 1532–1543, 2014.
+
+Matthew E Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. Deep contextualized word representations. arXiv preprint arXiv:1802.05365, 2018.
+
+Steven T. Piantadosi. Zipf’s word frequency law in natural language: A critical review and future directions. Psychonomic bulletin & review, 21(5):1112–1130, 2014.
+
+Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training, 2018.
+
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. OpenAI Blog, 1(8):9, 2019.
+
+Emily Reif, Ann Yuan, Martin Wattenberg, Fernanda B Viegas, Andy Coenen, Adam Pearce, and Been Kim. Visualizing and measuring the geometry of bert. In Advances in Neural Information Processing Systems, pp. 8594–8603, 2019.
+
+Peter J Rousseeuw. Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of computational and applied mathematics, 20:53–65, 1987.
+
+Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter. arXiv preprint arXiv:1910.01108, 2019.
+
+Yu Sun, Shuohuan Wang, Yukun Li, Shikun Feng, Xuyi Chen, Han Zhang, Xin Tian, Danxiang Zhu, Hao Tian, and Hua Wu. Ernie: Enhanced representation through knowledge integration. arXiv preprint arXiv:1904.09223, 2019.
+
+# SUPPLEMENTARY: FULL RESULTS ON PTB AND WIKITEXT-2 DATASETS
+
+A RESULTS ON WIKITEXT-2 DATASET
+
+A.1 THE UNADJUSTED INTER AND INTRA COSINE SIMILARITY
+
+Note that ”dist” in the following legends represents DistilBERT model.
+
+![](images/d3b31b73869877931ca23d5335d8e559b86505992ae72bab6d01eea5408d73db.jpg)  
+(a) Inter-type cosine similarity. As layers goes deeper, inter-type cosine goes higher. All models’ last layer behaves slightly differently.
+
+![](images/fd112fcbdd7851f1e89a735db24e549230ca05a99c4286c76d3969c383abcae9.jpg)  
+(b) Intra-type cosine similarity. The intratype cosine decreases showing the same type’s embedding instances are spreading in deeper layers.
+
+# A.2 THE CENTER-SHIFTED AND CLUSTERED COSINE SIMILARITY
+
+The inter-type and intra-type cosines are adjusted using the proposed center-shifting and clustering methods. Now it reflects the isotropy in almost all layers in all models.
+
+![](images/27d062567129ed55049d928946d8298204602844f53ff6425e780483f68710d6.jpg)  
+(a) The clustered inter-type cosine (center shifted). It shows strong isotropy as the average cosine between different types is close to 0 across all layers in all models. The GPT2’s last layer still has slighly higher cosine compared with others.   
+(b) The clustered intra-type cosine (center shifted). The intra-type cosines are much more consistent than the unadjusted counterpart (the cosine decreases nearly monotonically as layers goes deeper).
+
+# A.3 THE APPROXIMATE LOCAL INTRINSIC DIMENSIONS
+
+![](images/9d2d424665076507dee26cec957ddc3a85a28d6236f015a5454e21a850ca6376.jpg)  
+Figure 12: Local Intrinsic Dimensions. The LID increases as layer goes deeper, reflecting embeddings spreading out in all models’ deeper layers (becoming more locally isotropic).
+
+# B FULL VISUALIZATION - PTB DATASET
+
+# B.1 BERT
+
+![](images/2e4eaef7513b51bb72fdd99fac9c371d18f71af1418a19c10e5c3bdd8fe2a32c.jpg)
+
+# B.2 DISTILBERT AND ELMO
+
+![](images/f5ab49376c5293ae84918964f8ec0d3832317d6dda68fbb0fb601df09cae06a8.jpg)
+
+# B.3 GPT
+
+![](images/658c078f041c7f147c5d6a9c00e6fc8005b4f72b6aa37c5876f641601e8ed051.jpg)
+
+# B.4 GPT2
+
+![](images/10dc438ddf4b4974ea74c1c03f1afb6172c2fa1415440f1b0694e3657180c496.jpg)
+
+# C FULL VISUALIZATION - WIKITEXT-2 DATASET
+
+# C.1 BERT
+
+![](images/494ce7bd1b5153c302f3e3bf4176a80030af640fe34fcb735798932f23eb4ef4.jpg)
+
+# C.3 GPT
+
+![](images/d3536c57f6da080aaec22a26a2c68bcd539e11588c0e371e08dd478c37f7686c.jpg)
+
+# C.4 GPT2
+
+![](images/71c302876402c4800c25f59fac14ac2d2292172d999f401b2fdcf9d5590b1d87.jpg)
+
+# D ADDITIONAL STUDIES
+
+# D.1 K-MEANS CLUSTERING ACCURACY
+
+We use K-Means to perform clustering, which raises two issues here. First, K-Means is very sensitive to initialization, different initialization could leads to different clustering results. However, note that in our task, we are not seeking for optimal clustering. Sub-optimal, e.g. treating two overlapping clusters as a big one, is totally fine.
+
+To illustrate this, we add another metric, Davies-Boulding (DB) index (Davies & Bouldin, 1979), to show that slightly different $K$ is fine. This DB index is the average similarity between each cluster and its closest cluster. The value closer to 0, the better clustering is done. We still search in [2, 15], and choose $K$ with the minimum DB index (MDB). MDB sometimes gives different $K$ than that by MMS metric. If MDB is $> 4$ , we discard MDB and treat all data as one single cluster. We provide the comparison of selecting $K$ using MMS (left) and MDB (right) here in Table 4. We can see that for less-distinct clusters, e.g. in BERT, two metric could yield different $K$ values, due to merging or splitting. For very separated clusters, e.g. in GPT2, the two metric agrees. We plot the cosines using MDB’s $K$ values, in Figure 21. It is similar to Figure 4, which uses slightly different $K$ from MMS. The values are close to 0 indicating isotropy in the center-shifted clusters. This means that the procedure to reveal isotropy, is not sensitive to $K$ in K-Means.
+
+Table 4: $K$ by MMS(left) vs MDB(right)   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=2>BERT D-BERT</td><td rowspan=1 colspan=3>GPT GPT2ELMo</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7   9</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>1015</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>412</td><td rowspan=1 colspan=1>155</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td><td rowspan=10 colspan=1></td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>415</td><td rowspan=1 colspan=1>1411</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>314</td><td rowspan=1 colspan=1>102</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>1413</td><td rowspan=1 colspan=1>24</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>614</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>215</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>27</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>116</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>24</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>110</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>93</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>22</td></tr></table>
+
+![](images/c74b9f11688be8777c2d718b7ae66e0014c01a49d8b1ef2041c8c50ca7353255.jpg)  
+Figure 21: The adjusted inter-type cosines, computed using $K$ from the criteria of minimizing DB index. The values are still close to 0.
+
+Another issue is that K-Means implicitly assumes convex clusters, which often does not hold. In fact, it assumes isotropic convex clusters because we simply use $\ell _ { 2 }$ distance. However, density-based clustering such as DBSCAN, is too slow thus cannot handle these datasets (million level). This is a trade-off to use K-Means, and empirical results above show that it is efficient and very useful to distinguish separated clusters.
+
+# D.2 CLUSTERS AND WORDS
+
+We study the tokens and their relationship to the clusters existed in the contextual embedding spaces. We picked some representative tokens to see how they are distributed. We also study the very unique small cluster in GPT2, and how it connects to the main cluster that is far away. We obtain the following observations:
+
+• For BERT, high frequent words (e.g. ’the’) stays at one side of the main big cluster, while low-frequent words are at the other side.   
+• For BERT, punctuation are random but occupy distinct islands: ’!’ is a small cluster close to the main island; ’‘’ and ’are distinct islands far away; ’?’ and some others are on the main ´ cluster.   
+• For GPT2, almost all single letters (a to z) and mid-to-high frequent would occupy both the left (big) and right (small) islands.   
+• For GPT2, we didn’t find any token that only appears in the right small island. It seems the token in the small island always has mirrors in the left big cluster.
+
+• For word types, e.g. noun, verb, etc, we didn’t find a clear pattern. We suspect word frequency affects more than categories.
+
+We provide a few examples. Figure 22, 23 show BERT layer 3, Figure 24, 25 show GPT2 layer 3.
+
+![](images/aa6919e04668010698db22808030aee83973b3ffbe8e31660f5f094e7c307d5b.jpg)  
+Figure 22: BERT Layer 3 Punctuation
+
+![](images/e9503c3fe8b70d306c1306c53809c6e81e2a7c7e621f9de2a1515cde4c47d270.jpg)  
+Figure 23: BERT Layer 3 Words
+
+(a) Frequent words and infrequent words are on the main cluster, but at two sides. An evidence that words are distributed based on the frequency.
+
+![](images/5a26b55f9580f90ab89586cd6bef05c01ca251fb8c049898500383a8088fd153.jpg)  
+Figure 24: GPT2 Layer 3 Punctuation
+
+Based on these observations, we have concluded that frequency plays an important role in the token distributions. High frequent words and low frequent words are often taking opposite sides of the space. This is also revealed in Section 4.3. We are yet not clear what causes this, but we suspect it is related to the training process. During training, high frequent words are updated more times. Also, since they are used in many many different context, they play a role as some shared embedding across context. Similar to the XLM model, the shared embedding are often more isotropic and more concentrated. However, this is early hypothesis and due to future research.
+
+![](images/61bcc139cf80ad60418f5affb2f714d3232f2857d1ad86733ec3149d77f18162.jpg)  
+Figure 25: GPT2 Layer 3 Words
+
+(a) Mid-to-high frequent words often occupy both distinct islands (notice that the right small cluster is also colored), where a roll-shaped alignment can be observed on the larger island.
+
+# D.3 EMBEDDING OF TRANSLATION LANGUAGE MODEL XLM
+
+We also perform analysis and visualization on the XLM model (Conneau & Lample, 2019). BERT is mask language model (MLM), GPT is causal language model (CLM), and XLM is translation language model (TLM). We provide visualization of XLM’s 6 layers embeddings here. This is on WikiText-2 dataset.
+
+![](images/806dbebb04a4975490ba49bdc803f110776f93f63fded2c8151eadfdb3f3efcb.jpg)
+
+We try to establish a systematic view of embedding geometry for different types of deep LMs. We have hypothesis and very preliminary results here. BERT (an MLM) show spreading clusters, but not very distinct. GPT (an CLM) shows highly separated clusters. XLM (an TLM) does not demonstrate clustering effect, and the embedding are centered.
+
+One possible explanation for XLM’s behavior, is that this is a multi-lingual model, and the embedding space have to be shared between languages. This is forced during the training process of this translation language models. In that case, a single cluster residing in the center, would be a good shared embedding across languages. However, this is just hypothesis and requires further study on more models.
+
+# D.4 LID ESTIMATION ROBUSTNESS
+
+We follow (Aumuller & Ceccarello, 2019) to choose ¨ $K = 1 0 0$ for K-Nearest-Neighbor (K-NN) search for LID approximation, and make a direct comparison with them. It raises the concern that 100 samples might not be enough to effectively estimate the local dimension. We conduct additional experiments here to select $K = 2 0 0$ , 500, 1000, and demonstrate that the LID estimation is robust. They provide similar LID estimates across all layers, in all the models. Though using more samples indeed obtain very slightly higher values of LID (in Figure 27, we can see a little bit up-shifting from left-most plot to the right-most plot). This is expected, as less number of samples often tends to under-estimate, and over-smoothing of LID. Nevertheless, the LID is still much smaller than the original dimension 768, so using 100 samples is a good trade-off to efficiently approximate LID.
+
+![](images/a5cd6892ab66d2515b9ffc0312780f17a2e20d75b9fb0caf49ca9208b01539d5.jpg)  
+Figure 27: LID estimate using different number of samples for nearest neighbor search.
+
+As layer goes deeper, the LID increases. In other words, the local space dimension expands, at a cost of losing density. For example, the spiral band (1-D) in GPT’s front layer, becomes a Swiss Roll (2-D), and the roll surface get thickness (3-D), as layer increases. But we are not clear about the reasons yet, only suspect that data is spreading as more context info is added in later layers (the embedding for a token in deeper layer is based on summation of all embeddings in the context, due to attention). This is due to future study.
+
+# D.5 ABLATION ANALYSIS ON CLUSTERING
+
+To better study the clustering effect, we conduct experiment that computes the inter-type cosines, on clusterd-only embeddings and clustered plus center-shifted embeddings. The following figure shows GPT2’s cosine on original embeddings without adjustment (blue), the clustering-only embeddings (orange), and full (clustering $^ +$ centering) adjustment (green).
+
+![](images/3dfdde2bfa21ba167cb9517485ea0e877cedf95bdaa8fc62274a36eb7a729fa2.jpg)
+
+In the original embedding without adjustment, we see inconsistent behavior in the last layers. However, if we perform clustering and measure the $S _ { \mathrm { i n t e r } }$ within the clusters (orange), we can see much more consistent behavior across layers (more flat curve). This indicates that the clustering effect exists in all the layers, which is also verified by layer-wise visualization in Appendix B.4 and C.4.
+
+Meanwhile, the large values of cosines in the orange curve are expected. Now cosines are only computed within clusters, where those clusters are not at the origin. The higher values here, the more concentrated clusters are. These indicate that after clustering, the subspace within each cluster are now consistent, across all the layers. Finally, we shift those clusters to the origin, and get the green curve (values near 0), indicating isotropic cluster shapes.
+
+# D.6 POSITIONAL ENCODING IN THE GEOMETRY
+
+It is very interesting to investigate whether the positional encoding affects the geometry in the contextual embedding spaces. In particular, since GPT/GPT2 have a unique Swiss-Roll shaped manifold, that is not observed in other models, we look at how the manifold is related to the positional encoding in GPT2. Note that we truncate the whole PTB text into 512-length segments, and feed those segments into the models. The positional encoding is applied to each 512-length segment. We pick a few punctuation and words, and draw them in the space labeled by their relative positions in their corresponding segments. The position ID ranges from 0 to 511.
+
+![](images/4d8857bcef41a545cb24d43fde5ecda6c5151e4852d99c5f604a03bc29255186.jpg)  
+Figure 29: GPT2 Layer 3 Punctuation. The position ID is monotonically increasing along the manifold.
+
+![](images/b7340eac72344ac4942d9c8ade5da4b2fd722cdc2dc7ef6bf94ffabee7f3a5f6.jpg)  
+Figure 30: GPT2 Layer 3 Words. The position ID is monotonically increasing along the manifold.
+
+We select 4 punctuation, “, ’ & $\$ 7$ , and four words “the first super man”, draw them in Figure 29 30. The color bar on the side indicates the relative position ID in their segment. Darker color is smaller IDs and lighter color is bigger IDs. Clearly, for both punctuation and words, the center of the Swiss-Roll corresponds to lower position IDs, where the other end of the manifold are high IDs. Also, the distribution is monotonic. From the center to the far end, the position ID increases. This suggests that the positional encoding is indeed highly correlated with the Swiss-Roll manifold for GPT models. The reason causing this is deferred to future study.
+
+Note that this finding is consistent with that reported in (Reif et al., 2019), where they found that positions of the token matters (tokens take all neighbors’ information indiscriminately, rather than only attending up to their semantic boundaries) in the BERT embedding geometry. We also study the context/semantic influence of the embeddings in the next subsection.
+
+# D.7 CONTEXT IN THE GEOMETRY
+
+We also look at how the context information influences the geometry. It is more sophisticated to analyze the context, so we pick a few examples to look at their context and corresponding positions in the embedding spaces. In particular, we choose the common polysemous words “like” and “interest”, as two examples. The word “like” often has two different use cases: 1. favor; 2. similar to. There are also some fixed phrases such as “would like”. The word “interest” has two senses as well: 1. like to do something; 2. the money sense.
+
+We identified the target word token (“like” or “interest”), and then print out 5 tokens before and after the target, as the context for illustration in Figure 31. From the figure we are not able to identify a clear pattern that word sense is correlated with the geometric space. However, this is only inspected by manually checking a few samples. A full statistical analysis should be carried out in the future work.
+
+![](images/52c0ef7a1a1af05fee8d2a5073039265f2911d975831698d802a0a2d86d83e76.jpg)  
+Figure 31: The context and positions in the embedding space.