Original: Neural networks learn an environment’s geometry in latent space by performing predictive coding on visual scenes
Revised: Neural networks learn an environment’s geometry in latent space by performing predictive coding on visual scenes

Original: AﬃliationAddress

Original: Abstract
Revised: Abstract

Original: Humans navigate complex environments using only visual cues and self-motion.
Revised: Humans navigate complex environments using only visual cues and self-motion.

Original: Mapping an environment is an essential task for navigation within a physical space;neuroscientists and cognitive scientists also postulate that mapping algorithmsunderlie cognition by mapping concepts, memories, and other nonspatial variables.
Revised: Mapping an environment is an essential task for navigation within a physical space; neuroscientists and cognitive scientists also postulate that mapping algorithms underlie cognition by mapping concepts, memories, and other nonspatial variables.

Original: Despite the broad importance of mapping algorithms in neuroscience, it is notclear how neural networks can build spatial maps exclusively from sensor observations without access to the environment’s coordinates through reinforcementlearning or supervised learning.
Revised: Despite the broad importance of mapping algorithms in neuroscience, it is not clear how neural networks can build spatial maps exclusively from sensor observations without access to the environment’s coordinates through reinforcement learning or supervised learning.

Original: Path integration, for example, implicitly needs theenvironment’s coordinates to predict how past velocities translate into the currentposition.
Revised: Path integration, for example, implicitly needs the environment’s coordinates to predict how past velocities translate into the current position.

Original: Here we show that predicting sensory observations—called predictivecoding—extends path integration from implicitly requiring the environment’s coordinates.
Revised: Here we show that predicting sensory observations—called predictive coding—extends path integration from implicitly requiring the environment’s coordinates.

Original: Speciﬁcally, a neural network constructs an environmental map in its latentspace by predicting visual input.
Revised: Speciﬁcally, a neural network constructs an environmental map in its latent space by predicting visual input.

Original: As the network traverses complex environmentsin Minecraft, spatial proximity between object positions aﬀects distances in thenetwork’s latent space.
Revised: As the network traverses complex environments in Minecraft, spatial proximity between object positions aﬀects distances in the network’s latent space.

Original: The relationship depends on the uniqueness of the environment’s visual scene as measured by the mutual information between the images andspatial position.
Revised: The relationship depends on the uniqueness of the environment’s visual scene as measured by the mutual information between the images and spatial position.

Original: Predictive coding extends to any sequential dataset.
Revised: Predictive coding extends to any sequential dataset.

Original: Observationsfrom paths traversing a manifold can generate such sequential data.
Revised: Observations from paths traversing a manifold can generate such sequential data.

Original: We anticipateneural networks that perform predictive coding identify the underlying manifoldwithout requiring the manifold’s coordinates.
Revised: We anticipate neural networks that perform predictive coding identify the underlying manifold without requiring the manifold’s coordinates.

Original:  Introduction
Revised:  Introduction

Original: After traversing a few paths, humans can learn to map complex environments using only visual scenesand self-motion.
Revised: After traversing a few paths, humans can learn to map complex environments using only visual scenes and self-motion.

Original: Experimental evidence in neuroscience also indicates that humans use mappingto build relationships between concepts given only a few examples.
Revised: Experimental evidence in neuroscience also indicates that humans use mapping to build relationships between concepts given only a few examples.

Original: Despite the signiﬁcance ofmapping algorithms in neuroscience, it is not clear how neural networks can build spatial mapsexclusively from sensor observations without access to the environment’s coordinates.
Revised: Despite the signiﬁcance of mapping algorithms in neuroscience, it is not clear how neural networks can build spatial maps exclusively from sensor observations without access to the environment’s coordinates.

Original: For example,path integration, also called dead-reckoning, estimates the current position 𝑥 𝑡 given the past velocity𝑣 𝑡 .
Revised: For example, path integration, also called dead-reckoning, estimates the current position 𝑥 𝑡 given the past velocity 𝑣 𝑡 .

Original: Path integration implicitly uses the environment’s coordinates in how the velocity inﬂuences thetransition function ℙ [ 𝑥 𝑡 | 𝑣 𝑡 ] .
Revised: Path integration implicitly uses the environment’s coordinates in how the velocity inﬂuences the transition function ℙ [ 𝑥 𝑡 | 𝑣 𝑡 ] .

Original: Recent approaches such as Banino et al.
Revised: Recent approaches such as Banino et al.

Original: (2018) use coordinates as labelsin supervised training.
Revised: (2018) use coordinates as labels in supervised training.

Original: Moreover, studies such as Gupta et al.
Revised: Moreover, studies such as Gupta et al.

Original: (2019) use the coordinates to determinewhether an agent reaches a goal.
Revised: (2019) use the coordinates to determine whether an agent reaches a goal.

Original: We show that predicting sensory observations—called predictive coding—is suﬃcient to map anenvironment.
Revised: We show that predicting sensory observations—called predictive coding—is suﬃcient to map an environment.

Original: Predictive coding extends path integration from requiring the environment’s coordinates.
Revised: Predictive coding extends path integration from requiring the environment’s coordinates.

Original: Speciﬁcally, a neural network constructs an environmental map in its latent space by traversing anenvironment and predicting visual scenes.
Revised: Speciﬁcally, a neural network constructs an environmental map in its latent space by traversing an environment and predicting visual scenes.

Original: The agent traverses various Minecraft environments with
Revised: The agent traverses various Minecraft environments with

Original: [Figure]
Revised: [Figure]

Original:  translational motion and receives only visual data.
Revised: Training translational motion and receives only visual data.

Original: As the agent traverses the environment, distancesin the network’s latent space depend on the spatial proximity of object positions.
Revised: As the agent traverses the environment, distances in the network’s latent space depend on the spatial proximity of object positions.

Original: Moreover, wedemonstrate that the network’s latent space predicts spatial position: a linear model can predictthe environment’s coordinates from the predictive coding network’s latent space.
Revised: Moreover, we demonstrate that the network’s latent space predicts spatial position: a linear model can predict the environment’s coordinates from the predictive coding network’s latent space.

Original: The prediction’saccuracy depends on the uniqueness of the environment’s visual scene as measured by the mutualinformation between images and spatial position.
Revised: The prediction’s accuracy depends on the uniqueness of the environment’s visual scene as measured by the mutual information between images and spatial position.

Original:  Related work
Revised:  Related work

Original: The hippocampal system not only develops a map of a spatial environment but also organizes sequentialexperiences (Buzsáki & Tingley, 2018).
Revised: The hippocampal system not only develops a map of a spatial environment but also organizes sequential experiences (Buzsáki & Tingley, 2018).

Original: Experimental evidence suggests that the hippocampalmaps sequences and abstract concepts by their relationships on an abstract map.
Revised: Experimental evidence suggests that the hippocampal maps sequences and abstract concepts by their relationships on an abstract map.

Original: Several studiesestablish that the hippocampal system also maps nonspatial variables—time (MacDonald et al., 2011;
Revised: Several studies establish that the hippocampal system also maps nonspatial variables—time (MacDonald et al., 2011;

Original: Pastalkova et al., 2008) and auditory frequency (Aronov et al., 2017).
Revised: Pastalkova et al., 2008) and auditory frequency (Aronov et al., 2017).

Original: Nieh et al.
Revised: Nieh et al.

Original: ﬁnd that thehippocampal system integrates sensory input to determine the environment’s state—not just position.
Revised: ﬁnd that the hippocampal system integrates sensory input to determine the environment’s state—not just position.

Original: Constantinescu et al.
Revised: Constantinescu et al.

Original: (2016) extends the principle that the hippocampus is a mapping unit, indicatinghumans organize abstract concepts using a gridlike neural representation
Revised: (2016) extends the principle that the hippocampus is a mapping unit, indicating humans organize abstract concepts using a gridlike neural representation.

Original: Neurological and behavioral evidence indicate that path integration develops a map of an environment
Revised: Neurological and behavioral evidence indicate that path integration develops a map of an environment.

Original: However it is not  conceptually how humans map an environment from visual cues and selfmotion—without knowledge of their own kinematics or the environment’s coordinates.
Revised: However, it is not clear conceptually how humans map an environment from visual cues and selfmotion—without knowledge of their own kinematics or the environment’s coordinates.

Original: Banino et al.(2018)
Revised: Banino et al.

Original:  Cueva & Wei (2018) and Zhang et al.
Revised: (2018), Cueva & Wei (2018), and Zhang et al.

Original: (2021) demonstrate that artiﬁcial neural networks can perform path integration and use a map to navigate.
Revised: (2021) demonstrate that artiﬁcial neural networks can perform path integration and use a map to navigate.

Original: However, these studies require supervised trainingon the environment’s coordinates
Revised: However, these studies require supervised training on the environment’s coordinates.

Original: Similarly, Duan et al.
Revised: Similarly, Duan et al.

Original:  Gupta et al.
Revised: , Gupta et al.

Original:  Mirowski et al.(2018)
Revised: (2019), Mirowski et al.

Original:  Chaplot et al.
Revised: (2018), and Singh Chaplot et al.

Original: (2020) use visual input for mapping but encode the environment’scoordinates either through a goal reward or through supervised training on coordinates.
Revised: (2020) use visual input for mapping but encode the environment’s coordinates either through a goal reward or through supervised training on coordinates.

Original: Previous studies implicitly introduce the environment’s coordinates into the network by goal rewards or supervised training
Revised: Previous studies implicitly introduce the environment’s coordinates into the network by goal rewards or supervised training.

Original: It is unclear whether networks can map an environment exclusively from sensory 
Revised: It is unclear whether networks can map an environment exclusively from sensory data.

Original:  Predictive coding
Revised:  Predictive coding

Original: Predictive inference involves framing the problem of learning from a sequence of actions and observations.
Revised: Predictive inference involves framing the problem of learning from a sequence of actions and observations.

Original: The learner and decision maker are called the agent .
Revised: The learner and decision maker are called the agent .

Original: With everything the agent interacts iscalled the environment .
Revised: With everything the agent interacts is called the environment .

Original: The state space 𝑆 and the action space 𝐴 are both topologies.
Revised: The state space 𝑆 and the action space 𝐴 are both topologies.

Original: The observationspace O are random variables 𝑂 ( 𝑠 ) indexed by states 𝑠 in the state space 𝑆 .
Revised: The observation space O are random variables 𝑂 dependent only on the current state 𝑠 in the state space 𝑆 .

Original: For a sequence ( 𝑠 𝑘 , … , 𝑠 1 )of states, we assume the observations are continuous in distribution[Equation]
Revised: For a sequence ( 𝑠 1 , … , 𝑠 𝑇 ) of states, we assume the observations are continuous in distribution

Original: and Markovian

Original: [Equation]
Revised: [Equation]

Original: The learning problem is the following.
Revised: The learning problem is the following.

Original: An agent constructs a sequence of actions (( 𝑎 1) , … ,( 𝑎 𝑁 ) ) ,and observes samples ( 𝑂 ( 𝑠 1 ) , … , 𝑂 ( 𝑠 𝑁 ) ) .
Revised: An agent constructs a sequence of actions ( 𝑎 1 , … , 𝑎 𝑇 ) , and observes samples ( 𝑂 ( 𝑠 1 ) , … , 𝑂 ( 𝑠 𝑇 ) ) .

Original: The goal is for the person to learn the stochastic process
Revised: The goal is for the person to learn the stochastic process

Original: [Equation]
Revised: [Equation]

Original: for every state 𝑠 in 𝑆 and action 𝑎 in 𝐴 .
Revised: for every state 𝑠 in 𝑆 and action 𝑎 in 𝐴 .

Original: Consider the example of a person walking through an open ﬁeld (Figure 1. a ).
Revised: Consider the example of a person walking through an open ﬁeld (Figure 1. a ).

Original: The person has aposition ( 𝑥, 𝑦 ) .
Revised: The person has a position ( 𝑥, 𝑦 ) .

Original: The person can walk around facing north.
Revised: The person can walk around facing north.

Original: The state space is the Euclidean space ℝ 2 .
Revised: The state space is the Euclidean space ℝ 2 .

Original: The action space is the person’s velocity ( 𝑣 𝑥 , 𝑣 𝑦 ) .
Revised: The action space is the person’s velocity ( 𝑣 𝑥 , 𝑣 𝑦 ) .

Original: Every action ( 𝑣 𝑥 , 𝑣 𝑦 ) deterministically maps a paststate ( 𝑥 1 , 𝑦 1 ) to a current position ( 𝑥 2 , 𝑦 2 ) with the equations
Revised: Every action ( 𝑣 𝑥𝑡 −1 , 𝑣 𝑦𝑡 −1 ) deterministically maps a past state ( 𝑥 𝑡 −1 , 𝑦 𝑡 −1 ) to a current position ( 𝑥 𝑡 , 𝑦 𝑡 ) with the equations

Original: [Equation]
Revised: [Equation]

Original: At each position in ℝ 2 there exists the random variable observations 𝑂 ( 𝑥, 𝑦 ) the person can seegiven their state ( 𝑥, 𝑦 ) .
Revised: At each position in ℝ 2 there exists the random variable observations 𝑂 ( 𝑥, 𝑦 ) the person can see given their state ( 𝑥, 𝑦 ) .

Original: A path, as a sequence of actions, generates a sequence of observations( 𝑂 1 , … , 𝑂 𝑁 ) ∼ ( ℙ [ 𝑂 ( 𝑥 1 , 𝑦 1 )] , … , ℙ [ 𝑂 ( 𝑥 𝑁 , 𝑦 𝑁 )]) , which satisﬁes a stochastic process.
Revised: A path, as a sequence of actions, generates a sequence of observations ( 𝑂 1 , … , 𝑂 𝑇 ) ∼ ( ℙ [ 𝑂 ( 𝑥 1 , 𝑦 1 )] , … , ℙ [ 𝑂 ( 𝑥 𝑇 , 𝑦 𝑇 )]) , which satisﬁes a stochastic process.

Original: The goal isfor the person to learn the stochastic process
Revised: The goal is for the person to learn the stochastic process

Original: [Equation]
Revised: [Equation]

Original:  for every position ( 𝑥, 𝑦 ) and action ( 𝑣 𝑥 , 𝑣 𝑦 ) .
Revised: ) | 𝑂 ( 𝑥 𝑡 −1 , 𝑦 𝑡 −1 ) , … , 𝑂 ( 𝑥 1 , 𝑦 1 ) , ( 𝑣 𝑥𝑡 −1 , 𝑣 𝑦𝑡 −1 ) , … , ( 𝑣 𝑥 1 , 𝑣 𝑦 1 )] for every position ( 𝑥, 𝑦 ) and action ( 𝑣 𝑥 , 𝑣 𝑦 ) .

Original:  A predictive coding neural network’s latent space constructs anenvironmental map
Revised:  A predictive coding neural network’s latent space constructs an environmental map

Original: We study how a neural network—without the environment’s coordinates—maps an environment byperforming predictive coding using the Malmo environment in Minecraft (Johnson et al., 2016).
Revised: We study how a neural network—without the environment’s coordinates—maps an environment by performing predictive coding using the Malmo environment in Minecraft (Johnson et al., 2016).

Original: Weconstruct an open ﬁeld (Figure 1. a ) enclosed by a fence of 20×20 blocks.
Revised: We construct an open ﬁeld (Figure 1. a ) enclosed by a fence of 20×20 blocks.

Original: Columns of varying heightsand colors provide the agent with a unique visual scene at each position.
Revised: Columns of varying heights and colors provide the agent with a unique visual scene at each position.

Original: The agent can traverse theenvironment by moving forwards, backwards, left, and right by one block.
Revised: The agent can traverse the environment by moving forwards, backwards, left, and right by one block.

Original: We use an encoder-decoderconvolutional neural network † with an embedding from the action to the latent space (Figure 1. a ).
Revised: We use an encoder-decoder convolutional neural network † with an embedding from the action to the latent space (Figure 1. a ).

Original: Tosolve the predictive coding problem we assume the predictive coding distribution is a multivariate
Revised: To solve the predictive coding problem we assume the predictive coding distribution is a multivariate

Original: Gaussian.
Revised: Gaussian.

Original: We estimate the density 𝑝 by solving the maximum likelihood estimator
Revised: We estimate the density 𝑝 by solving the maximum likelihood estimator

Original: [Equation]
Revised: [Equation]

Original: The neural network approximates predictive coding by minimizing the mean-squared error betweenthe current visual scene and its prediction given the previous image and action.
Revised: The neural network approximates predictive coding by minimizing the mean-squared error between the current visual scene and its prediction given the previous image and action.

Original: We ﬁrst show the latent space’s capability to estimate the environment’s distances within one block,demonstrating the latent space contains signiﬁcant information about the physical space.
Revised: We ﬁrst show the latent space’s capability to estimate the environment’s distances within one block, demonstrating the latent space contains signiﬁcant information about the physical space.

Original: To determinethe latent axes that capture the environment’s geometry, we ﬁt a linear model that estimates theenvironment’s coordinates from the latent space.
Revised: To determine the latent axes that capture the environment’s geometry, we ﬁt a linear model that estimates the environment’s coordinates from the latent space.

Original: We calculate the pairwise distance between thepredicted locations as well as the actual locations (Figure 1. c ).
Revised: We calculate the pairwise distance between the predicted locations as well as the actual locations (Figure 1. c ).

Original: For the 20 × 20 block lattice, themean error between predicted distances and actual distances is 0 .
Revised: For the 20 × 20 block lattice, the mean error between predicted distances and actual distances is 0 .

Original: 743 ± 0 .
Revised: 743 ± 0 .

Original: 019 blocks ( 95% conﬁdenceinterval).
Revised: 019 blocks ( 95% conﬁdence interval).

Original: Furthermore, the latent space captures a linear representation of physical coordinates.
Revised: Furthermore, the latent space captures a linear representation of physical coordinates.

Original: Figure 1.
Revised: Figure 1.

Original: cshows the linear model’s coordinate estimates ( ∙ ) from the latent space.
Revised: c shows the linear model’s coordinate estimates ( ∙ ) from the latent space.

Original: The grey lines show theerror from the physical coordinates ( ∙ ) with prediction error of 0 .
Revised: The grey lines show the error from the physical coordinates ( ∙ ) with prediction error of 0 .

Original: 779 ± 0 .
Revised: 779 ± 0 .

Original: 186 blocks.
Revised: 186 blocks.

Original: The predictionerror increases as the agent travels to the front, indicating the latent space contains less positionalinformation toward the front.
Revised: The prediction error increases as the agent travels to the front, indicating the latent space contains less positional information toward the front.

Original: [Figure]
Revised: [Figure]

Original: g h e f c d
Revised: g h e f c d

Original:  Predictability of spatial location from the latent space depends on the 
Revised:  Predictability of spatial location from the latent space depends on the uniqueness of the environment’s visual scene

Original: uniqueness of the environment’s visual scene

Original: In section 4, we demonstrate a neural network can map the environment in Figure 1.
Revised: In section 4, we demonstrate a neural network can map the environment in Figure 1.

Original: a by learningpredictive coding.
Revised: a by learning predictive coding.

Original: In this environment (Environment A), positions are relatively identiﬁable fromvisual landmarks.
Revised: In this environment (Environment A), positions are relatively identiﬁable from visual landmarks.

Original: Many environments contain areas  no identiﬁable landmarks.
Revised: Many environments contain areas with no identiﬁable landmarks.

Original: Introducing anenvironment (Environment B) with sparse landmarks (Figure 2.
Revised: Introducing an environment (Environment B) with sparse landmarks (Figure 2.

Original: b ), we show the neural networkreconciles these areas in latent space by minimizing prediction errors of the physical coordinates.
Revised: b ), we show the neural network reconciles these areas in latent space by minimizing prediction errors of the physical coordinates.

Original: Inthis environment, many locations are not identiﬁable from landmarks.
Revised: In this environment, many locations are not identiﬁable from landmarks.

Original: Between EnvironmentsA and
Revised: Between Environments

Original:  B, we calculate the  log diﬀerence  log 𝑝 (  𝑋, 𝑂 ) − log 𝑝 (  𝑋 ) 𝑝 ( 𝑂 )  between the joint probability 𝑝 ( 𝑋, 𝑂 )and marginal probabilities 𝑝 ( 𝑋 ) and 𝑝 ( 𝑂 ) between the location  𝑋 and the visual scene 𝑂 —based onthe presence of visual landmarks (Figure 2.
Revised: A and B, we calculate the expected log diﬀerence 𝔼 𝑂 ∼ ℙ [ 𝑂 ] [ log 𝑝 ( 𝑥 ∈ 𝑋, 𝑂 ) − log 𝑝 ( 𝑥 ∈ 𝑋 ) 𝑝 ( 𝑂 ) ] between the joint probability 𝑝 ( 𝑋, 𝑂 ) and marginal probabilities 𝑝 ( 𝑋 ) and 𝑝 ( 𝑂 ) between the location 𝑥 ∈ 𝑋 and the visual scene 𝑂 —based on the presence of visual landmarks (Figure 2.

Original: c-d ).
Revised: c-d ).

Original: The mutual information 𝐼 ( 𝑋 ; 𝑂 ) is the expectationof this statistic, i.e., 𝔼 [log 𝑝 ( 𝑋, 𝑂 ) − log 𝑝 ( 𝑋 ) 𝑝 ( 𝑂 )] .
Revised: The mutual information 𝐼 ( 𝑋 ; 𝑂 ) is the expectation of this parameter, i.e., 𝔼 [log 𝑝 ( 𝑋, 𝑂 ) − log 𝑝 ( 𝑋 ) 𝑝 ( 𝑂 )] .

Original: Intuitively, areas in Environment B with a lowlog diﬀerence have locations that are less predictable from visual scenes—with a value of zero if thevisual scene and positions are independent.
Revised: Intuitively, areas in Environment B with a low log diﬀerence have locations that are less predictable from visual scenes—with a value of zero if the visual scene and positions are independent.

Original: We visualize the latent space units as the agent traverses the environment (Figure 2.
Revised: We visualize the latent space units as the agent traverses the environment (Figure 2.

Original: e-f ).
Revised: e-f ).

Original: The neuralnetwork in Environment B has notably less non-zero latent units as compared to the neural networkin Environment A. Similar to section 4, we use a linear model to estimate the predictability of theenvironment’s positions from the network’s latent space (Figure 2.
Revised: The neural network in Environment B has notably more zero latent units as compared to the neural network in Environment A. Similar to section 4, we use a linear model to estimate the predictability of the environment’s positions from the network’s latent space (Figure 2.

Original: g-h ).
Revised: g-h ).

Original: The prediction error dependson the log probability diﬀerence  log 𝑝 ( 𝑋, 𝑂 ) − log 𝑝 ( 𝑋 ) 𝑝 ( 𝑂 )  between the visual scene and the agent’sposition; the network reconciles positions without landmarks by guessing in latent space.
Revised: The prediction error depends on the log probability diﬀerence 𝔼 𝑂 ∼ ℙ [ 𝑂 ] [ log 𝑝 ( 𝑋, 𝑂 ) − log 𝑝 ( 𝑋 ) 𝑝 ( 𝑂 ) ] between the visual scene and the agent’s position; the network reconciles positions without landmarks by guessing in latent space.

Original:  Discussion
Revised:  Discussion

Original: Humans can navigate complex environments using only visual landmarks and self-motion; however,current algorithms require the environment’s coordinates or knowledge of their own kinematics.
Revised: Humans can navigate complex environments using only visual landmarks and self-motion; however, current algorithms require the environment’s coordinates or knowledge of their own kinematics.

Original: Wedemonstrated that predictive coding is suﬃcient to map an environment.
Revised: We demonstrated that predictive coding is suﬃcient to map an environment.

Original: By predicting visual scenes,predictive coding generalizes the prediction task of path integration to arbitrary visual scenes.
Revised: By predicting visual scenes, predictive coding generalizes the prediction task of path integration to arbitrary visual scenes.

Original: In thispaper, we treat predictive coding with the state space  as a Euclidean space.
Revised: In this paper, we treat predictive coding with the state space  as a Euclidean space.

Original: The state space  canbe any topology, most notably a diﬀerentiable manifold.
Revised: The state space  can be any topology, most notably a diﬀerentiable manifold.

Original: Observations ( 𝑂 ( 𝛾 ( 𝑡 ))
Revised: Observations ( 𝑂 ( 𝛾 ( 𝑡 ))

Original: 𝑡 ∈ 𝑇 from paths 𝛾 ( 𝑡 )along the manifold generate a stochastic process.
Revised: 𝑡 ∈ 𝑇 from a path 𝛾 ( 𝑡 ) along the manifold generates a stochastic process.

Original: We anticipateto formulate a majority of sequentiallearning problems  as predictive coding on manifolds.
Revised: We anticipate a majority of sequential learning problems can be formulated as predictive coding on manifolds.

Original: By learning predictive coding, neural networksmay identify the underlying manifold without requiring the manifold’s coordinates.
Revised: By learning predictive coding, neural networks may identify the underlying manifold without requiring the manifold’s coordinates.

Original:  6.1 Detailed experimental setup
Revised: Supplemental Information 6.1 Detailed experimental setup

Original: For all experiments, the Malmo framework in Minecraft (Johnson et al., 2016) simulate the environments.
Revised: For all experiments, the Malmo framework in Minecraft (Johnson et al., 2016) simulate the environments.

Original: The agent traverses the environment by a directed random walk, which is discretized onthe lattice.
Revised: The agent traverses the environment by a directed random walk, which is discretized on the lattice.

Original: The actions are represented by a one-hot vector (left, right, forward, backward).
Revised: The actions are represented by a one-hot vector (left, right, forward, backward).

Original: Visualobservations are represented by 64 × 64 RGB images.
Revised: Visual observations are represented by 64 × 64 RGB images.

Original: For training, we use an U-Net architecture (Ronneberger et al., 2015) with a ResNet38 encoder (Heet al., 2015) encoder and the corresponding ResNet38 with transposed convolutions for the decoder.
Revised: For training, we use an U-Net architecture (Ronneberger et al., 2015) with a ResNet38 encoder (He et al., 2015) encoder and the corresponding ResNet38 with transposed convolutions for the decoder.

Original: Alinear transform maps the action one-hot vector {0 , 1} 4 to 512 channels of ℝ 2×2 units.
Revised: A linear transform maps the action one-hot vector {0 , 1} 4 to 512 channels of ℝ 2×2 units.

Original: The ℝ 2×2×512output units are concatenated to the output of the ResNet38 encoder to form a ℝ 2×2×1024 array.
Revised: The ℝ 2×2× output units are concatenated to the output of the ResNet38 encoder to form a ℝ 2×2×1024 array.

Original: Theℝ 2×2×1024 is input into the corresponding ResNet38 decoder.
Revised: The ℝ 2×2×1024 is input into the corresponding ResNet38 decoder.

Original: For Environment A and Environment B, the neural network trained on 20 , 000 batches with a batchsize of 128 .
Revised: For Environment A and Environment B, the neural network trained on 20 , 000 batches with a batch size of 128 .

Original: The neural network minimizes the mean-squared error between the predicted and actualimage using RMSprop with a learning rate of 10 −3 with a decay of 0 .
Revised: The neural network minimizes the mean-squared error between the predicted and actual image using RMSprop with a learning rate of 10 −3 with a decay of 0 .

Original: 1 every 5 , 000 batches and a 𝓁 2weight decay of 5 × 10 −6 .
Revised: 1 every 5 , 000 batches and a 𝓁weight decay of 5 × 10 −6 .

Revised: 6.2 Extended ﬁgures

Original: [Figure]
Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]