SPACE: S PATIALLY P ARALLEL A TTENTION AND C OMPONENT E XTRACTION FOR S CENE D ECOMPOSITION
Anonymous authors
Paper under double-blind review
A BSTRACT
We propose a generative latent variable model for unsupervised scene decomposition.
Our model, SPACE, provides a uniﬁed probabilistic modeling framework to combine the best of previous models.
SPACE can explicitly provide factorized object representation per foreground object while also decomposing background segments of complex morphology.
Previous models are good at either of these, but not both.
With the proposed parallel-spatial attention, SPACE resolves the scalability problem of previous methods and thus makes the model applicable to scenes with a much larger number of objects without performance degradation.
Besides, the foreground/background distinction of SPACE is more effective and intuitive than other methods because unlike other methods SPACE can detect static objects that have been difﬁcult to detect as foreground.
In experiments on Atari and 3D-Rooms, we show that SPACE achieves the above properties consistently in all experiments in comparison to SPAIR, IODINE, and GENESIS.
I NTRODUCTION
One of the unsolved key challenges in machine learning is unsupervised learning of structured representation for a visual scene containing many objects with occlusion, partial observability, and complex background.
When properly decomposed into meaningful symbolic entities such as objects and spaces, this structured representation brings many advantages of symbolic representation that contemporary deep learning with continuous vector representation has not been successful.
The advantages include sample efﬁciency of downstream tasks such as a deep reinforcement learning agent (Mnih et al., 2013), ability of visual variable binding (Sun, 1992) for reasoning and causal inference over the relationship between the objects and agents in a scene, and compositionality and transferability for generalization.
Recently, several methods have been proposed for this problem, unsupervised object-oriented scene decomposition, and these can be categorized into two approaches: mixture-scene models and spatial-attention models.
In the mixture-scene models (Greff et al., 2017; 2019; Burgess et al., 2019; Engelcke et al., 2019), a scene is explained by a mixture of K component images.
The main beneﬁt is to provide ﬂexible segmentation maps for objects and background segments with complex morphology for which spatial attention with bounding boxes may have difﬁculties.
However, this approach has limitations in scalability to scenes with many objects, and thus currently has been applied to scenes with only less than ten components.
This is because, to obtain a jointly complete scene, a component needs to refer to other components, and thus inference is inherently performed sequentially either by an RNN generating a component per step or by iterative inference (Marino et al., 2018).
Besides, because each component corresponds to a full-scale image, important physical features of objects like position and scale are only implicitly encoded in the scale of a full image, and thus further disentanglement is required to extract these useful features.
The spatial-attention models (Eslami et al., 2016; Crawford & Pineau, 2019) show the opposite properties to mixture-scene models.
It detects objects using spatial attention and thus can obtain not only the appearance representation but also fully disentangled explicit geometric representation of an object such as position and scale.
Such features grounded on the semantics of physics should be useful in many ways (e.g., sample efﬁciency, interpretability, geometric reasoning and inference, and transferability).
However, representing areas such as complex background segments that have too ﬂexible morphology to be captured by spatial attention (e.g., based on rectangular bounding boxes) is the main limitation of this approach.
It also shows scalability issues as objects are processed sequentially.
In this paper, we propose a method, called Spatially Parallel Attention and Component Extraction (SPACE), that combines the best of both approaches.
SPACE learns to process foreground objects, which can be captured efﬁciently by bounding boxes, by using parallel spatial-attention while decomposing the remaining area that includes both morphologically complex objects and background segments by the component mixture.
Thus, SPACE provides object-wise disentangled representation of foreground objects along with explicit properties like where per object while also providing decomposed representations of complex background components.
Besides, we resolve the scalability issue of existing spatial attention methods by developing fully parallel foreground-object processing.
In experiments on 3D-room scenes and Atari game scenes, we quantitatively and qualitatively compare the representation of SPACE to other models and show that SPACE combines the beneﬁts of both approaches in addition to signiﬁcant speed-ups due to the parallel foreground processing.
The contributions of the paper are as follows.
First, we introduce a model that uniﬁes the beneﬁts of two existing approaches in a principled framework of probabilistic latent variable modeling.
Second, we introduce a spatially parallel multi-object processing module and demonstrate that it can significantly mitigate the scalability problem of previous methods.
Third, we demonstrate SPACE can detect static objects (e.g., the key in Montezuma’s revenge) that always appear in the same position in all training images and thus are typically treated as a part of the background in other models.
This problem has been a key challenge in unsupervised object detection.
Lastly, we provide extensive analysis on and comparisons to previous models.
T HE P ROPOSED M ODEL : SPACE
In this section, we describe our proposed model, Spatially Parallel Attention and Component Extraction (SPACE).
The main idea of SPACE, presented in Figure 1, is to propose a uniﬁed probabilistic generative model that combines the beneﬁts of the spatial-attention models and mixture-scene models.
2. G ENERATIVE P ROCESS
SPACE assumes that a scene x is decomposed into two latents: foreground z fg and background z bg .
The foreground is further decomposed into a set of independent foreground objects z fg = { z fg i } and the background is also decomposed further into a sequence of background segments z
bg
=
z bg 1: K .
The foreground is ﬁrst generated and then the generation of the background is conditioned on the foreground.
The image distributions of the foreground objects and the background components are combined together into the following pixel-wise mixture model to produce the complete image distribution.
[Equation]
Here, the foreground mixing probability α is computed as α = f α ( z fg ) .
This way, the foreground is given precedence in assigning its own mixing weight and the remaining is apportioned to the background.
The mixing weight assigned to the background is further sub-divided among the K background components.
These weights are computed as π k = f π k
( z bg 1: k ) and (cid:80)
k
π k = 1 .
With these notations, the complete generative model can be described as follows.
[Equation]
We now describe the foreground and background models in more detail.
Foreground.
SPACE implements z fg as a structured latent.
In this structure, an image is treated as if it were divided into H × W cells and each cell is tasked with modeling at most one (nearby) object in the scene.
This type of structuring has been used in (Redmon et al., 2016;
Santoro et al., 2017;
Crawford & Pineau, 2019).
Similar to SPAIR, in order to model an object, each cell i is associated with a set of latents ( z pres i , z where i , z depth i , z what i ) .
In this notation, z pres is a binary random variable denoting if the cell models any object or not, z where denotes the size of the object and its location relative to the cell, z depth denotes the depth of the object to resolve occlusions and z what models the object appearance and its mask.
These latents may then be used to compute the foreground image component p ( x | z fg ) which is modeled as a Gaussian distribution N ( µ fg , σ 2 fg ) .
In practice, we treat σ 2 fg as a hyperparameter and decode only the mean image µ fg .
In this process, SPACE reconstructs the objects associated to each cell having z pres = 1 .
For each such cell, the model uses the z what i to decode the object glimpse and its mask and the glimpse is then positioned on a full-resolution canvas using z where i via the Spatial Transformer Network (Jaderberg et al., 2015).
Using the object masks and z depth i , all the foreground objects are combined into a single foreground mean-image µ fg and the foreground mask α
(See Appendix D for more details).
SPACE imposes a prior distribution on these latents as follows:
[Equation]
Here, only z pres i is modeled using a Bernoulli distribution while the remaining are modeled as Gaussian.
Background.
To model the background, SPACE implements z bg k , similar to GENESIS, as ( z mk , z ck ) where z m k models the mixing probabilities π k of the components and z c k models the RGB distribution p ( x |
z bg k ) of the k th background component as a Gaussian N ( µ bg i , σ 2 bg ) .
The following prior is imposed upon these latents.
[Equation]
2. I NFERENCE AND T RAINING
Since we cannot analytically evaluate the integrals in equation 2 due to the continuous latents z fg and z bg 1: K , we train the model using a variational approximation.
The true posterior on these variables is approximated as follows.
[Equation]
This is used to derive the following ELBO to train the model using the reparameterization trick and
SGD (Kingma & Welling, 2013).
[Equation]
See Appendix B for the derivation of the ELBO and the related details.
Parallel Inference of Cell Latents.
SPACE uses mean-ﬁeld approximation when inferring the cell latents, so z fg i =
( z pres i , z where i , z depth i , z what i ) for each cell does not depend on other cells.
[Equation]
As shown in Figure 1, this allows each cell to act as an independent object detector, spatially attending to its own local region in parallel.
This is in contrast to inference in SPAIR, where each cell’s latents auto-regressively depend on some or all of the previously traversed cells in a row-major order i.e., q ( z fg | x ) = (cid:81) HWi =1 q ( z fg i | z fg <i , x ) .
However, this method becomes prohibitively expensive in practice as the number of objects increases.
While Crawford & Pineau (2019) claim that these lateral connections are crucial for performance since they model dependencies between objects and thus prevent duplicate detections, we challenge this assertion by observing that 1) due to the bottom-up encoding conditioning on the input image, each cell should have information about its nearby area without explicitly communicating with other cells, and 2) in (physical) spatial space, two objects cannot exist at the same position.
Thus, the relation and interference between objects should not be severe and the mean-ﬁeld approximation is a good choice in our model.
In our experiments, we verify empirically that this is indeed the case and observe that SPACE shows comparable detection performance to SPAIR while having signiﬁcant gains in training speeds and efﬁciently scaling to scenes with many objects.
Preventing Box-Splitting.
If the prior for the bounding box size is set to be too small, then the model could split a large object by multiple bounding boxes and when the size prior is too large, the model may not capture small objects in the scene, hence, causing a trade-off between the prior values of the bounding box size.
To alleviate this problem, we found it helpful to introduce an auxiliary loss which we call the boundary loss .
In the boundary loss, we construct a boundary of thickness b pixels along the borders of each glimpse.
Then, we restrict an object to be inside this boundary and penalize the model if an object’s mask overlaps with the boundary area.
Thus, the model is penalized if it tries to split a large object by multiple smaller bounding boxes.
A detailed implementation of the boundary loss is mentioned in Appendix C.
R ELATED W ORKS
Our proposed model is inspired by several recent works in unsupervised object-oriented scene decomposition.
The Attend-Infer-Repeat (AIR) (Eslami et al., 2016) framework uses a recurrent neural network to attend to different objects in a scene and each object is sequentially processed one at a time.
An object-oriented latent representation is prescribed that consists of ‘what’, ‘where’, and ‘presence’ variables.
The ‘what’ variable stores the appearance information of the object, the ‘where’ variable represents the location of the object in the image, and the ‘presence’ variable controls how many steps the recurrent network runs and acts as an interruption variable when the model decides that all objects have been processed.
Since the number of steps AIR runs scales with the number of objects it attends to, it does not scale well to images with many objects.
Spatially Invariant Attend, Infer, Repeat (SPAIR) (Crawford &
Pineau, 2019) attempts to address this issue by replacing the recurrent network with a convolutional network.
Similar to YOLO (Redmon et al., 2016), the locations of objects are speciﬁed relative to local grid cells rather than the entire image, which allow for spatially invariant computations.
In the encoder network, a convolutional neural network is ﬁrst used to map the image to a feature volume with dimensions equal to a pre-speciﬁed grid size.
Then, each cell of the grid is processed sequentially to produce objects.
This is done sequentially because the processing of each cell takes as input feature vectors and sampled objects of nearby cells that have already been processed.
SPAIR therefore scales with the pre-deﬁned grid size which also represents the maximum number of objects that can be detected.
Our model uses an approach similar to SPAIR to detect foreground objects, but importantly we make the foreground object processing fully parallel to scale to large number of objects without performance degradation.
Works based on Neural Expectation Maximization (Van Steenkiste et al., 2018; Greff et al., 2017) do achieve unsupervised object detection but do not explicitly model the presence , appearance , and location of objects.
These methods also suffer from the problem of scaling to images with a large number of objects.
For unsupervised mixture-scene models, several recent models have shown promising results.
MONet (Burgess et al., 2019) leverages a deterministic recurrent attention network that outputs pixel-wise masks for the scene components.
A variational autoencoder (VAE) (Kingma & Welling, 2013) is then used to model each component.
IODINE (Greff et al., 2019) approaches the problem from a spatial mixture model perspective and uses amortized iterative reﬁnement of latent object representations within the variational framework.
GENESIS (Engelcke et al., 2019) also uses a spatial mixture model which is encoded by component-wise latent variables.
Relationships between these components are captured with an autoregressive prior, allowing complete images to be modeled by a collection of components.
E VALUATION
We evaluate our model on two datasets: 1) an Atari (Bellemare et al., 2013) dataset that consists of random images from a pretrained agent playing the games, and 2) a generated 3D-room dataset that consists of images of a walled enclosure with a random number of objects on the ﬂoor.
In order to test the scalability of our model, we use both a small 3D-room dataset that has 4-8 objects and a large 3D-room dataset that has 18-24 objects.
Each image is taken from a random camera angle and the colors of the objects, walls, ﬂoor, and sky are also chosen at random.
Additional details of the datasets can be found in the Appendix E.
Baselines.
We compare our model against two mixture-scene models (IODINE and GENESIS) and one spatial-attention model (SPAIR).
Since SPAIR does not have an explicit background component, we add an additional VAE for processing the background.
Additionally, we test against two implementations of SPAIR: one where we train on the entire image using a 16 × 16 grid and another where we train on random 32 × 32 pixel patches using a 4 × 4 grid.
We denote the former model as SPAIR and the latter as SPAIR-P. SPAIR-P is consistent with the SPAIR’s alternative training regime on Space Invaders demonstrated in Crawford & Pineau (2019) to address the slow training of SPAIR on the full grid size because of its sequential inference.
Lastly, for performance reasons, unlike the original SPAIR implementation, we use parallel processing for rendering the objects from their respective latents onto the canvas 1 for both SPAIR and SPAIR-P.
Thus, our SPAIR implementation can be seen as a stronger baseline in terms of the speed than the original SPAIR.
The complete details of the architecture used is given in Appendix D.
4. Q UALITATIVE C OMPARISON OF I NFERRED R EPRESENTATIONS
In this section, we provide a qualitative analysis of the generated representations of the different models.
For each model, we performed a hyperparameter search and present the results for the best settings of hyperparameters for each environment. Figure 2 shows sample scene decompositions of our baselines from the 3D-Room dataset and Figure 3 shows the results on Atari.
Note that SPAIR does not use component masks and IODINE and GENESIS do not separate foreground from background, hence the corresponding cells are left empty.
Additionally, we only show a few representative components for IODINE and GENESIS since we ran those experiments with larger
K than can be displayed.
More qualitative results of SPACE can be found in Appendix A.
IODINE & GENESIS.
In the 3D-Room environment, IODINE is able to segment the objects and the background into separate components.
However, it occasionally does not properly decompose objects (in the Large 3D-room results, the orange sphere on the right is not reconstructed) and may generate blurry objects.
GENESIS is able to segment the background walls, ﬂoor, and sky into multiple components. It is able to capture blurry foreground objects in the Small 3D-Room, but is not able to cleanly capture foreground objects with the larger number of objects in the Large 3DRoom.
In Atari, for all games, both IODINE and GENESIS fail to capture the foreground properly.
We believe this is because the objects in Atari games are smaller, less regular and lack the obvious latent factors like color and shape as in the 3D dataset, which demonstrates that detection-based approaches are more appropriate in this case.
SPAIR & SPAIR-P. SPAIR is able to detect tight bounding boxes in both 3D-Room and most Atari games (it does not work as well on dynamic background games, which we discuss below).
SPAIR-P, however, often fails to detect the foreground objects in proper bounding boxes, frequently uses multiple bounding boxes for one object and redundantly detects parts of the background as foreground objects.
This is a limitation of the patch training as the receptive ﬁeld of each patch is limited to a 32 × 32 glimpse, hence prohibiting it to detect objects larger than that and making it difﬁcult to distinguish the background from foreground.
These two properties are illustrated well in Space Invaders, where it is able to detect the small aliens, but it detects the long piece of background ground on the bottom of the image as foreground objects.
SPACE.
In 3D-Room, SPACE is able to accurately detect almost all objects despite the large variations in object positions, colors and shapes, while producing a clean segmentation of the background walls, ground, and sky.
This is in contrast to the SPAIR model, while being able to provide a similar foreground detection quality, encodes the whole background into a single component, which makes the representation less disentangled and the reconstruction more blurry.
Similarly in Atari, SPACE consistently captures all foreground objects while producing clean background segmentation across many different games.
Dynamic Backgrounds.
SPACE and SPAIR exhibit some very interesting behavior when trained on games with dynamic backgrounds.
For the most static game - Space Invaders, both SPACE and SPAIR work well.
For Air Raid, in which the background building moves, SPACE captures all objects accurately while providing a two-component segmentation, whereas SPAIR and SPAIRP produce splitting and heavy re-detections.
In the most dynamic games, SPAIR completely fails because of the difﬁculty to model dynamic background with a single VAE component, while SPACE is able to perfectly segment the blue racing track while accurately detecting all foreground objects.
Foreground vs Background.
Typically, foreground is the dynamic local part of the scene that we are interested in, and background is the relatively static and global part.
This deﬁnition, though intuitive, is ambiguous.
Some of the objects, such as the red shields near the bottom in Space Invaders and the key in Montezuma’s Revenge (Figure 5) are detected as foreground objects in SPACE, but are considered background in SPAIR.
Though these objects are static 2 , they are important elements of the games and should be considered as foreground objects.
Similar behavior is observed in Atlantis, where SPACE detects some foreground objects from the middle base that is above the water.
We believe this is an interesting property of SPACE and could be very important for providing useful representations for downstream tasks.
By using a spatial broadcast network (Watters et al., 2019) which is much weaker when compared to other decoders like sub-pixel convolutional nets (Shi et al.
(2016)), we limit the capacity of background module, which favors modeling static objects as foreground rather than background.
Boundary Loss.
We notice SPAIR sometimes splits objects into two whereas SPACE is able to create the correct bounding box for the objects (for example, see Air Raid).
This may be attributed to the addendum of the auxiliary boundary loss in the SPACE model that would penalize splitting an object with multiple bounding boxes.
4. Q UANTITATIVE C OMPARISON
In this section we compare SPACE with the baselines in several quantitative metrics.
We ﬁrst note that each of the baseline models has a different decomposition capacity ( C ), which we deﬁne as the capability of the model to decompose the scene into its semantic constituents such as the foreground objects and the background segmented components.
For SPACE, the decomposition capacity is equal to the number of grid cells H × W (which is the maximum number of foreground objects that can be detected) plus the number of background components K .
For SPAIR, the decomposition capacity is equal to the number of grid cells H × W plus 1 for background.
For IODINE and
GENESIS, it is equal to the number of components K .
For each experiment, we compare the metrics for each model with similar decomposition capacities.
This way, each model can decompose the image into the same number of components.
For a setting in SPACE with a grid size of H × W with K SPACE components, the equivalent settings in IODINE and GENESIS would be with C = ( H × W ) + K SPACE . The equivalent setting in SPAIR would be a grid size of H × W . As previously shown, SPAIR-P does not work well in many of our environments, so we do not include it in these quantitative experiments.
As in the qualitative section, we use the SPAIR implementation with sequential inference and parallel rendering in order to speed up the experiments.
Gradient Step Latency. The leftmost chart of Figure 4 shows the time taken to complete one gradient step (forward and backward propagation) for different decomposition capacities for each of the models.
We see that SPAIR’s latency grows with the number of cells because of the sequential nature of its latent inference step.
Similarly GENESIS and IODINE’s latency grows with the number of components K because each component is processed sequentially in both the models.
IODINE is the slowest overall with its computationally expensive iterative inference procedure.
Furthermore, both IODINE and GENESIS require storing data for each of the K components, so we were unable to run our experiments on 256 components or greater before running out of memory on our 22GB {+4 × 4 8 × 8 16 × 16 24 × 24 2 −2 − 1 22 1 22 3 
 Number of Cells ( H × W ) G r a d i e n t+}
S t e p L a t e n c y
( i n s ) Training Latency Plot   · 10    1, 1, Wall Clock Time (in s) R ec o n s t r u c t i o n
P i x e l M
S E
Convergence 3D Room (4 × 4) 0.2 0.4 0.6 0.
Wall Clock Time (in s)
Convergence 3D Room (8 × 8) 0.1 0.5 1 1.5 2 2.5· 10    1, 1,
Wall Clock Time (in s)
Convergence 3D Room (16 × 16)
[Table]
GPU.
On the other hand, SPACE employs parallel processing for the foreground which makes it scalable to large grid sizes, allowing it to detect a large number of foreground objects without any signiﬁcant performance degradation.
Although this data was collected for gradient step latency, this comparison implies a similar relationship exists with inference time which is a main component in the gradient step.
Time for Convergence.
The remaining three charts in Figure 4 show the amount of time each model takes to converge in different experimental settings.
We use the pixel-wise mean squared error (MSE) as a measurement of how close a model is to convergence.
We see that not only does
SPACE achieve the lowest MSE, it also converges the quickest out of all the models.
Average Precision and Error Rate.
In order to assess the quality of our bounding box predictions and the effectiveness of boundary loss, we measure the Average Precision and Object Count Error Rate of our predictions.
Our results are shown in Table 1.
We only report these metrics for 3DRoom since we have access to the ground truth bounding boxes for each of the objects in the scene.
All three models have very similar average precision and error rate. Despite being parallel in its inference, SPACE has a comparable count error rate to that of SPAIR. SPACE also achieves better average precision and count error rate compared to its variant without the boundary loss (SPACEWB), which shows the efﬁcacy of our proposed loss.
From our experiments, we can assert that SPACE can produce similar quality bounding boxes as SPAIR while 1) having orders of magnitude faster inference and gradient step time, 2) converging more quickly, 3) scaling to a large number of objects without signiﬁcant performance degradation, and 4) providing complex background segmentation.
C ONCLUSION
We propose SPACE, a uniﬁed probabilistic model that combines the beneﬁts of the object representation models based on spatial attention and the scene decomposition models based on component mixture.
SPACE can explicitly provide factorized object representation per foreground object while also decomposing complex background segments.
SPACE also achieves a signiﬁcant speed-up and thus makes the model applicable to scenes with a much larger number of objects without performance degradation.
Besides, the detected objects in SPACE are also more intuitive than other methods.
We show the above properties of SPACE on Atari and 3D-Rooms.
Interesting future directions are to replace the sequential processing of background by a parallel one and to improve the model for natural images.
Our next plan is to apply SPACE for object-oriented model-based reinforcement learning.
B ELBO D ERIVATIONS
In this section
[Equation]
we derive the ELBO for the log-likelihood log p ( x ) .
[Equation]
KL Divergence for the Foreground Latents Under the SPACE ’s approximate inference, the
D KL ( q ( z fg | x ) (cid:107) p ( z fg )) , is evaluated as follows.
[Equation]
KL Divergence for the Background Latents Under our GENESIS-like modeling of inference for the background latents, the KL term for the background is evaluated as follows.
[Equation]
Relaxed treatment of z pres In our implementation, we model the Bernoulli random variable z pres i using the Gumbel-Softmax distribution (Jang et al., 2016).
We use the relaxed value of z pres in the entire training and use hard samples only for the visualizations.
C B OUNDARY L OSS
In this section we elaborate on the implementation details of the boundary loss .
We construct a kernel of the size of the glimpse, gs × gs
(we use gs = 32 ) with a boundary gap of b = 6 having negative uniform weights inside the boundary and a zero weight in the region between the boundary and the glimpse.
This ensures that the model is penalized when the object is outside the boundary.
This kernel is ﬁrst mapped onto the global space via STN Jaderberg et al.
(2015) to obtain the global kernel.
This is then multiplied element-wise with global object mask α to obtain the boundary loss map .
The objective of the loss is to minimize the mean of this boundary loss map .
In addition to the
ELBO, this loss is also back-propagated via RMSProp (Tieleman & Hinton.
This loss, due to the boundary constraint, enforces the bounding boxes to be less tight and results in lower average precision, so we disable the loss and optimize only the ELBO after the model has converged well.
D I MPLEMENTATION D ETAILS D. A LGORITHMS
Algorithm 1 and Algorithm 2 present SPACE’s inference for foreground and background.
Algorithm 3 show the details of the generation process of the background module.
For foreground generation, we simply sample the latent variables from the priors instead of conditioning on the input.
Note that, for convenience the algorithms for the foreground module and background module are presented with for loops, but inference for all variables of the foreground module are implemented as parallel convolution operations and most operations of the background module (barring the LSTM module) are parallel as well.
Input: image x
Output: foreground mask α , appearance µ fg e img = ImageEncoderFg
( x ) for i ← 1 to HW do /*
The following is performed in parallel */ ρ i
= ZPresNet( e img i )
[ µ depth i , σ depth i ] =
ZDepthNet( e img i ) [ µ scale i , σ scale i ] =
ZScaleNet( e img i )
[ µ shift i , σ shift i ] =
ZShiftNet( e img i )
z pres i ∼ Bern( ρ i )
z depth i ∼ N ( µ depth i , σ depth , 2 i )
z scale i ∼ N ( µ scale i , σ scale , 2 i )
z shift i ∼ N ( µ shift i , σ shift , 2 i ) /
* rescale local shift to global shift */ z where i =
[ σ ( z scale i ) , rescale (tanh( z shift i ))]
/* Extract glimpses with a Spatial Transformer */ ˆ x i =
ST( x , z where i ) [ µ what i , σ what i ] =
GlimpseEncoder( ˆ x i )
z what i ∼ N ( µ what i , σ what , 2 i ) /* Foreground mask and appearance */ [ α att i ,
o att i ] =
GlimpseDecoder( z what i )
ˆ
α att i
= α att i (cid:12) z pres i
y att i
=
ˆ α att i (cid:12)
o att i end /
* Compute weights for each component */
[Equation]
/ * Compute global weighted mask and foreground appearance [Equation] / α [Equation]
sum
[Equation]
µ [Equation] sum [Equation]
D. T RAINING R EGIME AND H YPERPARAMETERS
For all experiments we use an image size of 128 × 128 and a batch size of 12 to 16 depending on memory usage.
We use the RMSProp (Tieleman & Hinton.
(2012)) optimizer with a learning rate of 1 × 10 − 5 for the foreground module and Adam (Kingma & Ba (2014)) optimizer with a learning rate of 1 × 10 − 3 for the background module.
We use gradient clipping with a maximum norm of 1.0.
For Atari games, we ﬁnd it beneﬁcial to set α to be ﬁxed for the ﬁrst 1000-2000 steps, and vary the actual value and number of steps for different games.
This allows both the foreground as well as the background module to learn in the early stage of training.
For 3D Room datasets, we found that the quality of background segmentation would degrade if trained for too long, so we set the background learning rate to zero at appropriate steps.
Input: image x initial 0 , c mask z m
Output: background masks π k , appearance µ bg k , for k
, K e img = ImageEncoderBg(
) for k ← to K do h k , c k = LSTM
([ z mk − 1 , e img ] , c k − 1 , h k − 1 ) [ µ mk , σ mk ] = PredictMask( h k ) z m ∼ N ( µ mk , σ m, 2 k ) /
* Actually decoded in parallel */ ˆ π k =
MaskDecoder( z mk ) /
* Stick breaking process as described in GENESIS */ π k = SBP(ˆ π 1: k ) [ µ ck , σ ck ] =
CompEncoder( [ π k , x ])
z c ∼ N ( µ ck , σ c, 2 k ) µ bg k =
CompDecoder( z ck ) end
Input z
Output: background masks π k appearance µ bg for k ← 1 to K do h k , c k = LSTM
( z mk − 1 , c k − 1 , h k − 1 ) [ µ mk , σ mk ] = PredictMaskPrior( h k )
z m ∼ N ( µ mk , σ m, 2 k ) /
* Actually decoded in parallel */ ˆ π k =
MaskDecoder( z mk ) /
* Stick breaking process as described in GENESIS */ π k = SBP(ˆ π 1: k ) [ µ ck , σ ck ] =
PredictComp( z mk ) z
ck ∼ N ( µ ck , σ c, 2 k ) µ bg k =
CompDecoder( z ck ) end
We list out our hyperparameters for 3D large dataset and joint training for 10 static Atari games below.
Hyperparameters for other experiments are similar, but are ﬁnetuned for each dataset individually.
In the tables below, ( m → n ) : ( p → q ) denotes annealing the hyperparameter value from m to n , starting from step p and till step q .
[Table]
Joint Training on 10 Atari Games
[Table]
D. M ODEL A RCHITECTURE
Here we describe the architecture of our 16 × 16 SPACE model.
The model for 8 × 8 grid cells is the same but with a stride-2 convolution for the last layer of the image encoder.
All modules that output distribution parameters are implemented with either one single fully connected layer or convolution layer, with the appropriate output size.
Image encoders are fully convolutional networks that output a feature map of shape H × W , and the glimpse encoder comprises of convolutional layers followed by a ﬁnal linear layer that computes the parameters of a Gaussian distribution.
For the glimpse decoder of the foreground module and the mask decoder of the background module we use the sub-pixel convolution layer (Shi et al.
On the lines of GENESIS (Engelcke et al.
and IODINE (Greff et al.
, we adopt Spatial Broadcast Network (Watters et al.
as the component decoder to decode z ck into background components.
For inference and generation of the background module, the dependence of z m k on z m 1: k − 1 is implemented with LSTMs, with hidden sizes of 64.
Dependence of z ck on z mk is implemented with a
MLP with two hidden layers with 64 units per layer.
We apply softplus when computing standard deviations for all Gaussian distributions, and apply sigmoid when computing reconstruction and masks.
We use either Group Normalization (GN) (Wu & He (2018)) and CELU (Barron (2017)) or
Batch Normalization (BN) (Ioffe & Szegedy (2015)) and ELU (Clevert et al.
depending on the module type.
The rest of the architecture details are described below.
In the following tables, ConvSub(n) denotes a sub-pixel convolution layer implemented as a stride-1 convolution and a PyTorch PixelShuﬄe( n ) layer, and GN( n ) denotes Group Normalization with n groups.
Name
Value
Comment z depth dim z pres dim z scale dim for x and y
axis
z shift dim for x and y axis
z what dim z m dim
z
c dim glimpse shape (32 , 32) for o att , α att image shape (128 , 128)
Background Image Encoder
Layer
Size/Ch.
Stride
Norm./Act.
Input
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Flatten
Linear
ELU
Glimpse Encoder
Layer
Size/Ch.
Stride
Norm./Act.
Input
Conv 3 ×
GN(4)/CELU
Conv 4 ×
GN(8)/CELU
Conv 3 ×
GN(4)/CELU
Conv 4 ×
GN(8)/CELU
Conv 4 ×
GN(8)/CELU
Conv 4 ×
GN(16)/CELU
Linear 32 +
Glimpse Decoder
Layer
Size/Ch.
Stride
Norm./Act.
Input
Conv 1 ×
GN(16)/CELU
ConvSub(2)
GN(16)/CELU
Conv 3 ×
GN(16)/CELU
ConvSub(2)
GN(16)/CELU
Conv 3 ×
GN
[Table]
Background Image Encoder
Layer
Size/Ch. Stride Norm./Act. Input
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
FlattenLinear
ELU
Mask Decoder
Layer
Size/Ch.
Stride
Norm./Act.
Input
Conv 1 ×
GN(16)/CELU
ConvSub(4)
GN(16)/CELU
Conv 3 ×
GN(16)/CELU
ConvSub(2)
GN(16)/CELU
Conv 3 ×
GN(16)/CELU
ConvSub(4)
GN(8)/CELU
Conv 3 ×
GN(8)/CELU
ConvSub(4)
GN(4)/CELU
Conv 3 ×
GN(4)/CELU
Conv 3 ×
Component Encoder
Layer
Size/Ch.
Stride
Norm./Act.
Input 3+1 (RGB+mask)
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Flatten
Linear 32+
Component Decoder
Layer
Size/Ch.
Stride
Norm./Act.
Input 32 (1d)
Spatial Broadcast 32+2 (3d)
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
BN/ELU
Conv 3 ×
D. B ASELINE SPAIR
Here we give out the details of the background encoder and decoder in training of SPAIR (both full image as well as patch-wise training).
The image encoder is same as that of SPACE with the only difference that the inferred latents are conditioned on previous cells’ latents as described in Section 2.2. SPAIR Background Encoder - Full Image training Layer Size/Ch.
Norm./Act.
Input 3 × 128 × 128 Linear
GN(16)/CELU Linear GN(16)/CELU Linear
SPAIR Background Decoder
Layer Size/Ch. Stride Norm./Act.
Input
Conv 3 ×
GN(16)/CELU
ConvSub 3 ×
GN(16)/CELU
Conv 3 ×
GN(16)/CELU
ConvSub 3 ×
GN(8)/CELU
Conv 3 ×
GN(8)/CELU
ConvSub 3 ×
GN(4)/CELU
Conv 3 ×
GN(4)/CELU Conv 3 × GN(4)/CELU Conv 3 ×
SPAIR Background Encoder For Patch Training Layer Size/Ch. Stride
Norm./Act. Input
Conv 2 × GN(4)/CELU Conv 2 ×
GN(8)/CELU
Conv 2 ×
GN(8)/CELU
Conv 2 ×
GN(16)/CELU
Conv 2 ×
GN(4)/CELU
SPAIR Background Decoder For Patch Training Layer Size/Ch. Stride
Norm./Act. Input Conv 1 × GN(16)/CELU
Conv 1 ×
ConvSub(4)
GN(16)/CELU
Conv 3 ×
GN(16)/CELU
Conv 1 × ConvSub(2)
GN(8)/CELU
Conv 3 ×
[Table]
E D ATASET D ETAILS
Atari.
For each game, we sample 60,000 random images from a pretrained agent (Wu et al., 2016).
We split the images into 50,000 for the training set, 5,000 for the validation set, and 5,000 for the testing set.
Each image is preprocessed into a size of 128 × 128 pixels with BGR color channels.
We present the results for the following games: Space Invaders, Air Raid, River Raid, Montezuma’s
Revenge.
We also train our model on a dataset of 10 games jointly, where we have 8,000 training images, 1, validation images, and 1,000 testing images for each game.
We use the following games: Asterix,
Atlantis, Carnival, Double Dunk, Kangaroo, Montezuma Revenge, Pacman, Pooyan, Qbert, Space
Invaders. Room 3D. We use MuJoCo (Todorov et al., 2012) to generate this dataset.
Each image consists of a walled enclosure with a random number of objects on the ﬂoor.
The possible objects are randomly sized spheres, cubes, and cylinders.
The small 3D-Room dataset has 4-8 objects and the large 3DRoom dataset has 18-24 objects.
The color of the objects are randomly chosen from 8 different colors and the colors of the background (wall, ground, sky) are chosen randomly from 5 different colors.
The angle of the camera is also selected randomly.
We use a training set of 63,000 images, a validation set of 7,000 images, and a test set of 7,000 images.
We use a 2-D projection from the camera to determine the ground truth bounding boxes of the objects so that we can report the average precision of the different models.