Bias and Generalization in Deep Generative Models: An Empirical Study

1 Introduction

• Because unbiased, consistent density estimation is impossible and possible distributions grow doubly exponentially with dimensionality [3], inductive bias matters in practical data regimes.The paper notes that the inductive bias of complex deep generative models is difficult to characterize analytically.Source: §1 · p. 1
• The framework projects image space onto selected low-dimensional features, then uses crafted training datasets to probe learned distributions, such as whether red and yellow spheres plus red cubes yield yellow cubes.Figure 1 depicts probes of single-valued features, multiple modes, and combinations of multiple features.Source: §1 · p. 1, Figure 1 · p. 2
• Generalization patterns were consistent across CLEVR, colored dots, and MNIST; GAN and VAE objectives; convolutional and fully connected architectures; and hyperparameter choices.Some patterns also resembled previously reported experiments in cognitive psychology.Source: §1 · p. 2
• Training on exactly 3 objects typically produced 2–5 objects with a log-normal-shaped distribution; multiple object-count modes behaved like averaged filters, except nearby modes showed prototype enhancement.For modes at 2 and 4 objects, the learned distribution favored their mean, 3, despite no training images containing 3 objects.Source: §1 · p. 2
• With multiple features, models memorized a small set of training combinations (e.g., 20) but generated novel combinations when variety was greater (e.g., 80).The paper studies the amount of novel combinations as a function of the number of combinations in the training set.Source: §1 · p. 2

2 Density Estimation, Bias, and Generalization

• High-dimensional feature combinations make training datasets exponentially small relative to the distribution’s support, so density estimation must rely on strong prior assumptions to generalize.The paper calls producing the learned distribution from training data generalization and the assumptions used inductive bias.Source: §2 · p. 2
• Deep generative models combine inductive biases from neural-network architecture, adversarial or variational training, and stochastic gradient descent [13] [14].The paper notes that these combined effects are difficult to study theoretically.Source: §2 · p. 2
• Prior studies found meaningful novel image examples beyond pixel-space interpolation, but did not systematically explain generalization across datasets or identify the biases involved.The cited studies do not answer how an algorithm generalizes given a new dataset.Source: §2 · pp. 2–3
• Because learned-distribution closeness has no commonly accepted evaluation metric [18] [19], the paper proposes settings where training and output distributions can be exactly characterized and compared.The high dimensionality of both the input dataset and output distribution contributes to the difficulty of systematic study.Source: §2 · p. 3

• Projecting real and learned image distributions onto interpretable features lets the authors characterize generalization, including whether models combine attributes absent from training data.The framework is inspired by cognitive psychology and focuses on images, though the authors say it can also apply to other domains.Source: §3 · p. 3
• Unlike FID, mode score, and Inception score, which compare pretrained-classifier features or labels, the framework uses interpretable high-level features to study exact differences between p(z) and q(z).The cited measures use hidden features or labels of a pretrained CNN classifier [22] [23]; related feature-projection ideas were previously used.Source: §3 · p. 3
• Because the synthetic feature spaces are discrete, low-dimensional, and have support no larger than 500, datasets of 100k–1M examples accurately approximate p(z), yet learned q(z) remains inaccurate.This approximate infinite-data setting is sufficient to reveal inductive biases, according to the authors.Source: §3 · p. 3

Feature Selection and Evaluation

• The selected features—numerosity, shape, color, size, and object location—are perceptually important and studied in cognitive psychology, with numerosity and shape evaluated by three independent human evaluators.The other features are evaluated with automated algorithms.Source: §3 · p. 3

Models

• Experiments use WGAN-GP [26] and VAE [4] with varied architectures and hyperparameters, and the authors report fairly consistent results across these models and objectives.The main text presents convolutional WGAN-GP results; results for other architectures appear in the appendix, with differences noted in the main text.Source: §3 · p. 3

4.1.1 Numerosity

• Generated numerosity centers on training counts with slight overestimation; six-dot training yields four to nine dots, resembling human and primate numerosity-response patterns.The pattern appears in both colored-dot and CLEVR experiments.Source: §4.1.1 · pp. 4–5, Figure 3 · p. 4

4.1.2 Color Proportion

• Color-proportion outputs are symmetric Gaussians centered on training values, sharper at 10% and 90%, consistent with Weber’s law rather than absolute-difference sensitivity.Source: §4.1.2 · p. 5, Figure 5 · p. 5

4.1.3 Size and Location

• Size generalization is skewed toward larger objects, whereas location generalization is fairly symmetric; both size and location distributions are very sharp.Source: §4.1.3 · p. 5, Figure 5 · p. 5

4.2 Convolution Effect and Prototype Enhancement Effect

• For sufficiently close training modes, the modes snap together and their mean receives high probability, making unseen 35%-red objects most likely; when far apart, convolution predicts behavior very well across GAN/VAE architectures and hyperparameters.Source: §4.2 · p. 6, Figure 6 · p. 6

4.3 Independence of Features

• Each feature’s learned distribution remains fairly stable as other dimensions vary, though greater randomness slightly increases variance and can merge modes that previously resisted prototype enhancement.This pattern holds across GAN/VAE and CNN/FC networks.Source: §4.3 · p. 6

Characterizing Generalization on Multiple Features

• For red proportion, size, and location, learned distributions are largely independent of nuisance-feature mode counts, though more random nuisance modes increase variance and can merge modes.Figure 7 considers nuisance-feature counts from 1 to 50.Source: Figure 7 · p. 7
• With few combinations, models nearly memorize training data; as combinations increase, they begin generating novel ones, a pattern consistent across settings.On CLEVR, 16 combinations yielded few or no unseen combinations, while 81 led the shape to generalize across all colors; shape marginals remained approximately preserved.Source: §5 · pp. 7–8, Figure 9 · p. 8

Experimental Setup

• The experiments vary feature combinations across pie images, three-digit MNIST images, and two-object CLEVR scenes.The pie and MNIST datasets sample subsets of roughly 2,000 and 1,000 possible combinations, respectively; CLEVR restricts a selected shape and color to a quarter of the possible pairings.Source: §5 · p. 7

Evaluating Precision-Recall

• As training combinations increase, GANs and VAEs retain very high recall but lose precision, generating novel combinations; about 100 combinations can suffice to produce them.Low precision appears inherent to these models in this task: increasing network size from 3M to 24M parameters or using half or twice as many examples did not remedy it.Source: §5 · p. 7, Figure 8 · p. 8

Visualizing Generalization

• On CLEVR, shapes observed with few colors generalize to other colors as the number of combinations grows, while each shape’s frequency stays approximately preserved.When a rare shape generalizes, it shares probability mass with colors absent from training.Source: §5 · pp. 7–8, Figure 9 · p. 8

6 Conclusion and Future Work

• The framework reveals distinct generalization patterns for individual features, including convolution effects, prototype enhancement, and independence, with some patterns resembling human and primate cognitive experiments.The study examines GANs and VAEs with different architectures, training objectives, and hyperparameter choices.Source: §6 · p. 8
• Novel feature combinations depend strongly on how many combinations appear in training, and generated combinations preserve each individual feature’s marginal distribution.The learned distribution was visualized to examine these combinations.Source: §6 · p. 8
• A key open question is which ingredients produce the observed behaviors, given the study’s exploration of GANs and VAEs with different architectures, objectives, and hyperparameters.The paper points to Appendix A for these model differences.Source: §6 · p. 8
• Future work must address how to organize and partition thousands of features, since the study characterizes generalization only in low-dimensional feature spaces.Generative models should capture distributions in the natural world.Source: §6 · p. 8

A Extended Figures for Different Objectives and Architectures

• The models showed qualitatively similar behavior under substantially different architecture and hyperparameter choices, including WGAN-FC and VAE-FC variants.The appendix presents these alternatives as checks that conclusions are not sensitive to modeling choices.Source: A Extended Figures for Different… · p. 11, Table 1 · p. 11

A.1 Extended Figures for WGAN

• The WGAN setting is the primary one used in the main paper, with its architecture and training details almost identical to [26].Figure 10 provides an extended plot corresponding to Figure 6.Source: §A.1 · p. 11, Figure 10 · p. 11

A.2 Extended Figures for VAE

• Figures 11, 12, 13, and 14 are identified as the VAE corresponding figures for Figures 3, 5, 6, and 7, respectively.Source: Figure 11 · p. 12, Figure 12 · p. 12, Figure 13 · p. 12

A.3 Extended Figures for WGAN-FC

• The WGAN-FC variant uses a fully connected architecture and Bernoulli latent noise, while retaining the WGAN-GP objective because DCGANs had severe mode missing.The authors report that these hyperparameter choices do not affect their results; Figures 15–18 provide corresponding plots.Source: §A.3 · p. 13, Figure 15 · p. 13, Figure 16 · p. 13

A.4 Extended Figures for VAE-FC

• The VAE-FC setting uses fully connected encoders and decoders, a larger learning rate, and a coefficient of 3 on DKL(q(z|x)∥p(z)).The appendix supplies extended figures for this additional setting.Source: §A.4 · p. 14

A.5 Extended Figures for CLEVR

• The CLEVR color generalization experiment examines 4x4 and 9x9 possible configurations, showing the training set and generated-feature frequencies for each setting.Figure 23 extends the color-feature experiment; its left panels show training sets and right panels show generated feature frequencies.Source: §A.5 · p. 15, Figure 23 · p. 15

A.6 Samples for Three MNIST Dataset

• Figure 24 shows four sample images from the Three MNIST Dataset.The dataset is introduced in Section 5.Source: Figure 24 · p. 15, §A.6 · p. 15

A.7 Invariance to Architecture Size

• Memorization is independent of deep-network size (Figure 25).Source: §A.7 · p. 15

B.1 Datasets

• Dots images contain k uniformly sampled dots with random colors and locations, rejecting any sample in which two dots overlap.The sampling procedure enforces non-overlap.Source: §B.1 · p. 15
• CLEVR images contain k objects with uniformly random shapes, colors, locations, and sizes; images are rejected if any objects overlap by more than 10%.The implementation is the public one cited as.Source: §B.1 · p. 16
• Pie images begin with a specified red percentage and three random non-red colors, then undergo four swaps of equal-angle slices at random locations to increase variation.Each color initially occupies a fixed slice.Source: §B.1 · p. 16

B.2 Evaluation

• Circle size is measured from non-background area and converted to radius, location from the mean non-background pixel position, and red proportion from pixels whose R component exceeds G and B.White backgrounds make non-background pixels identifiable in the single-circle images.Source: §B.2 · p. 16
• A feature combination is counted within the support of q(z) when q assigns it at least 10% of the uniform probability assigned to each training-set combination.This threshold is intended to reduce sensitivity to possible error and noise when computing φ.Source: §B.2 · p. 16
• >95% accuracy is achieved by the shift- and rotation-augmented CNN on generated MNIST samples, based on human judgment of 500 samples.Source: §B.2 · p. 16