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Understanding and Improving Interpolation in Autoencoders via an Adversarial Regularizer
David Berthelot, Colin Raffel, Aurko Roy, Ian Goodfellow
TL;DR
Autoencoder interpolation is useful but lacks a precise definition and is not explicitly enforced by standard objectives. The paper introduces ACAI, which uses an adversarial critic and a quantitative benchmark; ACAI substantially outperformed common autoencoders and improved downstream feature learning and clustering.
Problem
Interpolation is widely used but ill-defined because “semantically meaningful combination” is vague, while standard autoencoder objectives do not explicitly enforce it.
Method
ACAI trains a critic to recover the mixing coefficient from interpolated data while training the autoencoder to fool the critic into predicting zero.
Results
ACAI substantially outperformed common autoencoder models on a synthetic interpolation benchmark and improved feature learning and unsupervised clustering performance.
Takeaways & Limitations
The results link improved interpolation behavior with more effective latent representations for downstream tasks within the evaluated setting.
Takeaways & Limitations
The study primarily focused on image datasets and used the regularizer with a standard vanilla autoencoder; extension to other autoencoders and non-image datasets remains future work.
Abstract
from arXiv · showhide
Autoencoders provide a powerful framework for learning compressed representations by encoding all of the information needed to reconstruct a data point in a latent code. In some cases, autoencoders can "interpolate": By decoding the convex combination of the latent codes for two datapoints, the autoencoder can produce an output which semantically mixes characteristics from the datapoints. In this paper, we propose a regularization procedure which encourages interpolated outputs to appear more realistic by fooling a critic network which has been trained to recover the mixing coefficient from interpolated data. We then develop a simple benchmark task where we can quantitatively measure the extent to which various autoencoders can interpolate and show that our regularizer dramatically improves interpolation in this setting. We also demonstrate empirically that our regularizer produces latent codes which are more effective on downstream tasks, suggesting a possible link between interpolation abilities and learning useful representations.
1 Introduction
The paper formalizes interpolation as a meaningful but previously ill-defined behavior in autoencoders, then proposes ACAI, a benchmark, and evaluations showing improved interpolation and downstream representation learning.
- Autoencoders learn lower-dimensional latent codes that reconstruct data and can support representation learning, generative modeling, and dimensionality reduction.
- Interpolation decodes mixtures of latent codes into outputs that can semantically combine characteristics of two datapoints.The authors also view interpolation as evidence that an autoencoder captures data structure rather than merely memorizing reconstructions.
- Interpolation is difficult to define because “semantically meaningful combination” is vague, and standard autoencoder objectives do not explicitly enforce it.
- ACAI introduces adversarial regularization to explicitly encourage high-quality interpolations in autoencoders.
- The paper develops a benchmark with quantifiable interpolation and reports superior interpolation behavior for ACAI over common autoencoder models.
- ACAI’s regularizer benefits representation learning for downstream tasks.
2 An Adversarial Regularizer for Improving Interpolations
ACAI improves autoencoder interpolation by training a critic to recover mixing coefficients while training the autoencoder to make interpolations appear non-interpolated and realistic.
- An autoencoder encodes x as a latent code z and decodes z into an approximate reconstruction.
- Interpolation decodes the convex combination αz1+(1−α)z2, producing outputs intended to morph from one datapoint toward the other as α varies.
- High-quality interpolation requires realistic intermediate points and semantically smooth morphing, but semantic similarity is difficult to codify explicitly.
- The critic predicts the mixing coefficient α from interpolated outputs, while the autoencoder is trained to fool it into predicting α = 0.The critic receives α values in [0, 0.5] to avoid ambiguity between α and 1−α.
- The critic loss predicts α and regularizes outputs on non-interpolated or data-space interpolated inputs; the latter helps stabilize adversarial learning.
- The autoencoder’s regularized loss gives it access to critic gradients, encouraging interpolated points to become indistinguishable from reconstructed data.In practice, this behavior also produces semantically smooth interpolations and improved representation-learning performance.
3 Autoencoders, and How They Interpolate
The paper makes interpolation measurable with a synthetic lines benchmark and compares how common autoencoders behave under this evaluation. ACAI achieves the strongest interpolation performance, while other models reveal distinct failures between realism and smoothness.
- 3.1 Autoencoding Lines: The benchmark represents each 32 × 32 line image with an angle Λ, making correct interpolation a smooth, shortest-path adjustment of that angle.Mean Distance measures proximity to real datapoints, while Smoothness measures whether interpolated angles follow a linear trajectory.
- 3.1 Autoencoding Lines: Interpolation is evaluated on a separate randomly sampled test set, although no model is explicitly trained to produce optimal interpolations.Good interpolation is treated as an emergent property of the architecture, loss, and training procedure.
- 3.2 Autoencoders: The baseline autoencoder had the worst Mean Distance, with intermediate interpolation points typically looking unrealistic despite roughly comparable Smoothness.This indicates that reconstruction ability alone does not guarantee realistic interpolations.
- 3.2 Autoencoders: The denoising autoencoder often produced data-space interpolations, resulting in comparatively poor Mean Distance and Smoothness scores.Its corruption process used additive isotropic Gaussian noise with σ = 1.0 after tuning.
- 3.2 Autoencoders: The VAE generated realistic intermediate points and accurately reconstructed inputs, but its line angles followed neither a smooth nor short path.Consequently, it achieved a very good Mean Distance score but a very poor Smoothness score.
- 3.2 Autoencoders: ACAI achieved the best Mean Distance and Smoothness scores, typically producing perfect interpolations while occasionally taking a longer-than-necessary path.The VQ-VAE instead produced poor interpolations because many codebook entries mapped to unrealistic datapoints.
4 Improved Representation Learning
The paper evaluates whether improved interpolation yields more useful latent representations through supervised classification and unsupervised clustering. ACAI improves classifier performance across settings and achieves the best or second-best clustering results on MNIST and SVHN.
- The study tests whether ACAI’s regularizer produces latent representations that perform better on supervised learning and clustering tasks.
- 4.1 Single-Layer Classifier: A single-layer classifier evaluates whether latent representations support class prediction despite a simple downstream model.
- 4.1 Single-Layer Classifier: 85.14% accuracy on SVHN with latent dimensionality 256 was achieved by ACAI, compared with 22.74% for the baseline autoencoder.
- 4.1 Single-Layer Classifier: ACAI achieved the best accuracy by a significant margin on MNIST and SVHN, while its CIFAR-10 performance was similar to the denoising autoencoder.
- 4.2 Clustering: Clustering uses PCA whitening followed by 1,000 random-initialized K-Means runs, with clustering accuracy measured using the optimal mapping between cluster IDs and classes.
- 4.2 Clustering: On MNIST and SVHN, ACAI achieved the best or second-best clustering performance for latent dimensionalities 32 and 256; CIFAR-10 results were omitted because accuracy was near-random.
5 Conclusion
The paper introduces ACAI, a critic-based regularizer for realistic interpolation, and evaluates it with a synthetic benchmark. ACAI substantially outperforms common autoencoders on interpolation and improves feature learning and unsupervised clustering, while broader model and dataset applicability remains future work.
- ACAI uses a critic to encourage interpolated datapoints to appear more realistic.
- A synthetic benchmark makes interpolation quantifiable, and ACAI substantially outperformed common autoencoder models on it.
- ACAI led to improved performance for feature learning and unsupervised clustering on downstream tasks.
- The study primarily focused on image datasets and applied the regularizer to a standard vanilla autoencoder, leaving other autoencoder types and non-image datasets for future work.
A Line Benchmark Evaluation Metrics
The synthetic lines benchmark evaluates interpolation by matching decoded intermediate images to nearby real line images, then measuring realism-oriented distance and angular smoothness. Smoothness is defined relative to a shortest, linearly progressing angle path.
- Each interpolation contains N decoded images produced by mixing two input images’ latent codes with α = (n−1)/(N−1).
- For every intermediate image, the evaluation finds the closest true line datapoint using cosine distance from a finite collection of evenly spaced angles.
- Mean Distance averages the cosine distances between interpolated images and their closest matching real line datapoints.
- Smoothness compares the largest normalized angular change with the minimum possible change, 1/(N−1).
- Angles are unwrapped by correcting discontinuities larger than π with multiples of ±2π before smoothness is computed.
B Base Model Architecture and Training Procedure
The baseline models use convolutional encoder and decoder blocks with pooling or upsampling, channel changes across resolution levels, and zero-padded convolutions. Training uses Gaussian initialization, batches of 64, and Adam with learning rate 0.0001.
- The encoder uses pairs of 3 × 3 convolutions followed by 2 × 2 average pooling, doubling channels before each pooling layer.
- Convolutions are zero-padded to preserve input and output height and width, with final convolutions producing the desired color channels.
- The decoder uses pairs of 3 × 3 convolutions followed by 2 × 2 nearest-neighbor upsampling, halving channels after each upsampling layer.
- Parameters are initialized as zero-mean Gaussian variables with standard deviation 1/√fan_in(1+0.22).
- Models train on 224 samples in batches of 64 using Adam with learning rate 0.0001 and default β1, β2, and ϵ.
C VAE Samples on the Line Benchmark
The VAE produces realistic samples and appears to cover the synthetic line data distribution, but its interpolations are not high quality.
- The VAE generally generates realistic samples and seems to cover the synthetic line benchmark’s data distribution despite poor interpolations.
D Interpolation Examples on Real Data
Figures 5–10 present example interpolations for seven autoencoder variants across MNIST, SVHN, and CelebA, using latent dimensionalities of 32 and 256. Further discussion of these interpolation results appears in section 3.3.
- Interpolation Examples: Figures 5–10 show interpolation behavior for the different autoencoders studied.The figures collectively cover MNIST, SVHN, and CelebA examples.
- MNIST: MNIST interpolations are shown at latent dimensionalities of 32 and 256 for Baseline, Dropout, Denoising, VAE, AAE, VQ-VAE, and ACAI autoencoders.These comparisons appear in Figures 5 and 6.
- SVHN: SVHN interpolations are shown at latent dimensionalities of 32 and 256 for Baseline, Dropout, Denoising, VAE, AAE, VQ-VAE, and ACAI autoencoders.These comparisons appear in Figures 7 and 8.
- CelebA: CelebA interpolations are shown at latent dimensionalities of 32 and 256 for Baseline, Dropout, Denoising, VAE, AAE, VQ-VAE, and ACAI autoencoders.These comparisons appear in Figures 9 and 10.