Source-linked AI summary
Better Mixing via Deep Representations
Yoshua Bengio, Grégoire Mesnil, Yann Dauphin, Salah Rifai
TL;DR
The paper asks whether better-disentangled representations can make Markov chains mix faster, addressing the difficulty of moving between separated modes. It tests this using deep representation-learning models, sampling and interpolation experiments, and finds better samples and mixing at higher levels. The authors relate these results to disentangling, manifold unfolding, and expanded plausible volume, while leaving links to training and tempering for future work.
Problem
The paper examines limited understanding of whether better-disentangled deep representations can produce faster-mixing Markov chains and why mixing might improve at higher levels.
Method
The paper tests its hypotheses experimentally with DBNs and CAEs by measuring sampling quality, mixing, interpolation, and noise-based generation across representation levels.
Results
Higher representation levels produced better samples and mixing, including better samples from interpolation and isotropic-noise experiments, consistent with the proposed disentangling and manifold hypotheses.
Takeaways & Limitations
Transporting MCMC sampling to deeper representations could yield better and faster results for applications requiring generated samples.
Takeaways & Limitations
The paper does not propose a new sampling or learning algorithm and identifies links to MCMC-dependent training and parallel tempering as future work.
Abstract
from arXiv · showhide
It has previously been hypothesized, and supported with some experimental evidence, that deeper representations, when well trained, tend to do a better job at disentangling the underlying factors of variation. We study the following related conjecture: better representations, in the sense of better disentangling, can be exploited to produce faster-mixing Markov chains. Consequently, mixing would be more efficient at higher levels of representation. To better understand why and how this is happening, we propose a secondary conjecture: the higher-level samples fill more uniformly the space they occupy and the high-density manifolds tend to unfold when represented at higher levels. The paper discusses these hypotheses and tests them experimentally through visualization and measurements of mixing and interpolating between samples.
1 Introduction and Background
The paper links deep representations’ potential to disentangle factors of variation with faster mixing for Markov-chain sampling. It investigates whether sampling becomes easier at higher representation levels and why.
- Deep representations may help separate underlying factors of variation, making subsequent processing such as decision-making easier.
- Markov-chain samplers can mix slowly when modes are separated by large low-density regions, producing correlated samples and delayed transitions.
- Slow mixing can make MCMC-dependent learning slower or poorer, and may limit learning when sampling stalls.
- The paper investigates whether sampling at higher representation levels enables faster mixing using existing deep learning algorithms rather than proposing a new sampler.
2 Hypotheses
The hypotheses propose that depth improves mixing partly through disentangling, which unfolds high-density manifolds and expands the volume of plausible representation-space configurations.
- H1: Depth vs Better Mixing: H1 proposes that successfully trained deeper architectures can yield representation spaces where Markov chains mix faster.
- H2: Depth vs Disentangling: H2 proposes that deeper representations partly improve mixing by better disentangling the underlying factors of variation.
- H2: Depth vs Disentangling: A disentangled representation can encode a global change, such as reverse video, by flipping one high-level bit instead of changing many input pixels or hidden units.
- H2: Depth vs Disentangling: The paper distinguishes disentangling from invariant-feature learning: disentangling separates factors without requiring prior identification of nuisance factors.
- H3: Disentangling Unfolds and Expands: H3 proposes that disentangling unfolds high-density manifolds and expands the relative volume occupied by high-probability points.
- H3: Disentangling Unfolds and Expands: More unfolded manifolds and expanded plausible regions would make local representation-space configurations more uniform and convex interpolations more likely.
3 Representation-Learning Algorithms
The experiments use DBNs built by stacking RBMs and CAEs whose encoders and decoders define representation mappings. Both models can be composed into deeper architectures and paired with sampling procedures.
- The study uses Deep Belief Networks trained by stacking Restricted Boltzmann Machines and Contractive Auto-Encoders with a proposed sampling algorithm.
- An auto-encoder maps an input vector x to representation h with an encoder and maps h to reconstruction r with a decoder.
- The CAE uses sigmoid encoder and decoder mappings, tied weights, reconstruction loss, and a contractive Jacobian-based regularization penalty.
- CAE layers can be stacked into deeper models, viewed either as deep auto-encoders or as top-level samplers coupled with encoding and decoding functions.
- CAE sampling alternates auto-encoder reconstruction with Gaussian noise added along variation directions captured by the encoder Jacobian.
4 Experiments
Experiments on MNIST and TFD test whether deeper representations improve sampling, interpolation, local sample quality, and class mixing while preserving discriminative ability. Across these tests, deeper levels produce more plausible samples, visit more classes, occupy more likely local volume, and remain useful for classification.
- Experimental setup: Experiments use MNIST and TFD with DBNs and CAEs, including architectures with two learned upper layers.The tested DBNs and CAEs differ in layer sizes across the two datasets.
- Better samples at higher levels: Deeper-layer samples are visually more likely and mix faster than samples obtained at lower levels.On TFD, deeper representations move between woman and man samples through facial expressions; on MNIST, they mix among classes instead of remaining stuck.
- Interpolating between neighbors: Higher-level interpolation produces more visually plausible input samples, including smoother transitions between examples from different classes.Lower-level interpolations can create implausible patterns, whereas deeper representations nearly jump between high-density regions.
- Measuring mixing: Deeper architectures visit more classes in MCMC sequences, and the CAE mixes faster than the DBN.Mixing is measured by the number of different classes visited over sequences of length 10, 20, or 100.
- Occupying more volume around data points: Deeper representations occupy a larger proportion of local volume with likely configurations in both neighbor-interpolation and random-noise experiments.The neighbor-based result supports manifold unfolding and volume expansion; the random-direction result primarily supports volume expansion.
- Discriminative ability versus volume expansion: Increasing representation depth improves linear separability and retains strong discriminative performance despite improved generative abilities and mixing.The paper interprets this compatibility through better disentangling and local unfolding of class-related manifolds.
5 Conclusion
The experiments supported the hypotheses that deeper representations improve samples and mixing, with benefits linked to disentangling and manifold unfolding. These findings suggest transporting MCMC sampling to deeper representations could yield better and faster results, while several related directions remain for future work.
- Conclusion: The experiments supported deeper representations yielding better samples and better mixing.The results also included better interpolated samples and better samples after adding isotropic noise at higher levels.
- Conclusion: The results associated these improvements with better disentangling, manifold unfolding, and expansion of the volume occupied by good samples.The paper also confirmed better discrimination on the tested models and explained its apparent conflict with volume expansion through better disentangling.
- Conclusion: Deeper representations could provide better and faster results for applications requiring MCMC samples.This conclusion is presented as a potential application of transporting the sampling problem to deeper representations.
- Conclusion: Future work should investigate links between deeper-representation mixing and training deep learners that use MCMC for log-likelihood gradients.The paper also identifies the relation between parallel tempering and better-mixing chains from deeper layers as an open direction.