Understanding Boltzmann Machine and Deep Learning via A Confident Information First Principle

TL;DR

This paper introduces the Confident-Information-First principle for dimensionality reduction in parameter spaces, providing a theoretical foundation for Boltzmann machines and developing CIF-based algorithms for density estimation.

Contribution

It formalizes the CIF principle using Fisher information, connects it to Boltzmann machines, and proposes new CIF-based algorithms for model training.

Findings

CIF helps identify essential parameters in Boltzmann machines.

CIF-based algorithms improve density estimation performance.

Theoretical analysis clarifies the role of layers in deep networks.

Abstract

Typical dimensionality reduction methods focus on directly reducing the number of random variables while retaining maximal variations in the data. In this paper, we consider the dimensionality reduction in parameter spaces of binary multivariate distributions. We propose a general Confident-Information-First (CIF) principle to maximally preserve parameters with confident estimates and rule out unreliable or noisy parameters. Formally, the confidence of a parameter can be assessed by its Fisher information, which establishes a connection with the inverse variance of any unbiased estimate for the parameter via the Cram\'{e}r-Rao bound. We then revisit Boltzmann machines (BM) and theoretically show that both single-layer BM without hidden units (SBM) and restricted BM (RBM) can be solidly derived using the CIF principle. This can not only help us uncover and formalize the essential parts of the target density that SBM and RBM capture, but also suggest that the deep neural network consisting of several layers of RBM can be seen as the layer-wise application of CIF. Guided by the theoretical analysis, we develop a sample-specific CIF-based contrastive divergence (CD-CIF) algorithm for SBM and a CIF-based iterative projection procedure (IP) for RBM. Both CD-CIF and IP are studied in a series of density estimation experiments.