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Theoretically informed selection of latent activation in autoencoder based recommender systems

Aviad Susman

TL;DR

This work identifies three key mathematical properties that the encoder in an autoencoder should exhibit to improve recommendation accuracy and demonstrates that common activation functions, such as ReLU and tanh, cannot fulfill these properties jointly within a generalizable framework.

Abstract

Autoencoders may lend themselves to the design of more accurate and computationally efficient recommender systems by distilling sparse high-dimensional data into dense lower-dimensional latent representations. However, designing these systems remains challenging due to the lack of theoretical guidance. This work addresses this by identifying three key mathematical properties that the encoder in an autoencoder should exhibit to improve recommendation accuracy: (1) dimensionality reduction, (2) preservation of similarity ordering in dot product comparisons, and (3) preservation of non-zero vectors. Through theoretical analysis, we demonstrate that common activation functions, such as ReLU and tanh, cannot fulfill these properties jointly within a generalizable framework. In contrast, sigmoid-like activations emerge as suitable choices for latent activations. This theoretically informed approach offers a more systematic method for hyperparameter selection, enhancing the efficiency of model design.

Theoretically informed selection of latent activation in autoencoder based recommender systems

TL;DR

This work identifies three key mathematical properties that the encoder in an autoencoder should exhibit to improve recommendation accuracy and demonstrates that common activation functions, such as ReLU and tanh, cannot fulfill these properties jointly within a generalizable framework.

Abstract

Autoencoders may lend themselves to the design of more accurate and computationally efficient recommender systems by distilling sparse high-dimensional data into dense lower-dimensional latent representations. However, designing these systems remains challenging due to the lack of theoretical guidance. This work addresses this by identifying three key mathematical properties that the encoder in an autoencoder should exhibit to improve recommendation accuracy: (1) dimensionality reduction, (2) preservation of similarity ordering in dot product comparisons, and (3) preservation of non-zero vectors. Through theoretical analysis, we demonstrate that common activation functions, such as ReLU and tanh, cannot fulfill these properties jointly within a generalizable framework. In contrast, sigmoid-like activations emerge as suitable choices for latent activations. This theoretically informed approach offers a more systematic method for hyperparameter selection, enhancing the efficiency of model design.

Paper Structure

This paper contains 4 sections, 2 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Desired mathematical behavior
  3. Results
  4. Conclusion

Key Result

Lemma 1

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a function that preserves orthogonality. That is, $\langle u,v\rangle = 0 \implies \langle f(u),f(v)\rangle = 0$ for all $u,v\in\mathbb{R}^n$. Additionally, $f$ preserves non-zero vectors namely $x\neq 0\implies f(x) \neq 0$ for all $x\in\mathbb{R}^n$. Then

Figures (1)

  • Figure 1: An autoencoder used within a recommender system for decreased computational complexity and increased accuracy.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Proposition 1
  • proof