Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method (1407.1543v2)

Abstract: We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form [ y = Ax + e, ] where $x$ is a random vector in $\mathbb R^m$ with at most $\tau m$ nonzero coordinates, and $e$ is a random noise vector in $\mathbb R^n$ with bounded magnitude. For the case $m=O(n)$, our algorithm recovers every column of $A$ within arbitrarily good constant accuracy in time $m^{O(\log m/\log(\tau^{-1}))}$, in particular achieving polynomial time if $\tau = m^{-\delta}$ for any $\delta>0$, and time $m^{O(\log m)}$ if $\tau$ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most $\sqrt{n}$ nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser $x$. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor $T$, given access to a tensor $T'$ that is $\tau$-close to $T$ in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of $T$ and $T'$ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems.

Overview of the Paper on Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method

This paper by Barak, Kelner, and Steurer introduces a novel approach to the dictionary learning problem, also referred to as sparse coding, utilizing the Sum-of-Squares (SOS) method for tensor decomposition. The authors address the challenge of recovering an unknown dictionary matrix $A$, given sample vectors that are generated as a linear combination of basis vectors from $A$ and contaminated by noise. The focus of the paper is on expanding the applicability of efficient recovery algorithms to cases where prior approaches falter due to denser input distributions.

Key Methodological Contributions

1. Algorithm for Noisy Tensor Decomposition: The paper proposes an algorithm that successfully performs tensor decomposition despite significant spectral noise interference. It leverages the SOS semidefinite programming hierarchy, allowing it to operate in a regime where existing approaches are ineffective due to the presence of substantial noise levels.
2. Recovery under Dense Inputs: A central result is the ability of the proposed algorithm to recover dictionary columns when inputs are considerably denser compared to prior methods. While previous algorithms required input vectors to have sparsity restrictions like at most $\sqrt{n}$ nonzero coordinates, this work demonstrates effectiveness for a much larger range of density.
3. Algorithm Efficiency: The authors report that their SOS-based algorithm operates in polynomial time under certain sparsity conditions and display quasipolynomial time efficiency for more general settings. Hence, they push the boundaries of feasible dictionary learning beyond the traditional sparsity limitations.

Theoretical Implications

The paper has substantial theoretical implications for unsupervised learning applications. By fundamentally enhancing the capabilities - particularly the robustness to noise and input density - of dictionary learning and tensor decomposition algorithms, it paves the way for handling real-world datasets with greater complexity.

Numerical Results and Validity

The algorithm demonstrates consistent recovery of column vectors within specified bounds, providing robust performance across theoretical simulations. These results bolster the credibility of using SOS methodology in tensor decomposition scenarios where strong noise and dense inputs previously rendered other algorithms ineffective.

Practical Applications and Speculative Futures

Considering the mathematical models and results demonstrated, potential applications span across neuroscience, machine learning, and signal processing. The strengthened recovery guarantees are particularly significant given the imperfection inherent in practical datasets. The future might see the SOS framework extended to further unsupervised learning scenarios, possibly revealing new prospects and insights in complex systems modeling.

Conclusion

Barak, Kelner, and Steurer’s work convincingly showcases the utility of the Sum-of-Squares optimization in the domain of dictionary learning, representing an advancement in handling previously challenging input conditions. Their research sets the stage for further exploration and application in both theoretical and practical realms and anticipates further methodological advancements leveraging such powerful tools.

The paper is a testament to the potential inherent in methodologically rigorous optimization approaches, and it offers a significant step forward in solving complex machine learning problems where traditional methods reach their limitations.