Riemannian Optimization on Relaxed Indicator Matrix Manifold (2503.20505v2)

Abstract: The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and prove that this relaxation forms a manifold, which we call the Relaxed Indicator Matrix Manifold (RIM manifold). Based on Riemannian geometry, we develop a Riemannian toolbox for optimization on the RIM manifold. Specifically, we provide several methods of Retraction, including a fast Retraction method to obtain geodesics. We point out that the RIM manifold is a generalization of the double stochastic manifold, and it is much faster than existing methods on the double stochastic manifold, which has a complexity of ( \mathcal{O}(n^3) ), while RIM manifold optimization is ( \mathcal{O}(n) ) and often yields better results. We conducted extensive experiments, including image denoising, with millions of variables to support our conclusion, and applied the RIM manifold to Ratio Cut, we provide a rigorous convergence proof and achieve clustering results that outperform the state-of-the-art methods. Our Code in \href{https://github.com/Yuan-Jinghui/Riemannian-Optimization-on-Relaxed-Indicator-Matrix-Manifold}{here}.

Riemannian Optimization on Relaxed Indicator Matrix Manifold: An Analytical Overview

The presented paper introduces a novel framework for optimizing indicator matrices utilized extensively in clustering, classification, and other machine learning tasks. Characteristically, these indicator matrices pose an NP-hard optimization challenge due to their binary nature coupled with column and row constraints. Addressing this complexity, the paper pioneers a manifold-based relaxation approach termed the Relaxed Indicator Matrix Manifold (RIM manifold). This leads to a more streamlined optimization process over the manifold, advancing beyond traditional methods such as the double stochastic manifold commonly used in related computations.

Key Contributions

The paper introduces a manifold relaxation method wherein the classical double stochastic constraints are expanded through flexible bounds on the column sums of the indicator matrix. This relaxation facilitates working within $\{ X \mid X 1_c = 1_n, l < X^T 1_n < u, X > 0 \}$, constructing an embedded submanifold that maintains both computational feasibility and quality of results, especially when compared to traditional approaches like the Stiefel or double stochastic manifold.

To underpin this novel method, the authors employ the apparatus of Riemannian geometry. They develop a comprehensive Riemannian optimization toolbox tailored for the RIM manifold, equipped with multiple Retraction methods which are fundamental to conducting gradient descent effectively on manifolds. One particular Retraction method stands out due to its capability to efficiently approximate geodesics on the manifold, easing computations and reducing time complexity to $\mathcal{O}(n)$. This dramatic decrease, compared to the $\mathcal{O}(n^3)$ complexity of double stochastic optimization, is pivotal for handling large-scale datasets with millions of variables.

Experimental Validation

Extensive experimental validation underlines the algorithm's efficiency, highlighted by comparisons with state-of-the-art techniques on tasks like image denoising and clustering. Particularly, the RIM manifold enhances the performance of the Ratio Cut model, a frequently employed clustering method, by yielding superior results over competitive alternatives, showcasing lower loss values and expeditious run times. Such performance underscores the practical applicability of the proposed manifold in complex, high-dimensional settings.

Theoretical and Practical Implications

Theoretically, the paper broadens the landscape of possible optimizations on relaxed indicator matrix manifolds by grounding its framework in robust mathematical structures. The generalized nature of the RIM manifold serves as a bridging framework between traditional manifold types, proposing a flexible alternative in data-driven applications. This flexible framework fosters use-case adaptability, especially in scenarios where precise prior knowledge varies or isn't available.

From a practical standpoint, the reduction in computational complexity paves the way for real-time deployment in large-scale enterprise systems and applications, where computational overhead is a decisive factor. Moreover, the adaptability in the definition of $l$ and $u$ allows practitioners to incorporate as much domain-specific knowledge as feasible into their models, optimizing their efficacy further.

Future Directions

Future explorations may explore enhancing the Retraction map even further, possibly finding closed-form geodesics where now approximations exist, facilitating even more efficient computations. Additionally, expanding applications of the RIM framework into areas such as hypergraph partitioning or in unsupervised learning paradigms can highlight potential suite extensions.

In summation, the framework and discoveries introduced lay promising groundwork for further advancements in Riemannian optimization, showcasing meaningful improvements over existing methodologies both in theory and application. The empirical evidence presented substantiates the usability and versatility of the RIM manifold, reiterating its relevance and potential over a broad spectrum of machine learning challenges.