Theory and Applications of the Sum-Of-Squares Technique by Francis Bach, Elisabetta Cornacchia, Luca Pesce, and Giovanni Piccioli

Core Concepts

The authors present an overview of the Sum-of-Squares (SOS) method used in optimization problems to transform non-convex global optimization problems into solvable semidefinite programs.

Abstract

The Sum-of-Squares (SOS) approximation method is a powerful technique used in optimization problems to derive lower bounds on objective functions. By representing functions as sums of squares in feature spaces, the SOS method simplifies complex optimization tasks. The authors explore applications in finite-dimensional and infinite-dimensional feature spaces using reproducing kernels. Additionally, they discuss the utilization of SOS for estimating quantities in information theory like the log-partition function. The content delves into convex duality formulations, sum-of-squares representations of non-negative functions, and tightness of approximations. It also extends the discussion to optimal control problems and connections to information theory through log partition functions and kernel KL divergence.

Stats

h(x) = φ(x)∗Hφ(x) H ∈ Hd, set of Hermitian matrices in Cd×d Proposition 1: H ≽ 0 and H ∈ Hd Proposition 2: If h is a SOS, then h is non-negative. Proposition 3: h(x) = φ(x)∗Hφ(x) is a SOS if there exists H′ ∈ V⊥ such that H - H′ ≽ 0. Kernel KL divergence formula: D(Σp∥Σq) = tr(Σp(log Σp - log Σq))

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Theory and applications of the Sum-Of-Squares technique

by Francis Bach... at arxiv.org 03-12-2024

https://arxiv.org/pdf/2306.16255.pdf

Deeper Inquiries

How does the application of the Sum-of-Squares technique impact computational efficiency in solving optimization problems

The application of the Sum-of-Squares (SOS) technique significantly impacts computational efficiency in solving optimization problems. By representing non-negative functions as sums of squares, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This transformation allows for efficient manipulation of non-negative functions in a computationally tractable manner. The use of SOS relaxations enables the conversion of complex optimization tasks into convex feasibility problems that can be efficiently solved using numerical methods on computers. Additionally, by leveraging subsampling techniques and regularization, the SOS approach reduces the computational complexity associated with handling infinite-dimensional spaces, making it feasible to solve large-scale optimization problems effectively.

What are the implications of using kernel methods for extending feature spaces to infinite dimensions

Utilizing kernel methods for extending feature spaces to infinite dimensions offers several implications and advantages. One key benefit is the ability to represent functions in high-dimensional or even infinite-dimensional spaces through implicit mappings induced by positive definite kernels. By defining a feature map from the input space to a reproducing kernel Hilbert space (RKHS), kernel methods allow for flexible representations that capture complex relationships within data without explicitly computing high-dimensional transformations. This extension facilitates more expressive models capable of capturing intricate patterns and structures present in data while maintaining computational efficiency through kernel tricks like Mercer's theorem. Furthermore, extending feature spaces using reproducing kernels provides a natural way to incorporate domain knowledge or prior information about data into machine learning algorithms. Kernel methods offer a principled framework for nonlinear mapping and enable powerful tools such as support vector machines (SVMs) and Gaussian processes that rely on RKHS properties for effective learning and generalization capabilities across various domains.

How can the concept of Von Neumann divergence enhance our understanding of information theory beyond traditional metrics like KL divergence

The concept of Von Neumann divergence, also known as kernel KL divergence when applied in information theory contexts, enhances our understanding beyond traditional metrics like Kullback-Leibler (KL) divergence by offering several valuable insights: Joint Convexity: The Von Neumann divergence is jointly convex in covariance matrices Σp and Σq derived from feature maps induced by positive definite kernels. Non-Negativity: It ensures non-negativity with equality only when p equals q based on properties related to trace operations on covariance matrices. Universality Property: When utilizing universal kernels generating feature maps ϕ(x), D(Σp∥Σq) becomes zero if and only if p equals q due to its inherent characteristics. Generalizability: Beyond finite sets or Euclidean spaces like Rd, this divergence measure extends applicability to structured objects allowing versatile applications across diverse domains requiring advanced information-theoretic analysis. By incorporating these properties into information theory frameworks, researchers can gain deeper insights into probabilistic inference tasks involving distributions defined over complex datasets represented through higher-dimensional features generated via positive definite kernels' embeddings within RKHS settings.