Locality Regularized Reconstruction: Structured Sparsity and Delaunay Triangulations

• URL: http://arxiv.org/abs/2405.00837v1
• Date: Wed, 1 May 2024 19:56:52 GMT
• Title: Locality Regularized Reconstruction: Structured Sparsity and Delaunay Triangulations
• Authors: Marshall Mueller, James M. Murphy, Abiy Tasissa,
• Abstract summary: Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. In this work we seek $mathbfw$ that forms a local reconstruction of $mathbfy$ by solving a regularized least squares regression problem. We prove that, for all levels of regularization and under a mild condition that the columns of $mathbfX$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$.
• Score: 7.148312060227714
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points $[\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n] = \mathbf{X} \in \mathbb{R}^{d \times n}$ and a vector $\mathbf{y} \in \mathbb{R}^d$, the goal is to find coefficients $\mathbf{w} \in \mathbb{R}^n$ so that $\mathbf{X} \mathbf{w} \approx \mathbf{y}$, subject to some desired structure on $\mathbf{w}$. In this work we seek $\mathbf{w}$ that forms a local reconstruction of $\mathbf{y}$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $\mathbf{X}$ that are close to $\mathbf{y}$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $\mathbf{X}$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$, thereby providing local sparse solutions when $d \ll n$. Under the same condition we also show that for any $\mathbf{y}$ contained in the convex hull of $\mathbf{X}$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $\mathbf{y}$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $\mathbf{X}$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex.

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