Clustered Matrix Approximation

Dimensionality reduction techniques can be employed to produce robust, cost-effective predictive models, and to enhance interpretability in exploratory data analysis. However, the models produced by many of these methods are formulated in terms of abstract factors or are too high-dimensional to facilitate insight and fit within low computational budgets.In this paper we explore an alternative approach to interpretable dimensionality reduction. Given a data matrix, we study the following question: are there subsets of variables that can be primarily explained by a single factor?We formulate this challenge as the problem of finding submatrices close to rank one. Despite its potential, this topic has not been sufficiently addressed in the literature, and there exist virtually no algorithms for this purpose that are simultaneously effective, efficient and scalable.We formalize the task as two problems which we characterize in terms of computational complexity, and propose efficient, scalable algorithms with approximation guarantees. Our experiments demonstrate how our approach can produce insightful findings in data, and show our algorithms to be superior to strong baselines. CCS CONCEPTS• Information systems → Data mining.

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“…Furthermore, these methods offer no quality guarantees. Recent works approach similar problems, such as clustered matrix approximation [27].…”

Section: Background and Related Workmentioning

confidence: 99%

Insightful Dimensionality Reduction with Very Low Rank Variable Subsets

Ordozgoiti¹,

Pai

Kołczyńska

2021

Proceedings of the Web Conference 2021

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“…In this subsection, we show how the tensor structure can be used to derive block-structured low-rank factorizations. Previous work in [27,28,33,2] also considered block low-rank factorizations but our work is novel in two different ways: we consider the additional structure present in the matrix and we leverage tensor-based algorithms to exploit this structure. « Fig.…”

Section: Block Structured Factorizationsmentioning

confidence: 99%

Structured Matrix Approximations via Tensor Decompositions

Kilmer¹,

Saibaba²

2021

Preprint

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We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric block Toeplitz). The goal is to uncover additional latent structure that will in turn lead to computationally efficient algorithms when the new structured matrix approximations are employed in the place of the original operator. Our approach has three steps: map the structured matrix to tensors, use tensor compression algorithms, and map the compressed tensors back to obtain two different matrix representations-sum of Kronecker products and block low-rank format. The use of tensor decompositions enables us to uncover latent structure in the problem and leads to compressed representations of the original matrix that can be used efficiently in applications. The resulting matrix approximations are memory efficient, easy to compute with, and preserve the error that is due to the tensor compression in the Frobenius norm. Our framework is quite general. We illustrate the ability of our method to uncover block-low-rank format on structured matrices from two applications: system identification, space-time covariance matrices. In addition, we demonstrate that our approach can uncover sum of structured Kronecker products structure on several matrices from the SuiteSparse collection. Finally, we show that our framework is broad enough to encompass and improve on other related results from the literature, as we illustrate with the approximation of a three-dimensional blurring operator.

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“…By considering matrix paths as continuous/differentiable analogies of numerical linear algebra algorithms in the sense of [7], we will work on the adaptation and application of the results in sections §3 and §4 to the structure-preserving approximation/perturbation of families of structured matrices with some particular spectral behavior in the sense of [1,2,18,17,21].…”

Section: Hints and Future Directionsmentioning

confidence: 99%

Connecting Commuting Normal Matrices

Vides

2017

Preprint

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In this document we study the local path connectivity of sets of m-tuples of commuting normal matrices with some additional geometric constraints in their joint spectra. In particular, given ε > 0 and any fixed but arbitrary m-tuple X ∈ Mn(C) m in the set of m-tuples of pairwise commuting normal matrix contractions, we prove the existence of paths between arbitrary m-tuples in the intersection of the previously mentioned sets of m-tuples in Mn(C) m and the δ-ball B ð (X, δ) centered at X for some δ > 0, with respect to some suitable metric ð in Mn(C) m induced by the operator norm. Two of the key features of these matrix paths is that δ can be chosen independent of n, and that the paths stay in the intersection of B ð (X, ε), and the set pairwise commuting normal matrix contractions with some special geometric structure on their joint spectra.We apply these results to study the local connectivity properties of matrix * -representations of some universal commutative C * -algebras. Some connections with the local connectivity properties of completely positive linear maps on matrix algebras are studied as well.

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Clustered Matrix Approximation

Cited by 7 publications

References 39 publications

Insightful Dimensionality Reduction with Very Low Rank Variable Subsets

Insightful Dimensionality Reduction with Very Low Rank Variable Subsets

Structured Matrix Approximations via Tensor Decompositions

Connecting Commuting Normal Matrices

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