Identification of Matrix Joint Block Diagonalization

• URL: http://arxiv.org/abs/2011.01111v1
• Date: Mon, 2 Nov 2020 16:42:32 GMT
• Title: Identification of Matrix Joint Block Diagonalization
• Authors: Yunfeng Cai and Ping Li
• Abstract summary: The matrix blind joint block diagonalization problem (BJBDP) plays an important role in independent subspace analysis (ISA) We propose a bi-block diagonalization'' method to solve BJBDP, and establish sufficient conditions under which the method is able to accomplish the task.
• Score: 28.83358353043287
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=A\Sigma_iA^\text{T}$ for all $i$, where $\Sigma_i$'s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis (ISA). This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $\Sigma_i$, especially when there is noise in $C_i$'s. In this paper, we propose a ``bi-block diagonalization'' method to solve BJBDP, and establish sufficient conditions under which the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors' knowledge, existing numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does.

Related papers

• Polynomial Algorithms for Simultaneous Unitary Similarity and Equivalence [0.0]
  We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem.<n>The algorithms have a complexity of $O(pn4)$.<n>This work finds application in Quantum Evolution, Quantum gate design and Canonical Simulation.
  arXiv  Detail & Related papers  (2025-11-08T09:24:59Z)
• Equality condition for a matrix inequality by partial transpose [4.115763274264731]
  We study the equality condition for a matrix inequality generated by partial transpose.<n>We show that $sumK_j=1 A_j otimes B_j$ is locally equivalent to an elegant block-diagonal form.
  arXiv  Detail & Related papers  (2025-08-26T03:42:46Z)
• Effective criteria for entanglement witnesses in small dimensions [0.0]
  We present an effective set of necessary and sufficient criteria for block-positivity of order $4$ sequences over $mathbbC$.<n>The method can be generalized to $mathcalHotimesmathcalH_d$ systems for $d>2$ to provide necessary but not sufficient criterion for block-positivity.
  arXiv  Detail & Related papers  (2025-06-10T23:14:52Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Optimal level set estimation for non-parametric tournament and crowdsourcing problems [49.75262185577198]
  Motivated by crowdsourcing, we consider a problem where we partially observe the correctness of the answers of $n$ experts on $d$ questions. In this paper, we assume that the matrix $M$ containing the probability that expert $i$ answers correctly to question $j$ is bi-isotonic up to a permutation of it rows and columns. We construct an efficient-time algorithm that turns out to be minimax optimal for this classification problem.
  arXiv  Detail & Related papers  (2024-08-27T18:28:31Z)
• S-FABLE and LS-FABLE: Fast approximate block-encoding algorithms for unstructured sparse matrices [0.0]
  The Fast Approximate BLock-Lazy algorithm (FABLE) is a technique to block-encode arbitrary $Ntimes N$ dense matrices into quantum circuits. We describe two modifications of FABLE to efficiently encode sparse matrices.
  arXiv  Detail & Related papers  (2024-01-08T20:57:16Z)
• Block perturbation of symplectic matrices in Williamson's theorem [0.0]
  We show that any symplectic matrix $tildeS$ diagonalizing $A+H$ in Williamson's theorem is of the form $tildeS=S Q+mathcalO(|H|)$. Our results hold even if $A$ has repeated symplectic eigenvalues.
  arXiv  Detail & Related papers  (2023-07-03T14:56:19Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Scalable Differentially Private Clustering via Hierarchically Separated Trees [82.69664595378869]
  We show that our method computes a solution with cost at most $O(d3/2log n)cdot OPT + O(k d2 log2 n / epsilon2)$, where $epsilon$ is the privacy guarantee. Although the worst-case guarantee is worse than that of state of the art private clustering methods, the algorithm we propose is practical.
  arXiv  Detail & Related papers  (2022-06-17T09:24:41Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Chi-square and normal inference in high-dimensional multi-task regression [7.310043452300736]
  The paper proposes chi-square and normal methodologies for the unknown coefficient matrix $B*$ of size $ptimes T$ in a Multi-Task (MT) linear model.
  arXiv  Detail & Related papers  (2021-07-16T11:19:49Z)
• A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations [0.0]
  We describe a non-commutative extension of Lee-Seung's algorithm, which we call the Matrix Multiplicative Update (MMU) algorithm, for computing Positive Semidefinite (PSD) factorizations. We show that under our update scheme the squared loss objective is non-increasing and fixed points correspond to critical points.
  arXiv  Detail & Related papers  (2021-06-01T07:55:09Z)
• Clustering, multicollinearity, and singular vectors [2.055949720959582]
  Let $A$ be a matrix with its pseudo-matrix $Adagger$ and set $S=I-AdaggerA$. We prove that, after re-ordering the columns of $A$, the matrix $S$ has a block-diagonal form where each block corresponds to a set of linearly dependent columns.
  arXiv  Detail & Related papers  (2020-08-07T20:32:34Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.