[Paper Review] Cut-norms and spectra of matrices

Vladimir Nikiforov|ArXiv.org|Dec 2, 2009

Graph theory and applications16 references3 citations

TL;DR

This paper resolves an open problem posed by Lovász on bounding spectral differences between graphs using cut-distance by introducing and analyzing two matrix norms—cut-norm ($\|\cdot\|_\square$) and $\|\cdot\|_\boxdot$)—and establishing tight inequalities between these norms and singular/eigenvalues. It proves that the spectral difference between two matrices is bounded by a constant multiple of the square root of their cut-distance, leading to a spectral sampling theorem: almost all principal submatrices of a real symmetric matrix are spectrally similar to the original matrix.

ABSTRACT

One of the aims of this paper is to solve an open problem of Lovasz about relations between graph spectra and cut-distance. The paper starts with several inequalities between two versions of the cut-norm and the two largest singular values of arbitrary complex matrices, exteding, in particular, the well-known graph-theoretical Expander Mixing Lemma and giving a hitherto unknown converse of it. Next, cut-distance is defined for Hermitian matrices, and, separately, for arbitrary complex matrices; using these extensions, we give upper bounds on the difference of corresponding eigenvalues and singular values of two matrices, thus solving the problem of Lovasz. Finally, we deduce a spectral sampling theorem, which informally states that almost all principal submatrices of a real symmetric matrix are spectrally similar to it.

Motivation & Objective

• To resolve Lovász's open problem on bounding spectral differences between graphs using cut-distance.
• To extend the Expander Mixing Lemma and its converse to arbitrary complex matrices using cut-norms.
• To define cut-distance for Hermitian and general complex matrices and derive eigenvalue/singular value difference bounds.
• To establish a spectral sampling theorem showing that most principal submatrices of a real symmetric matrix are spectrally similar to the full matrix.

Proposed method

• Introduces two versions of the cut-norm: $\|A\|_\square = \max_{X,Y} \frac{1}{mn}|\Sigma(A[X,Y])|$ and $\|A\|_\boxdot = \max_{X,Y \neq \emptyset} \frac{1}{\sqrt{|X||Y|}} |\Sigma(A[X,Y])|$.
• Derives tight upper bounds on the largest singular value $\sigma_1(A)$ in terms of $\|A\|_\square$, $\|A\|_\boxdot$, and matrix dimensions, with constants depending on whether $A$ is real or complex.
• Uses the inequality $|\sigma_i(X) - \sigma_i(Y)| \leq \sigma_1(X - Y)$ to relate singular value differences to the cut-norm of the difference matrix.
• Defines a normalized cut-distance $\widehat{\delta}_{\boxminus}(A^{(rk,sk)}, B^{(mk,nk)})$ for scaled matrices and uses it to bound singular value differences asymptotically.
• Applies Cauchy’s Interlacing Theorem and probabilistic bounds to show that random principal submatrices of a real symmetric matrix are spectrally close to the original matrix with high probability.
• Uses approximation arguments and known results (e.g., Borgs et al.) to establish concentration bounds on cut-distance for random submatrices.

Experimental results

Research questions

• RQ1 What is the best possible upper bound on the spectral difference between two graphs in terms of their cut-distance, as posed by Lovász?
• RQ2 Can the Expander Mixing Lemma and its converse be generalized to arbitrary complex matrices using cut-norms?
• RQ3 How can cut-distance be meaningfully extended to Hermitian and general complex matrices to bound eigenvalue and singular value differences?
• RQ4 To what extent are principal submatrices of a real symmetric matrix spectrally similar to the original matrix?
• RQ5 Can the spectral behavior of matrices be controlled via cut-distance in a way that enables sampling-based spectral approximation?

Key findings

• The largest singular value satisfies $\sigma_1(A) \leq 4\sqrt{|A|_\infty \|A\|_\square mn}$ for complex matrices and $\sigma_1(A) \leq 2\sqrt{|A|_\infty \|A\|_\square mn}$ for real matrices, with tightness up to constant factors.
• For any matrix $A$, $\sigma_1(A) \geq \|A\|_\boxdot$, and $\sigma_1(A) \geq \|A\|_\square \sqrt{mn}$, providing a converse to the upper bounds.
• The difference in corresponding singular values of two matrices is bounded by $6\sqrt{\widehat{\delta}_{\boxminus}(A^{(rk,sk)}, B^{(mk,nk)})}$, where $\widehat{\delta}_{\boxminus}$ is a normalized cut-distance.
• For a random principal submatrix $B$ of size $k \times k$ from a real symmetric $n \times n$ matrix $A$, the spectral difference satisfies $\left|\frac{\sigma_i(A)}{\sqrt{n}} - \frac{\sigma_i(B)}{\sqrt{k}}\right| \leq 6\sqrt{\widehat{\delta}_{\boxminus}}$ with high probability.
• With high probability, $\delta_\square(A,B) \leq 10|A|_\infty (\log_2 k)^{-1/2}$, leading to $\left|\frac{\mu_i(A)}{n} - \frac{\mu_i(B)}{k}\right| \leq 30|A|_\infty (\log_2 k)^{-1/4}$ for eigenvalues.
• The spectral sampling theorem holds: almost all principal submatrices of a real symmetric matrix are spectrally similar to the original matrix, as the spectral difference decays as $O((\log k)^{-1/4})$.