Kadison-Singer Problem Research Articles

Fast Algorithms for Maximizing the Minimum Eigenvalue in Fixed Dimension

In the minimum eigenvalue problem, we are given a collection of vectors in Rd, and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix ∑i∈Bvivi⊤. We give a O(nO(dlog⁡(d)/ϵ2))-time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least (1−ϵ) times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.

Improved bounds in Weaver's KSr conjecture for high rank positive semidefinite matrices

Recently Marcus, Spielman and Srivastava proved Weaver's KSr conjecture, which gives a positive solution to the Kadison-Singer problem. In [13,10], Cohen and Brändén independently extended this result to obtain the arbitrary-rank version of Weaver's KSr conjecture. In this paper, we present a new bound in Weaver's KSr conjecture for the arbitrary-rank case. To do that, we introduce the definition of (k,m)-characteristic polynomials and employ it to improve the previous estimate on the largest root of the mixed characteristic polynomials. For the rank-one case, our bound agrees with the Bownik-Casazza-Marcus-Speegle bound when r=2[8] and with the Ravichandran-Leake bound when r>2[21]. For the higher-rank case, we sharpen the previous bounds from Cohen and Brändén.

Paving property for real stable polynomials and strongly Rayleigh processes

One of the equivalent formulations of the Kadison–Singer problem which was resolved in 2013 by Marcus, Spielman and Srivastava, is the “paving conjecture”. In this paper, we first extend this result to real stable polynomials. We prove that for every multi-affine real stable polynomial satisfying a simple condition, it is possible to partition its set of variables to a small number of subsets such that the “restriction” of the polynomial to each subset has small roots. Then, we derive a probabilistic interpretation of this result. We show that there exists a partition of the underlying space of every strongly Rayleigh process into a small number of sets such that the restriction of the point process to each set has “almost independent” points. This result implies that the dependence structure of strongly Rayleigh processes is constrained—a phenomenon that is to be expected in negatively dependent measures. To prove this result, we introduce and study the notion of kernel polynomial for strongly Rayleigh processes. This notion is a natural generalization of the kernel of determinantal processes. We also derive an entropy bound for strongly Rayleigh processes in terms of the roots of the kernel polynomial which is interesting on its own.

Existence and Exactness of Exponential Riesz Sequences and Frames for Fractal Measures

We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ℝd generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the Rϵ-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, when the measure is the middle-third Cantor measure, or more generally for self-similar measures with no-overlap condition, there are always exponential Riesz sequences of maximal possible Beurling dimension.

A New Upper Bound for Sampling Numbers

We provide a new upper bound for sampling numbers (g_n)_{nin mathbb {N}} associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that gn2≤Clog(n)n∑k≥⌊cn⌋σk2,n≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g^2_n \\le \\frac{C\\log (n)}{n}\\sum \\limits _{k\\ge \\lfloor cn \\rfloor } \\sigma _k^2,\\quad n\\ge 2, \\end{aligned}$$\\end{document}where (sigma _k)_{kin mathbb {N}} is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding mathrm {Id}:H(K) rightarrow L_2(D,varrho _D). The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H^s_{text {mix}}(mathbb {T}^d) in L_2(mathbb {T}^d) with s>1/2. We obtain the asymptotic bound gn≤Cs,dn-slog(n)(d-1)s+1/2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g_n \\le C_{s,d}n^{-s}\\log (n)^{(d-1)s+1/2}, \\end{aligned}$$\\end{document}which improves on very recent results by shortening the gap between upper and lower bound to sqrt{log (n)}. The result implies that for dimensions d>2 any sparse grid sampling recovery method does not perform asymptotically optimal.

A non-diagonalizable pure state

We construct a pure state on the C*-algebra [Formula: see text] of all bounded linear operators on [Formula: see text], which is not diagonalizable [i.e., it is not of the form [Formula: see text] for any orthonormal basis [Formula: see text] of [Formula: see text] and an ultrafilter u on N]. This constitutes a counterexample to Anderson's conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah, and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison-Singer problem due to A. Marcus, D. Spielman, and N. Srivastava that the restriction of our pure state to any atomic masa [Formula: see text] of diagonal operators with respect to an orthonormal basis [Formula: see text] is not multiplicative on [Formula: see text].

Mixed determinants and the Kadison–Singer problem

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison–Singer problem to prove improved paving estimates. Working with Anderson’s paving formulation of Kadison–Singer instead of Weaver’s vector balancing version, we show that the machinery of interlacing polynomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic polynomials turn out to be related to the so called “mixed determinants” that have been previously studied in a different context by Borcea and Branden. This approach allows us to show that any projection with diagonal entries 1/2 can be 4 paved, yielding improvements over the best known current estimates of 12. This approach also allows us to show that any projection with diagonal entries strictly less than 1/4 can be 2 paved, matching recent results of Bownik, Casazza, Marcus and Speegle. We also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have several pleasing combinatorial properties.

Sparsification: In Theory and Practice

The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixtieth anniversary of the problem, we revisit the problem in its original formulation and also explore its transition to a result with wide ranging applications. We also describe how the notion of “sparsification” transcended various fields and how this notion led to resolution of the problem.

On syndetic Riesz sequences

Applying the solution to the Kadison–Singer problem, we show that every subset S of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials {eiλx}λ∈Λ such that Λ ⊂ ℤ is a set with gaps between consecutive elements bounded by $$\frac{C}{|\mathcal{S}|}$$ . In the case when $$\mathcal{S}$$ is an open set we demonstrate, using quasicrystals, how such Λ can be deterministically constructed.

OPTIMAL SPLITTING OF PARSEVAL FRAMES USING WALSH MATRICES

In 2014 Adam Marcus, Daniel Spielman and Nikhil Srivastava used random vectors to prove a key discrepancy theorem and in so doing gave a positive answer to the long-standing Kadison-Singer Problem. In this paper we use Walsh matrices to construct a class of natural frames in Euclidean space and discuss how these frames relate to the key discrepancy theorem.

Lyapunov’s theorem for continuous frames

Akemann and Weaver (2014) have shown a remarkable extension of Weaver’s K S r KS_r Conjecture (2004) in the form of approximate Lyapunov’s theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov’s theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem.

Approximating Matrices and Convex Bodies

We show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. Our proof uses the recent solution of the Kadison-Singer problem. We also develop a sort of tensorization technique to deal with constraint approximation problems. As an application, we provide a sparsification result with equal weights and an optimal approximate John's decomposition for non-symmetric convex bodies. This enables us to show that any convex body in $\mathbb{R}^n$ is arbitrary close to another one having $O(n)$ contact points and fills the gap left in the literature after the results of Rudelson and Srivastava by completely answering the problem. As a consequence, we also show that the method developed by Gu\'edon, Gordon and Meyer to establish the isomorphic Dvoretzky theorem yields to the best known result once we inject our improvements.

Partitions of equiangular tight frames

We present a new efficient algorithm to construct partitions of a special class of equiangular tight frames (ETFs) that satisfy the operator norm bound established by a theorem of Marcus, Spielman, and Srivastava (MSS), which they proved as a corollary yields a positive solution to the Kadison–Singer problem. In particular, we prove that certain diagonal partitions of complex ETFs generated by recursive skew-symmetric conference matrices yield a refinement of the MSS bound. Moreover, we prove that all partitions of ETFs whose largest subset has cardinality three or less also satisfy the MSS bound.

Graphs, Vectors, and Matrices

This survey accompanies the Josiah Williard Gibbs Lecture that I gave at the 2016 Joint Mathematics Meetings. It provides an introduction to three topics: algebraic and spectral graph theory, the sparsification of graphs, and the recent resolution of the Kadison–Singer Problem. The first and third are connected by the second.

Improved bounds in Weaver and Feichtinger conjectures

Abstract We sharpen the constant in the KS 2 {\mathrm{KS}_{2}} Conjecture of Weaver [31] that was given by Marcus, Spielman and Srivastava [28] in their solution of the Kadison–Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger Conjecture.

Reconciling quantum physics with math

Mathematicians explore the root of many problems in developing a proof for the Kadison-Singer problem.

Detecting Fourier Subspaces

Let G be a finite abelian group. We examine the discrepancy between subspaces of $$l^2(G)$$ which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier subspace whose dimension is small compared to $$|G| = \mathrm{dim}\left( l^2(G)\right) $$ tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison–Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is a standard subspace which is essentially indistinguishable from its orthogonal complement.

A II1 Factor Approach to the Kadison–Singer Problem

We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra $\ell^\infty\Bbb N\subset \Cal B(\ell^2\Bbb N)$ have unique state extensions to $\Cal B(\ell^2\Bbb N)$, is equivalent to a similar statement in II$_1$ factor framework, concerning the ultrapower inclusion $D^\omega \subset R^\omega$, where $D$ is the Cartan subalgebra of the hyperfinite II$_1$ factor $R$, and $\omega$ is a free ultraflter. While we do not settle the problem in this latter form, we prove that if $A$ is any singular maximal abelian subalgebra of $R$, then the inclusion $A^\omega \subset R^\omega$ does satisfy the Kadison-Singer property.

A Lyapunov-type theorem from Kadison-Singer

Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem by showing that if u_1, ..., u_m are column vectors in C^d such that \sum u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so that \sum_{i \in S} u_iu_i^* is approximately (1/2)I, with the approximation good in operator norm to order \epsilon^{1/2} where \epsilon = \max \|u_i\|^2. We extend their result to show that every linear combination of the matrices u_iu_i^* with coefficients in [0,1] can be approximated in operator norm to order \epsilon^{1/8} by a matrix of the form \sum_{i \in S} u_iu_i^*.

English

The Kadison-Singer Problem (K-S) has expanded since 1959 to a very large number of equivalent problems in various fields. In the present paper we will introduce the notion of weak paveability for positive elements of a von Neumann algebra M. This new formulation implies the traditional version of paveability iff K-S is affirmed. We show that the set of weakly paveable positive elements of $M^+$ is open and norm dense in $M^+$. Finally, we show that to affirm K-S it suffices to show that projections with compact diagonal are weakly paveable. Therefore weakly paveable matrices will either contain a counterexample, or else weak paveability must be an easier route to affirming K-S.