Projections and the Kadison-Singer Problem

The Kadison-Singer problem in discrepancy theory, II

We apply Srivastava's spectral sparsification technique to a vector balancing version of the Kadison-Singer problem. The result is a one-sided version of the conjectured solution.

The Kadison-Singer problem in discrepancy theory

We give a combinatorial form of the Kadison-Singer problem, a famous problem in C*-algebra. This combinatorial problem, which has several minor variations, is a discrepancy question about vectors in C^n. Some partial results can be easily deduced from known facts in discrepancy theory.

Hyperbolic polynomials and the Kadison-Singer problem

Recently Marcus, Spielman and Srivastava gave a spectacular proof of a theorem which implies a positive solution to the Kadison-Singer problem via Weaver's $KS_r$ conjecture. We extend this theorem to the realm of hyperbolic polynomials and hyperbolicity cones, as well as to arbitrary ranks. We also sharpen the theorem by providing better bounds, which imply better bounds in Weaver's $KS_r$ conjecture for each $r>2$. For $r=2$ our bound agrees with Bownik et al.

Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem

We use the method of interlacing families of polynomials introduced to prove two theorems known to imply a positive solution to the Kadison--Singer problem. The first is Weaver's conjecture $KS_{2}$ \cite{weaver}, which is known to imply Kadison--Singer via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza, et al., of Anderson's original paving conjecture(s), for which we are able to compute explicit paving bounds. The proof involves an analysis of the largest roots of a family of polynomials that we call the "mixed characteristic polynomials" o…

The Solution of the Kadison-Singer Problem

These lecture notes are meant to accompany two lectures given at the CDM 2016 conference, about the Kadison-Singer Problem. They are meant to complement the survey by the same authors (along with Spielman) which appeared at the 2014 ICM. In the first part of this survey we will introduce the Kadison-Singer problem from two perspectives ($C^*$ algebras and spectral graph theory) and present some examples showing where the difficulties in solving it lie. In the second part we will develop the framework of interlacing families of polynomials, and show how it is used to solve the problem. None of …

Minimal sequences and the Kadison-Singer problem

The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We consider the special case: Feichtinger's conjecture for exponentials and prove that the set of projections onto a measurable subset of the circle group of the set of exponential functions equals a union of a finite number of Reisz sequences if and only if there e…

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 ε^{1/2} where ε= \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 ε^{1/8} by a matrix of the form \sum_{i \in S} u_iu_i^*.

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 polyomials 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 carefully studied by Borcea and Branden. This technique allows us to show that any projection with diagona…

Approximating matrices and convex bodies through Kadison-Singer

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 sh…

Lyapunov's Theorem for continuous frames

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's $KS_r$ 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 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.

秩一矩阵差异与算法化 Kadison--Singer

Rank-One Matrix Discrepancy and Algorithmic Kadison--Singer

本文提出确定性多项式时间算法，解决秩一矩阵符号选择问题，并由此给出 Kadison--Singer 问题在 Weaver 差异理论表述下的算法，常数通用。

从自由概率到矩阵差异性的漫步 II：Weaver 问题与 Kadison-Singer 猜想

A Walk From Free Probability to Matrix Discrepancy II: Weaver's Problem and the Kadison-Singer Conjecture

本文提出一个多项式时间确定性算法，通过超立方体漫步和自由概率势函数，解决 Weaver 问题并实现差异性至多 $35\sqrt\varepsilon$，为 Kadison-Singer 猜想提供构造性证明。

Improved bounds in Weaver and Feichtinger Conjectures

We sharpen the constant in the $KS_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} 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.

Kadison--Singer 划分与 Bilu--Linial 图符号的多项式时间算法

Kadison--Singer partitions and Bilu--Linial graph signings in polynomial time

本文提出谱偏差的两个多项式时间算法：确定性舍入定理实现Kadison--Singer半划分与图符号，以及Las Vegas算法求解Bilu--Linial符号问题，达到接近最优的谱界。

A counterexample to a conjecture of Akemann and Anderson

Akemann and Anderson made a conjecture about ``paving'' projections in finite dimensional matrix algebras which, if true, would settle the well-known Kadison-Singer problem. We falsify their conjecture by an explicit seqence of counterexamples.

Asymptotically Optimal Multi-Paving

Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions $A^{(1)}, \cdots, A^{(k)} \in M_n(\mathbb{C})$ and $ε> 0$, one may find a paving $X_1 \amalg \cdots \amalg X_r = …

The Bourgain-Tzafriri conjecture and concrete constructions of non-pavable projections

It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the proofs of this fact are existence proofs. We will use variations of the discrete Fourier Transform matrices to construct concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense. In 1989, Bourgain and Tzafriri showed that the class of zero diagonal matrices with small entries (on the ord…

An analytical Lieb-Sokal lemma

A polynomial $p \in \mathbb{R}[z_1, \cdots, z_n]$ is called real stable if it is non-vanishing whenever all the variables take values in the upper half plane. A well known result of Elliott Lieb and Alan Sokal states that if $p$ and $q$ are $n$ variate real stable polynomials, then the polynomial $q(\partial)p := q(\partial_1, \cdots, \partial_n)p$, is real stable as well. In this paper, we prove analytical estimates on the locations on the zeroes of the real stable polynomial $q(\partial)p$ in the case when both $p$ and $q$ are multiaffine, an important special case, owing to connections to n…

Paving over arbitrary MASAs in von Neumann algebras

We consider a paving property for a maximal abelian *-subalgebra (MASA) $A$ in a von Neumann algebra $M$, that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison-Singer paving). If $A$ is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion $A^ω\subset M^ω$. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use [MSS13] to check this for all MASAs in $\mathcal B(\ell^2 \mathbb N)$, all Cartan subalgebras in amenable von Neumann algebras and in g…

The random paving property for uniformly bounded matrices

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systemati…