The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \ldots, u_n \in \mathbb R^d$ that are $\epsilon$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, \ldots, v_n \in \mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1, \ldots, u_n$, the squared distance (to the set of exact solutions) is defined as $\inf_{v} \sum_{i=1}^n \| u_i - v_i \|_2^2$ where the infimum is over the set of exact solutions. Previous results show that the squared distance of any $\epsilon$-nearly solution is at most $O({\rm{poly}}(d,n,\epsilon))$ and there are $\epsilon$-nearly solutions with squared distance at least $\Omega(d\epsilon)$. The fundamental open question is whether the squared distance can be independent of the number of vectors $n$. We answer this question affirmatively by proving that the squared distance of any $\epsilon$-nearly solution is $O(d^{13/2} \epsilon)$. Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any $\epsilon$-nearly solution is $O(d^2 n \epsilon)$. Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an $\epsilon$-nearly solution is $O(d^{5/2} \epsilon)$ when $n$ is large enough and $\epsilon$ is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor changes in various place

Geodesic Convex Analysis of Group Scaling for the Paulsen Problem and the Tensor Normal Model

The framework of scaling problems has recently had much interest in the theoretical computer science community due to its variety of applications, from algebraic complexity to machine learning. In this thesis, our main motivation will be two new applications: the Paulsen problem from frame theory, and the tensor normal model in statistical estimation. In order to give new results for these problems, we provide novel convergence analyses for matrix scaling and tensor scaling. Specifically, we will use the framework of geodesic convex optimization presented in Bürgisser et al. [20] and analyze two sufficient conditions (called strong convexity and pseudorandomness) for fast convergence of the natural gradient flow algorithm in this setting. This allows us to unify and improve many previous results [62], [63], and [36] for special cases of tensor scaling. In the first half of the thesis, we focus on the Paulsen problem where we are given a set of n vectors in d dimensions that ε-approximately satisfy two balance conditions, and asked whether there is a nearby set of vectors that exactly satisfy those balance conditions. This is an important question from frame theory [24] for which very little was known despite considerable attention. We are able to give optimal distance bounds for the Paulsen problem in both the worst-case and the average-case by improving the smoothed analysis approach of Kwok et al. [62]. Specifically, we analyze certain strong convergence conditions for frame scaling, and then show that a random perturbation of the input frame satisfies these conditions and can be scaled to a nearby solution. In the second half of the thesis, we study the matrix and tensor normal models, which are a family of Gaussian distributions on tensor data where the covariance matrix respects this tensor product structure. We are able to generalize our scaling results to higher- order tensors and give error bounds for the maximum likelihood estimator (MLE) of the tensor normal model with a number of samples only a single dimension factor above the existence threshold. This result relies on some spectral properties of random Gaussian tensors shown by Pisier [80]. We also give the first rigorous analysis of the Flip-Flop algorithm, showing that it converges exponentially to the MLE with high probability. This explains the empirical success of this well-studied heuristic for computing the MLE

Stable phase retrieval and perturbations of frames

A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ is said to do phase retrieval if for all distinct vectors $x,y\in H$ the magnitude of the frame coefficients $(|\langle x, x_j\rangle|)_{j\in J}$ and $(|\langle y, x_j\rangle|)_{j\in J}$ distinguish $x$ from $y$ (up to a unimodular scalar). A frame which does phase retrieval is said to do $C$-stable phase retrieval if the recovery of any vector $x\in H$ from the magnitude of the frame coefficients is $C$-Lipschitz. It is known that if a frame does stable phase retrieval then any sufficiently small perturbation of the frame vectors will do stable phase retrieval, though with a slightly worse stability constant. We provide new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are significant in that they are independent of the dimension of the Hilbert space and the number of vectors in the frame.Comment: 14 page

Temperature Dependence Of The Electronic Absorption Spectrum Of NO2

The nitrogen dioxide (NO2) radical is composed of the two most abundant elements in the atmosphere, where it can be formed in a variety of ways including combustion, detonation of energetic materials, and lightning. Relevant also to smog and ozone cycles, together these processes span a wide range of temperatures. Remarkably, high-resolution NO2 electronic absorption spectra have only been reported in a narrow range below about 300 K. Previously, we reported [ J. Phys. Chem. A 2021, 125, 5519−5533 ] the construction of quasi-diabatic potential energy surfaces (PESs) for the lowest four electronic states (X̃, Ã, B̃, and C̃) of NO2. In addition to three-dimensional PESs based on explicitly correlated MRCI(Q)-F12/VTZ-F12 ab initio data, the geometry dependence of each component of the dipoles and transition dipoles was also mapped into fitted surfaces. The multiconfigurational time-dependent Hartree (MCTDH) method was then used to compute the 0 K electronic absorption spectrum (from the ground rovibrational initial state) employing those energy and transition dipole surfaces. Here, in an extension of that work, we report an investigation into the effects of elevated temperature on the spectrum, considering the effects of the population of rotationally and vibrationally excited initial states. The calculations are complemented by new experimental measurements. Spectral contributions from hundreds of rotational states up to N = 20 and from 200 individually-characterized vibrational states were computed. A spectral simulation tool was developed that enables modeling the spectrum at various temperatures─by weighting individual spectral contributions via the partition function, or for pure excited initial states, which can be probed via transient absorption spectroscopy. We validate these results against experimental absorption spectroscopy data at high temperatures, as well as via a new measurement from the (1,0,1) initial vibrational state

On some problems and solutions in frame theory

We de ne frames for a nite dimensional Hilbert space HM as the complete systems in HM: The basic frame families are classi ed such as tight and Parseval frames,equal norm frames and equiangular frames. The statements of some problems that have already become famous in the theory of frames are given. Considerable progress has been made in addressing some of them in recent years.The first author is supported by the RFBR grant 17-01-00138

Sparsity Regularization in Diffuse Optical Tomography

The purpose of this dissertation is to improve image reconstruction in Diffuse Optical Tomography (DOT), a high contrast imaging modality that uses a near infrared light source. Because the scattering and absorption of a tumor varies significantly from healthy tissue, a reconstructed spatial representation of these parameters serves as tomographic image of a medium. However, the high scatter and absorption of the optical source also causes the inverse problem to be severely ill posed, and currently only low resolution reconstructions are possible, particularly when using an unmodulated direct current (DC) source. In this work, the well posedness of the forward problem and possible function space choices are evaluated, and the ill posed nature of the inverse problem is investigated along with the uniqueness issues stemming from using a DC source. Then, to combat the ill posed nature of the problem, a physically motivated additional assumption is made that the target reconstructions have sparse solutions away from simple backgrounds. Because of this, and success with a similar implementation in Electrical Impedance Tomography, a sparsity regularization framework is applied to the DOT inverse problem. The well posedness of this set up is rigorously proved through new regularization theory results and the application of a Hilbert space framework similar to recent work. With the sparsity framework justified in the DOT setting, the inverse problem is solved through a novel smoothed gradient and soft shrinkage algorithm. The effectiveness of the algorithm, and the sparsity regularization of DOT, is evaluated through several numerical simulations using a DC source with comparison to a Levenberg Marquardt implementation and published error results