Exponential algorithmic speedup by quantum walk

We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification

On some varieties associated with trees

This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be smooth and to admit sometimes free actions of algebraic tori. Some results are obtained on their number of points over finite fields and on their cohomology.Comment: 37 pages, 7 figure

Consistency and convergence rate of phylogenetic inference via regularization

It is common in phylogenetics to have some, perhaps partial, information about the overall evolutionary tree of a group of organisms and wish to find an evolutionary tree of a specific gene for those organisms. There may not be enough information in the gene sequences alone to accurately reconstruct the correct "gene tree." Although the gene tree may deviate from the "species tree" due to a variety of genetic processes, in the absence of evidence to the contrary it is parsimonious to assume that they agree. A common statistical approach in these situations is to develop a likelihood penalty to incorporate such additional information. Recent studies using simulation and empirical data suggest that a likelihood penalty quantifying concordance with a species tree can significantly improve the accuracy of gene tree reconstruction compared to using sequence data alone. However, the consistency of such an approach has not yet been established, nor have convergence rates been bounded. Because phylogenetics is a non-standard inference problem, the standard theory does not apply. In this paper, we propose a penalized maximum likelihood estimator for gene tree reconstruction, where the penalty is the square of the Billera-Holmes-Vogtmann geodesic distance from the gene tree to the species tree. We prove that this method is consistent, and derive its convergence rate for estimating the discrete gene tree structure and continuous edge lengths (representing the amount of evolution that has occurred on that branch) simultaneously. We find that the regularized estimator is "adaptive fast converging," meaning that it can reconstruct all edges of length greater than any given threshold from gene sequences of polynomial length. Our method does not require the species tree to be known exactly; in fact, our asymptotic theory holds for any such guide tree.Comment: 34 pages, 5 figures. To appear on The Annals of Statistic

Spectral problem on graphs and L-functions

The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf Q}_p)$. As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the finite subgraph-the reduced graph containing all loops of the generalized tree. We study the spectral problem on these graphs, for which we introduce the notion of spherical functions-eigenfunctions of a discrete Laplace operator acting on the graph. We define the S-matrix and prove its unitarity. We present a proof of the Hashimoto-Bass theorem expressing L-function of any finite (reduced) graph via determinant of a local operator $\Delta(u)$ acting on this graph and relate the S-matrix determinant to this L-function thus obtaining the analogue of the Selberg trace formula. The discrete spectrum points are also determined and classified by the L-function. Numerous examples of L-function calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur

Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

On the Computational Complexity of Vertex Integrity and Component Order Connectivity

The Weighted Vertex Integrity (wVI) problem takes as input an $n$-vertex graph $G$, a weight function $w:V(G)\to\mathbb{N}$, and an integer $p$. The task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ plus the weight of a heaviest component of $G-X$ is at most $p$. Among other results, we prove that: (1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight $1$; (2) wVI can be solved in $O(p^{p+1}n)$ time; (3) wVI admits a kernel with at most $p^3$ vertices. Result (1) refutes a conjecture by Ray and Deogun and answers an open question by Ray et al. It also complements a result by Kratsch et al., stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an $n$-vertex graph $G$, a weight function $w:V(G)\to \mathbb{N}$, and two integers $k$ and $l$, and the task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ is at most $k$ and the weight of a heaviest component of $G-X$ is at most $l$. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We prove, among other results, that: (4) wCOC can be solved in $O(\min\{k,l\}\cdot n^3)$ time on interval graphs, while the unweighted version can be solved in $O(n^2)$ time on this graph class; (5) wCOC is W[1]-hard on split graphs when parameterized by $k$ or by $l$; (6) wCOC can be solved in $2^{O(k\log l)} n$ time; (7) wCOC admits a kernel with at most $kl(k+l)+k$ vertices. We also show that result (6) is essentially tight by proving that wCOC cannot be solved in $2^{o(k \log l)}n^{O(1)}$ time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the conference proceedings of ISAAC 201

Graph Priors, Optimal Transport, and Deep Learning in Biomedical Discovery

Recent advances in biomedical data collection allows the collection of massive datasets measuring thousands of features in thousands to millions of individual cells. This data has the potential to advance our understanding of biological mechanisms at a previously impossible resolution. However, there are few methods to understand data of this scale and type. While neural networks have made tremendous progress on supervised learning problems, there is still much work to be done in making them useful for discovery in data with more difficult to represent supervision. The flexibility and expressiveness of neural networks is sometimes a hindrance in these less supervised domains, as is the case when extracting knowledge from biomedical data. One type of prior knowledge that is more common in biological data comes in the form of geometric constraints. In this thesis, we aim to leverage this geometric knowledge to create scalable and interpretable models to understand this data. Encoding geometric priors into neural network and graph models allows us to characterize the models’ solutions as they relate to the fields of graph signal processing and optimal transport. These links allow us to understand and interpret this datatype. We divide this work into three sections. The first borrows concepts from graph signal processing to construct more interpretable and performant neural networks by constraining and structuring the architecture. The second borrows from the theory of optimal transport to perform anomaly detection and trajectory inference efficiently and with theoretical guarantees. The third examines how to compare distributions over an underlying manifold, which can be used to understand how different perturbations or conditions relate. For this we design an efficient approximation of optimal transport based on diffusion over a joint cell graph. Together, these works utilize our prior understanding of the data geometry to create more useful models of the data. We apply these methods to molecular graphs, images, single-cell sequencing, and health record data