arXiv subjects

Antonios Varvitsiotis

Publications and source records attributed to Antonios Varvitsiotis.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Primal-Dual Framework for Symmetric Cone Programming

In this paper, we introduce a primal-dual algorithmic framework for solving Symmetric Cone Programs (SCPs), a versatile optimization model that unifies and extends Linear, Second-Order Cone (SOCP), and Semidefinite Programming (SDP). Our work generalizes the primal-dual framework for SDPs introduced by Arora and Kale, leveraging a recent extension of the Multiplicative Weights Update method (MWU) to symmetric cones. Going beyond existing works, our framework can handle SOCPs and mixed SCPs, exhibits nearly linear time complexity, and can be effectively parallelized. To illustrate the efficacy of our framework, we employ it to develop approximation algorithms for two geometric optimization problems: the Smallest Enclosing Sphere problem and the Support Vector Machine problem. Our theoretical analyses demonstrate that the two algorithms compute approximate solutions in nearly linear running time and with parallel depth scaling polylogarithmically with the input size. We compare our algorithms against CGAL as well as interior point solvers applied to these problems. Experiments show that our algorithms are highly efficient when implemented on a CPU and achieve substantial speedups when parallelized on a GPU, allowing us to solve large-scale instances of these problems.

math.OC↗

arXiv2023

Entangling capabilities and unitary quantum games

We consider a class of games between two competing players that take turns acting on the same many-body quantum register. Each player can perform unitary operations on the register, and after each one of them acts on the register the energy is measured. Player A aims to maximize the energy while player B to minimize it. This class of zero-sum games has a clear second mover advantage if both players can entangle the same portion of the register. We show, however, that if the first player can entangle a larger number of qubits than the second player (which we refer to as having quantum advantage), then the second mover advantage can be significantly reduced. We study the game for different types of quantum advantage of player A versus player B and for different sizes of the register, in particular, scenarios in which absolutely maximally entangled states cannot be achieved. In this case, we also study the effectiveness of using random unitaries. Last, we consider mixed initial preparations of the register, in which case the player with a quantum advantage can rely on strategies stemming from the theory of ergotropy of quantum batteries.

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arXiv2023

Multiplicative Updates for Online Convex Optimization over Symmetric Cones

We study online convex optimization where the possible actions are trace-one elements in a symmetric cone, generalizing the extensively-studied experts setup and its quantum counterpart. Symmetric cones provide a unifying framework for some of the most important optimization models, including linear, second-order cone, and semidefinite optimization. Using tools from the field of Euclidean Jordan Algebras, we introduce the Symmetric-Cone Multiplicative Weights Update (SCMWU), a projection-free algorithm for online optimization over the trace-one slice of an arbitrary symmetric cone. We show that SCMWU is equivalent to Follow-the-Regularized-Leader and Online Mirror Descent with symmetric-cone negative entropy as regularizer. Using this structural result we show that SCMWU is a no-regret algorithm, and verify our theoretical results with extensive experiments. Our results unify and generalize the analysis for the Multiplicative Weights Update method over the probability simplex and the Matrix Multiplicative Weights Update method over the set of density matrices.

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arXiv2021

Multiplicative updates for symmetric-cone factorizations

Given a matrix $X\in \mathbb{R}^{m\times n}_+$ with non-negative entries, the cone factorization problem over a cone $\mathcal{K}\subseteq \mathbb{R}^k$ concerns computing $\{ a_1,\ldots, a_{m} \} \subseteq \mathcal{K}$ and $\{ b_1,\ldots, b_{n} \} \subseteq~\mathcal{K}^*$ belonging to its dual so that $X_{ij} = \langle a_i, b_j \rangle$ for all $i\in [m], j\in [n]$. Cone factorizations are fundamental to mathematical optimization as they allow us to express convex bodies as feasible regions of linear conic programs. In this paper, we introduce and analyze the symmetric-cone multiplicative update (SCMU) algorithm for computing cone factorizations when $\mathcal{K}$ is symmetric; i.e., it is self-dual and homogeneous. Symmetric cones are of central interest in mathematical optimization as they provide a common language for studying linear optimization over the nonnegative orthant (linear programs), over the second-order cone (second order cone programs), and over the cone of positive semidefinite matrices (semidefinite programs). The SCMU algorithm is multiplicative in the sense that the iterates are updated by applying a meticulously chosen automorphism of the cone computed using a generalization of the geometric mean to symmetric cones. Using an extension of Lieb's concavity theorem and von Neumann's trace inequality to symmetric cones, we show that the squared loss objective is non-decreasing along the trajectories of the SCMU algorithm. Specialized to the nonnegative orthant, the SCMU algorithm corresponds to the seminal algorithm by Lee and Seung for computing Nonnegative Matrix Factorizations.

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arXiv2021

Analysis of Optimization Algorithms via Sum-of-Squares

We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms renders them particularly relevant for applications in machine learning and large-scale data analysis. Relying on sum-of-squares (SOS) optimization, we introduce a hierarchy of semidefinite programs that give increasingly better convergence bounds for higher levels of the hierarchy. Alluding to the power of the SOS hierarchy, we show that the (dual of the) first level corresponds to the Performance Estimation Problem (PEP) introduced by Drori and Teboulle [Math. Program., 145(1):451--482, 2014], a powerful framework for determining convergence rates of first-order optimization algorithms. Consequently, many results obtained within the PEP framework can be reinterpreted as degree-1 SOS proofs, and thus, the SOS framework provides a promising new approach for certifying improved rates of convergence by means of higher-order SOS certificates. To determine analytical rate bounds, in this work we use the first level of the SOS hierarchy and derive new result{s} for noisy gradient descent with inexact line search methods (Armijo, Wolfe, and Goldstein).

math.OC↗

arXiv2021

A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations

Given a matrix $X\in \mathbb{R}_+^{m\times n}$ with nonnegative entries, a Positive Semidefinite (PSD) factorization of $X$ is a collection of $r \times r$-dimensional PSD matrices $\{A_i\}$ and $\{B_j\}$ satisfying $X_{ij}= \mathrm{tr}(A_i B_j)$ for all $\ i\in [m],\ j\in [n]$. PSD factorizations are fundamentally linked to understanding the expressiveness of semidefinite programs as well as the power and limitations of quantum resources in information theory. The PSD factorization task generalizes the Non-negative Matrix Factorization (NMF) problem where we seek a collection of $r$-dimensional nonnegative vectors $\{a_i\}$ and $\{b_j\}$ satisfying $X_{ij}= a_i^\top b_j$, for all $i\in [m],\ j\in [n]$ -- one can recover the latter problem by choosing matrices in the PSD factorization to be diagonal. The most widely used algorithm for computing NMFs of a matrix is the Multiplicative Update algorithm developed by Lee and Seung, in which nonnegativity of the updates is preserved by scaling with positive diagonal matrices. In this paper, we describe a non-commutative extension of Lee-Seung's algorithm, which we call the Matrix Multiplicative Update (MMU) algorithm, for computing PSD factorizations. The MMU algorithm ensures that updates remain PSD by congruence scaling with the matrix geometric mean of appropriate PSD matrices, and it retains the simplicity of implementation that Lee-Seung's algorithm enjoys. Building on the Majorization-Minimization framework, we show that under our update scheme the squared loss objective is non-increasing and fixed points correspond to critical points. The analysis relies on Lieb's Concavity Theorem. Beyond PSD factorizations, we use the MMU algorithm as a primitive to calculate block-diagonal PSD factorizations and tensor PSD factorizations. We demonstrate the utility of our method with experiments on real and synthetic data.

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arXiv2020

Graph isomorphism: Physical resources, optimization models, and algebraic characterizations

In the $(G,H)$-isomorphism game, a verifier interacts with two non-communicating players (called provers) by privately sending each of them a random vertex from either $G$ or $H$, whose aim is to convince the verifier that two graphs $G$ and $H$ are isomorphic. In recent work along with Atserias, Šámal and Severini [Journal of Combinatorial Theory, Series B, 136:89--328, 2019] we showed that a verifier can be convinced that two non-isomorphic graphs are isomorphic, if the provers are allowed to share quantum resources. In this paper we model classical and quantum graph isomorphism by linear constraints over certain complicated convex cones, which we then relax to a pair of tractable convex models (semidefinite programs). Our main result is a complete algebraic characterization of the corresponding equivalence relations on graphs in terms of appropriate matrix algebras. Our techniques are an interesting mix of algebra, combinatorics, optimization, and quantum information.

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arXiv2020

Convergence to Second-Order Stationarity for Non-negative Matrix Factorization: Provably and Concurrently

Non-negative matrix factorization (NMF) is a fundamental non-convex optimization problem with numerous applications in Machine Learning (music analysis, document clustering, speech-source separation etc). Despite having received extensive study, it is poorly understood whether or not there exist natural algorithms that can provably converge to a local minimum. Part of the reason is because the objective is heavily symmetric and its gradient is not Lipschitz. In this paper we define a multiplicative weight update type dynamics (modification of the seminal Lee-Seung algorithm) that runs concurrently and provably avoids saddle points (first order stationary points that are not second order). Our techniques combine tools from dynamical systems such as stability and exploit the geometry of the NMF objective by reducing the standard NMF formulation over the non-negative orthant to a new formulation over (a scaled) simplex. An important advantage of our method is the use of concurrent updates, which permits implementations in parallel computing environments.

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arXiv2019

Local certification of programmable quantum devices of arbitrary high dimensionality

The onset of the era of fully-programmable error-corrected quantum computers will be marked by major breakthroughs in all areas of science and engineering. These devices promise to have significant technological and societal impact, notable examples being the analysis of big data through better machine learning algorithms and the design of new materials. Nevertheless, the capacity of quantum computers to faithfully implement quantum algorithms relies crucially on their ability to prepare specific high-dimensional and high-purity quantum states, together with suitable quantum measurements. Thus, the unambiguous certification of these requirements without assumptions on the inner workings of the quantum computer is critical to the development of trusted quantum processors. One of the most important approaches for benchmarking quantum devices is through the mechanism of self-testing that requires a pair of entangled non-communicating quantum devices. Nevertheless, although computation typically happens in a localized fashion, no local self-testing scheme is known to benchmark high dimensional states and measurements. Here, we show that the quantum self-testing paradigm can be employed to an individual quantum computer that is modelled as a programmable black box by introducing a noise-tolerant certification scheme. We substantiate the applicability of our scheme by providing a family of outcome statistics whose observation certifies that the computer is producing specific high-dimensional quantum states and implementing specific measurements.

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arXiv2019

Robust self-testing of quantum systems via noncontextuality inequalities

Characterising unknown quantum states and measurements is a fundamental problem in quantum information processing. In this Letter, we provide a novel scheme to self-test local quantum systems using non-contextuality inequalities. Our work leverages the graph-theoretic framework for contextuality introduced by Cabello, Severini, and Winter, combined with tools from mathematical optimisation that guarantee the unicity of optimal solutions. As an application, we show that the celebrated Klyachko-Can-Biniciogglu-Shumovsky inequality and its generalisation to contextuality scenarios with odd n-cycle compatibility relations admit robust self-testing.

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arXiv2019

Graph Homomorphisms via Vector Colorings

In this paper we study the existence of homomorphisms $G\to H$ using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number $t \ge 2$ for which there exists an assignment of unit vectors $i\mapsto p_i$ to its vertices such that $\langle p_i, p_j\rangle\le -1/(t-1),$ when $i\sim j$. Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for $n>2r$ the Kneser graph $K_{n:r}$ and the $q$-Kneser graph $qK_{n:r}$ are cores, and furthermore, that for $n/r = n'/r'$ there exists a homomorphism $K_{n:r}\to K_{n':r'}$ if and only if $n$ divides $n'$. In terms of new applications, we show that the even-weight component of the distance $k$-graph of the $n$-cube $H_{n,k}$ is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms $H_{n,k}\to H_{n',k'}$ when $n/k = n'/k'$. Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence's list of strongly regular graphs and found that at least 84% are cores.

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arXiv2018

Correlation matrices, Clifford algebras, and completely positive semidefinite rank

We introduce a notion of matrix valued Gram decompositions for correlation matrices whose study is motivated by quantum information theory. We show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra and thus, their size is exponential in terms of the rank of the correlation matrix. Using this we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank, recently derived in the literature. This fact also underlies and generalizes Tsirelson's seminal lower bound on the local dimension of a quantum system necessary to generate an extreme quantum correlation.

math.OC↗

arXiv2018

Structure of the set of quantum correlators via semidefinite programming

Quantum information leverages properties of quantum behaviors in order to perform useful tasks such as secure communication and randomness certification. Nevertheless, not much is known about the intricate geometric features of the set quantum behaviors. In this paper we study the structure of the set of quantum correlators using semidefinite programming. Our main results are (i) a generalization of the analytic description by Tsirelson-Landau-Masanes, (ii) necessary and sufficient conditions for extremality and exposedness, and (iii) an operational interpretation of extremality in the case of two dichotomic measurements, in terms of self-testing. We illustrate the usefulness of our theoretical findings with many examples and extensive computational work.

quant-ph↗

arXiv2018

Characterising the correlations of prepare-and-measure quantum networks

Prepare-and-measure (P&M) quantum networks are the basic building blocks of quantum communication and cryptography. These networks crucially rely on non-orthogonal quantum encodings to distribute quantum correlations, thus enabling superior communication rates and information-theoretic security. Here, we present a computational toolbox that is able to efficiently characterise the set of input-output probability distributions for any discrete-variable P&M quantum network, assuming only the inner-product information of the quantum encodings. Our toolbox is thus highly versatile and can be used to analyse a wide range of quantum network protocols, including those that employ infinite-dimensional quantum code states. To demonstrate the feasibility and efficacy of our toolbox, we use it to reveal new results in multipartite quantum distributed computing and quantum cryptography. Taken together, these findings suggest that our method may have implications for quantum network information theory and the development of new quantum technologies.

quant-ph↗

arXiv2018

Vector Coloring the Categorical Product of Graphs

A vector $t$-coloring of a graph is an assignment of real vectors $p_1, \ldots, p_n$ to its vertices such that $p_i^Tp_i = t-1$ for all $i=1, \ldots, n$ and $p_i^Tp_j \le -1$ whenever $i$ and $j$ are adjacent. The vector chromatic number of $G$ is the smallest real number $t \ge 1$ for which a vector $t$-coloring of $G$ exists. For a graph $H$ and a vector $t$-coloring $p_1,\ldots,p_n$ of a graph $G$, the assignment $(i,\ell) \mapsto p_i$ is a vector $t$-coloring of the categorical product $G \times H$. It follows that the vector chromatic number of $G \times H$ is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring. Furthermore, we prove a necessary and sufficient condition for when all of the optimal vector colorings of the product can be expressed in terms of the optimal vector colorings of the factors. The vector chromatic number is closely related to the well-known Lovász theta function, and both of these parameters admit formulations as semidefinite programs. This connection to semidefinite programming is crucial to our work and the tools and techniques we develop could likely be of interest to others in this field.

math.CO↗

arXiv2017

Quantum and non-signalling graph isomorphisms

We introduce a two-player nonlocal game, called the $(G,H)$-isomorphism game, where classical players can win with certainty if and only if the graphs $G$ and $H$ are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the $(G,H)$-isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. On the basis of this correspondence, we show that quantum isomorphic graphs are necessarily cospectral. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to produce quantum isomorphic graphs that are nevertheless not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our construction is related to the FGLSS reduction from inapproximability literature, as well as the CFI construction.

quant-ph↗

arXiv2016

Device-independent dimension tests in the prepare-and-measure scenario

Analyzing the dimension of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. In this paper, we consider this problem in the prepare-and-measure scenario. Specifically, we provide a lower bound on the dimension of the prepared quantum systems which is a function that only depends on the measurement statistics. Furthermore, we show that our bound performs well on several examples. {In particular}, we show that our bound provides new insights into the notion of dimension witness, and we also use it to show that the sets of restricted-dimensional prepare-and-measure correlations are not always convex.

quant-ph↗

arXiv2016

Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the non-convexity of certain restricted-dimensional quantum correlations.

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