17,197 research outputs found
Sparse Plus Low Rank Matrix Decomposition: A Discrete Optimization Approach
We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the
problem of decomposing a corrupted data matrix into a sparse matrix of
perturbations plus a low-rank matrix containing the ground truth. SLR is a
fundamental problem in Operations Research and Machine Learning which arises in
various applications, including data compression, latent semantic indexing,
collaborative filtering, and medical imaging. We introduce a novel formulation
for SLR that directly models its underlying discreteness. For this formulation,
we develop an alternating minimization heuristic that computes high-quality
solutions and a novel semidefinite relaxation that provides meaningful bounds
for the solutions returned by our heuristic. We also develop a custom
branch-and-bound algorithm that leverages our heuristic and convex relaxations
to solve small instances of SLR to certifiable (near) optimality. Given an
input -by- matrix, our heuristic scales to solve instances where
in minutes, our relaxation scales to instances where in
hours, and our branch-and-bound algorithm scales to instances where in
minutes. Our numerical results demonstrate that our approach outperforms
existing state-of-the-art approaches in terms of rank, sparsity, and
mean-square error while maintaining a comparable runtime
Robust Social Welfare Maximization via Information Design in Linear-Quadratic-Gaussian Games
Information design in an incomplete information game includes a designer with
the goal of influencing players' actions through signals generated from a
designed probability distribution so that its objective function is optimized.
We consider a setting in which the designer has partial knowledge on agents'
utilities. We address the uncertainty about players' preferences by formulating
a robust information design problem against worst case payoffs. If the players
have quadratic payoffs that depend on the players' actions and an unknown
payoff-relevant state, and signals on the state that follow a Gaussian
distribution conditional on the state realization, then the information design
problem under quadratic design objectives is a semidefinite program (SDP).
Specifically, we consider ellipsoid perturbations over payoff coefficients in
linear-quadratic-Gaussian (LQG) games. We show that this leads to a tractable
robust SDP formulation. Numerical studies are carried out to identify the
relation between the perturbation levels and the optimal information
structures
Computation of Green's function by local variational quantum compilation
Computation of the Green's function is crucial to study the properties of
quantum many-body systems such as strongly correlated systems. Although the
high-precision calculation of the Green's function is a notoriously challenging
task on classical computers, the development of quantum computers may enable us
to compute the Green's function with high accuracy even for
classically-intractable large-scale systems. Here, we propose an efficient
method to compute the real-time Green's function based on the local variational
quantum compilation (LVQC) algorithm, which simulates the time evolution of a
large-scale quantum system using a low-depth quantum circuit constructed
through optimization on a smaller-size subsystem. Our method requires shallow
quantum circuits to calculate the Green's function and can be utilized on both
near-term noisy intermediate-scale and long-term fault-tolerant quantum
computers depending on the computational resources we have. We perform a
numerical simulation of the Green's function for the one- and two-dimensional
Fermi-Hubbard model up to sites lattice (32 qubits) and demonstrate
the validity of our protocol compared to a standard method based on the Trotter
decomposition. We finally present a detailed estimation of the gate count for
the large-scale Fermi-Hubbard model, which also illustrates the advantage of
our method over the Trotter decomposition.Comment: 22 pages, 13 figure
Quantifying the Expressive Capacity of Quantum Systems: Fundamental Limits and Eigentasks
The expressive capacity of quantum systems for machine learning is limited by
quantum sampling noise incurred during measurement. Although it is generally
believed that noise limits the resolvable capacity of quantum systems, the
precise impact of noise on learning is not yet fully understood. We present a
mathematical framework for evaluating the available expressive capacity of
general quantum systems from a finite number of measurements, and provide a
methodology for extracting the extrema of this capacity, its eigentasks.
Eigentasks are a native set of functions that a given quantum system can
approximate with minimal error. We show that extracting low-noise eigentasks
leads to improved performance for machine learning tasks such as
classification, displaying robustness to overfitting. We obtain a tight bound
on the expressive capacity, and present analyses suggesting that correlations
in the measured quantum system enhance learning capacity by reducing noise in
eigentasks. These results are supported by experiments on superconducting
quantum processors. Our findings have broad implications for quantum machine
learning and sensing applications.Comment: 7 + 21 pages, 4 + 12 figures, 1 tabl
An Analysis Tool for Push-Sum Based Distributed Optimization
The push-sum algorithm is probably the most important distributed averaging
approach over directed graphs, which has been applied to various problems
including distributed optimization. This paper establishes the explicit
absolute probability sequence for the push-sum algorithm, and based on which,
constructs quadratic Lyapunov functions for push-sum based distributed
optimization algorithms. As illustrative examples, the proposed novel analysis
tool can improve the convergence rates of the subgradient-push and stochastic
gradient-push, two important algorithms for distributed convex optimization
over unbalanced directed graphs. Specifically, the paper proves that the
subgradient-push algorithm converges at a rate of for general
convex functions and stochastic gradient-push algorithm converges at a rate of
for strongly convex functions, over time-varying unbalanced directed
graphs. Both rates are respectively the same as the state-of-the-art rates of
their single-agent counterparts and thus optimal, which closes the theoretical
gap between the centralized and push-sum based (sub)gradient methods. The paper
further proposes a heterogeneous push-sum based subgradient algorithm in which
each agent can arbitrarily switch between subgradient-push and
push-subgradient. The heterogeneous algorithm thus subsumes both
subgradient-push and push-subgradient as special cases, and still converges to
an optimal point at an optimal rate. The proposed tool can also be extended to
analyze distributed weighted averaging.Comment: arXiv admin note: substantial text overlap with arXiv:2203.16623,
arXiv:2303.1706
Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients
In this article, we build on previous work to present an optimization
algorithm for nonlinearly constrained multi-objective optimization problems.
The algorithm combines a surrogate-assisted derivative-free trust-region
approach with the filter method known from single-objective optimization.
Instead of the true objective and constraint functions, so-called fully linear
models are employed, and we show how to deal with the gradient inexactness in
the composite step setting, adapted from single-objective optimization as well.
Under standard assumptions, we prove convergence of a subset of iterates to a
quasi-stationary point and if constraint qualifications hold, then the limit
point is also a KKT-point of the multi-objective problem
An iterative warping and clustering algorithm to estimate multiple wave-shape functions from a nonstationary oscillatory signal
Nonsinusoidal oscillatory signals are everywhere. In practice, the
nonsinusoidal oscillatory pattern, modeled as a 1-periodic wave-shape function
(WSF), might vary from cycle to cycle. When there are finite different WSFs,
, so that the WSF jumps from one to another suddenly, the
different WSFs and jumps encode useful information. We present an iterative
warping and clustering algorithm to estimate from a
nonstationary oscillatory signal with time-varying amplitude and frequency, and
hence the change points of the WSFs. The algorithm is a novel combination of
time-frequency analysis, singular value decomposition entropy and vector
spectral clustering. We demonstrate the efficiency of the proposed algorithm
with simulated and real signals, including the voice signal, arterial blood
pressure, electrocardiogram and accelerometer signal. Moreover, we provide a
mathematical justification of the algorithm under the assumption that the
amplitude and frequency of the signal are slowly time-varying and there are
finite change points that model sudden changes from one wave-shape function to
another one.Comment: 39 pages, 11 figure
A hybrid quantum algorithm to detect conical intersections
Conical intersections are topologically protected crossings between the
potential energy surfaces of a molecular Hamiltonian, known to play an
important role in chemical processes such as photoisomerization and
non-radiative relaxation. They are characterized by a non-zero Berry phase,
which is a topological invariant defined on a closed path in atomic coordinate
space, taking the value when the path encircles the intersection
manifold. In this work, we show that for real molecular Hamiltonians, the Berry
phase can be obtained by tracing a local optimum of a variational ansatz along
the chosen path and estimating the overlap between the initial and final state
with a control-free Hadamard test. Moreover, by discretizing the path into
points, we can use single Newton-Raphson steps to update our state
non-variationally. Finally, since the Berry phase can only take two discrete
values (0 or ), our procedure succeeds even for a cumulative error bounded
by a constant; this allows us to bound the total sampling cost and to readily
verify the success of the procedure. We demonstrate numerically the application
of our algorithm on small toy models of the formaldimine molecule
(\ce{H2C=NH}).Comment: 15 + 10 pages, 4 figure
Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals
We present an analysis and numerical study of an optimal control problem for
the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a
crucial component in modern technology. They exhibit long range orientational
order in their nematic phase, which is represented by a tensor-valued (spatial)
order parameter . Equilibrium LC states correspond to functions
that (locally) minimize an LdG energy functional. Thus, we consider an
-gradient flow of the LdG energy that allows for finding local minimizers
and leads to a semi-linear parabolic PDE, for which we develop an optimal
control framework. We then derive several a priori estimates for the forward
problem, including continuity in space-time, that allow us to prove existence
of optimal boundary and external ``force'' controls and to derive optimality
conditions through the use of an adjoint equation. Next, we present a simple
finite element scheme for the LdG model and a straightforward optimization
algorithm. We illustrate optimization of LC states through numerical
experiments in two and three dimensions that seek to place LC defects (where
) in desired locations, which is desirable in applications.Comment: 26 pages, 9 figure
Multidimensional adaptive order GP-WENO via kernel-based reconstruction
This paper presents a fully multidimensional kernel-based reconstruction
scheme for finite volume methods applied to systems of hyperbolic conservation
laws, with a particular emphasis on the compressible Euler equations.
Non-oscillatory reconstruction is achieved through an adaptive order weighted
essentially non-oscillatory (WENO-AO) method cast into a form suited to
multidimensional stencils and reconstruction. A kernel-based approach inspired
by Gaussian process (GP) modeling is presented here. This approach allows the
creation of a scheme of arbitrary order with simply defined multidimensional
stencils and substencils. Furthermore, the fully multidimensional nature of the
reconstruction allows a more straightforward extension to higher spatial
dimensions and removes the need for complicated boundary conditions on
intermediate quantities in modified dimension-by-dimension methods. In
addition, a new simple-yet-effective set of reconstruction variables is
introduced, as well as an easy-to-implement effective limiter for positivity
preservation, both of which could be useful in existing schemes with little
modification. The proposed scheme is applied to a suite of stringent and
informative benchmark problems to demonstrate its efficacy and utility.Comment: Submitted to Journal of Computational Physics April 202
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