618 research outputs found
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Improved Algorithms for White-Box Adversarial Streams
We study streaming algorithms in the white-box adversarial stream model,
where the internal state of the streaming algorithm is revealed to an adversary
who adaptively generates the stream updates, but the algorithm obtains fresh
randomness unknown to the adversary at each time step. We incorporate
cryptographic assumptions to construct robust algorithms against such
adversaries. We propose efficient algorithms for sparse recovery of vectors,
low rank recovery of matrices and tensors, as well as low rank plus sparse
recovery of matrices, i.e., robust PCA. Unlike deterministic algorithms, our
algorithms can report when the input is not sparse or low rank even in the
presence of such an adversary. We use these recovery algorithms to improve upon
and solve new problems in numerical linear algebra and combinatorial
optimization on white-box adversarial streams. For example, we give the first
efficient algorithm for outputting a matching in a graph with insertions and
deletions to its edges provided the matching size is small, and otherwise we
declare the matching size is large. We also improve the approximation versus
memory tradeoff of previous work for estimating the number of non-zero elements
in a vector and computing the matrix rank.Comment: ICML 202
Learning and Control of Dynamical Systems
Despite the remarkable success of machine learning in various domains in recent years, our understanding of its fundamental limitations remains incomplete. This knowledge gap poses a grand challenge when deploying machine learning methods in critical decision-making tasks, where incorrect decisions can have catastrophic consequences. To effectively utilize these learning-based methods in such contexts, it is crucial to explicitly characterize their performance. Over the years, significant research efforts have been dedicated to learning and control of dynamical systems where the underlying dynamics are unknown or only partially known a priori, and must be inferred from collected data. However, much of these classical results have focused on asymptotic guarantees, providing limited insights into the amount of data required to achieve desired control performance while satisfying operational constraints such as safety and stability, especially in the presence of statistical noise.
In this thesis, we study the statistical complexity of learning and control of unknown dynamical systems. By utilizing recent advances in statistical learning theory, high-dimensional statistics, and control theoretic tools, we aim to establish a fundamental understanding of the number of samples required to achieve desired (i) accuracy in learning the unknown dynamics, (ii) performance in the control of the underlying system, and (iii) satisfaction of the operational constraints such as safety and stability. We provide finite-sample guarantees for these objectives and propose efficient learning and control algorithms that achieve the desired performance at these statistical limits in various dynamical systems. Our investigation covers a broad range of dynamical systems, starting from fully observable linear dynamical systems to partially observable linear dynamical systems, and ultimately, nonlinear systems.
We deploy our learning and control algorithms in various adaptive control tasks in real-world control systems and demonstrate their strong empirical performance along with their learning, robustness, and stability guarantees. In particular, we implement one of our proposed methods, Fourier Adaptive Learning and Control (FALCON), on an experimental aerodynamic testbed under extreme turbulent flow dynamics in a wind tunnel. The results show that FALCON achieves state-of-the-art stabilization performance and consistently outperforms conventional and other learning-based methods by at least 37%, despite using 8 times less data. The superior performance of FALCON arises from its physically and theoretically accurate modeling of the underlying nonlinear turbulent dynamics, which yields rigorous finite-sample learning and performance guarantees. These findings underscore the importance of characterizing the statistical complexity of learning and control of unknown dynamical systems.</p
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Foundations of Node Representation Learning
Low-dimensional node representations, also called node embeddings, are a cornerstone in the modeling and analysis of complex networks. In recent years, advances in deep learning have spurred development of novel neural network-inspired methods for learning node representations which have largely surpassed classical \u27spectral\u27 embeddings in performance. Yet little work asks the central questions of this thesis: Why do these novel deep methods outperform their classical predecessors, and what are their limitations?
We pursue several paths to answering these questions. To further our understanding of deep embedding methods, we explore their relationship with spectral methods, which are better understood, and show that some popular deep methods are equivalent to spectral methods in a certain natural limit. We also introduce the problem of inverting node embeddings in order to probe what information they contain. Further, we propose a simple, non-deep method for node representation learning, and find it to often be competitive with modern deep graph networks in downstream performance.
To better understand the limitations of node embeddings, we prove some upper and lower bounds on their capabilities. Most notably, we prove that node embeddings are capable of exact low-dimensional representation of networks with bounded max degree or arboricity, and we further show that a simple algorithm can find such exact embeddings for real-world networks. By contrast, we also prove inherent bounds on random graph models, including those derived from node embeddings, to capture key structural properties of networks without simply memorizing a given graph
Efficient Computation of the Quantum Rate-Distortion Function
The quantum rate-distortion function plays a fundamental role in quantum
information theory, however there is currently no practical algorithm which can
efficiently compute this function to high accuracy for moderate channel
dimensions. In this paper, we show how symmetry reduction can significantly
simplify common instances of the entanglement-assisted quantum rate-distortion
problems. This allows for more efficient computation regardless of the
numerical algorithm being used, and provides insight into the quantum channels
which obtain the optimal rate-distortion tradeoff. Additionally, we propose an
inexact variant of the mirror descent algorithm to compute the quantum
rate-distortion function with provable sublinear convergence rates. We show how
this mirror descent algorithm is related to Blahut-Arimoto and
expectation-maximization methods previously used to solve similar problems in
information theory. Using these techniques, we present the first numerical
experiments to compute a multi-qubit quantum rate-distortion function, and show
that our proposed algorithm solves faster and to higher accuracy when compared
to existing methods.Comment: 37 pages, 2 figures, 2 tables. v2: Minor edits to introduction,
abstract, and notatio
Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Construction, and Error Analysis
Elementary quantum mechanics proposes that a closed physical system
consistently evolves in a reversible manner. However, control and readout
necessitate the coupling of the quantum system to the external environment,
subjecting it to relaxation and decoherence. Consequently, system-environment
interactions are indispensable for simulating physically significant theories.
A broad spectrum of physical systems in condensed-matter and high-energy
physics, vibrational spectroscopy, and circuit and cavity QED necessitates the
incorporation of bosonic degrees of freedom, such as phonons, photons, and
gluons, into optimized fermion algorithms for near-future quantum simulations.
In particular, when a quantum system is surrounded by an external environment,
its basic physics can usually be simplified to a spin or fermionic system
interacting with bosonic modes. Nevertheless, troublesome factors such as the
magnitude of the bosonic degrees of freedom typically complicate the direct
quantum simulation of these interacting models, necessitating the consideration
of a comprehensive plan. This strategy should specifically include a suitable
fermion/boson-to-qubit mapping scheme to encode sufficiently large yet
manageable bosonic modes, and a method for truncating and/or downfolding the
Hamiltonian to the defined subspace for performing an approximate but highly
accurate simulation, guided by rigorous error analysis. In this paper, we aim
to provide such an exhaustive strategy. Specifically, we emphasize two aspects:
(1) the discussion of recently developed quantum algorithms for these
interacting models and the construction of effective Hamiltonians, and (2) a
detailed analysis regarding a tightened error bound for truncating the bosonic
modes for a class of fermion-boson interacting Hamiltonians
Quantum computing for finance
Quantum computers are expected to surpass the computational capabilities of
classical computers and have a transformative impact on numerous industry
sectors. We present a comprehensive summary of the state of the art of quantum
computing for financial applications, with particular emphasis on stochastic
modeling, optimization, and machine learning. This Review is aimed at
physicists, so it outlines the classical techniques used by the financial
industry and discusses the potential advantages and limitations of quantum
techniques. Finally, we look at the challenges that physicists could help
tackle
Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms
Several quantum algorithms for linear algebra problems, and in particular
quantum machine learning problems, have been "dequantized" in the past few
years. These dequantization results typically hold when classical algorithms
can access the data via length-squared sampling. In this work we investigate
how robust these dequantization results are. We introduce the notion of
approximate length-squared sampling, where classical algorithms are only able
to sample from a distribution close to the ideal distribution in total
variation distance. While quantum algorithms are natively robust against small
perturbations, current techniques in dequantization are not. Our main technical
contribution is showing how many techniques from randomized linear algebra can
be adapted to work under this weaker assumption as well. We then use these
techniques to show that the recent low-rank dequantization framework by Chia,
Gily\'en, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework
for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based
on the Quantum Singular Value Transformation, can be generalized to the case of
approximate length-squared sampling access to the input. We also apply these
results to obtain a robust dequantization of many quantum machine learning
algorithms, including quantum algorithms for recommendation systems, supervised
clustering and low-rank matrix inversion.Comment: 55 page
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