We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices encompassed by the theory is measured in terms of Hausdorff and upper Minkowski dimensions. Our goal is the characterization of the number of linear measurements, with an emphasis on rank-$1$ measurements, needed for the existence of an algorithm that yields reconstruction, either perfect, with probability 1, or with arbitrarily small probability of error, depending on the setup. Concretely, we show that matrices taken from a set $\mathcal{U}$ such that $\mathcal{U}-\mathcal{U}$ has Hausdorff dimension $s$ can be recovered from $k>s$ measurements, and random matrices supported on a set $\mathcal{U}$ of Hausdorff dimension $s$ can be recovered with probability 1 from $k>s$ measurements. What is more, we establish the existence of recovery mappings that are robust against additive perturbations or noise in the measurements. Concretely, we show that there are $\beta$-H\"older continuous mappings recovering matrices taken from a set of upper Minkowski dimension $s$ from $k>2s/(1-\beta)$ measurements and, with arbitrarily small probability of error, random matrices supported on a set of upper Minkowski dimension $s$ from $k>s/(1-\beta)$ measurements. The numerous concrete examples we consider include low-rank matrices, sparse matrices, QR decompositions with sparse R-components, and matrices of fractal nature.
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This paper considers the problem of completing a rating matrix based on sub-sampled matrix entries as well as observed social graphs and hypergraphs. We show that there exists a \emph{sharp threshold} on the sample probability for the task of exactly completing the rating matrix -- the task is achievable when the sample probability is above the threshold, and is impossible otherwise -- demonstrating a phase transition phenomenon. The threshold can be expressed as a function of the ``quality'' of hypergraphs, enabling us to \emph{quantify} the amount of reduction in sample probability due to the exploitation of hypergraphs. This also highlights the usefulness of hypergraphs in the matrix completion problem. En route to discovering the sharp threshold, we develop a computationally efficient matrix completion algorithm that effectively exploits the observed graphs and hypergraphs. Theoretical analyses show that our algorithm succeeds with high probability as long as the sample probability exceeds the aforementioned threshold, and this theoretical result is further validated by synthetic experiments. Moreover, our experiments on a real social network dataset (with both graphs and hypergraphs) show that our algorithm outperforms other state-of-the-art matrix completion algorithms.
We present a result according to which certain functions of covariance matrices are maximized at scalar multiples of the identity matrix. This is used to show that experimental designs that are optimal under an assumption of independent, homoscedastic responses can be minimax robust, in broad classes of alternate covariance structures. In particular it can justify the common practice of disregarding possible dependence, or heteroscedasticity, at the design stage of an experiment.
Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is a challenging one. Given the practical difficulty in building a monolithic architecture for quantum computers based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected quantum processors. In such a setting, all syndrome measurements and logical operations must be performed using high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond DQC since entanglement purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder for distilling $3$-qubit GHZ states using a rate $0.118$ family of lifted product QLDPC codes and obtain an input threshold of $\approx 0.7974$ under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of $0.118$ for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of $3$-qubit GHZ states to construct a scalable GHZ purification protocol.
As is the case for many curved exponential families, the computation of maximum likelihood estimates in a multivariate normal model with a Kronecker covariance structure is typically carried out with an iterative algorithm, specifically, a block-coordinate ascent algorithm. In this article we highlight a setting, specified by a coprime relationship between the sample size and dimension of the Kronecker factors, where the likelihood equations have algebraic degree one and an explicit, easy-to-evaluate rational formula for the maximum likelihood estimator can be found. A partial converse of this result is provided that shows that outside of the aforementioned special setting and for large sample sizes, examples of data sets can be constructed for which the degree of the likelihood equations is larger than one.
Generative models of observations under interventions have been a vibrant topic of interest across machine learning and the sciences in recent years. For example, in drug discovery, there is a need to model the effects of diverse interventions on cells in order to characterize unknown biological mechanisms of action. We propose the Sparse Additive Mechanism Shift Variational Autoencoder, SAMS-VAE, to combine compositionality, disentanglement, and interpretability for perturbation models. SAMS-VAE models the latent state of a perturbed sample as the sum of a local latent variable capturing sample-specific variation and sparse global variables of latent intervention effects. Crucially, SAMS-VAE sparsifies these global latent variables for individual perturbations to identify disentangled, perturbation-specific latent subspaces that are flexibly composable. We evaluate SAMS-VAE both quantitatively and qualitatively on a range of tasks using two popular single cell sequencing datasets. In order to measure perturbation-specific model-properties, we also introduce a framework for evaluation of perturbation models based on average treatment effects with links to posterior predictive checks. SAMS-VAE outperforms comparable models in terms of generalization across in-distribution and out-of-distribution tasks, including a combinatorial reasoning task under resource paucity, and yields interpretable latent structures which correlate strongly to known biological mechanisms. Our results suggest SAMS-VAE is an interesting addition to the modeling toolkit for machine learning-driven scientific discovery.
In unitary property testing a quantum algorithm, also known as a tester, is given query access to a black-box unitary and has to decide whether it satisfies some property. We propose a new technique for proving lower bounds on the quantum query complexity of unitary property testing and related problems, which utilises the connection between unitary property testing and unitary channel discrimination. The main advantage of this technique is that all obtained lower bounds hold for any $\mathsf{C}$-tester with $\mathsf{C} \subseteq \mathsf{QMA}(\text{poly(n)} / \mathsf{qpoly}$, showing that even having access to both (unentangled) quantum proofs and advice does not help for many unitary problems. We apply our technique to prove lower bounds for problems like quantum phase estimation, the entanglement entropy problem, quantum Gibbs sampling and more, removing all logarithmic factors in the lower bounds obtained by the sample-to-query lifting theorem of Wang and Zhang (2023). As a direct corollary, we show that there exists a quantum oracle relative to which $\mathsf{QMA}(\text{poly(n)} / \mathsf{qpoly} \not\supset \mathsf{SBQP}$.
We provide a quantitative assessment of welfare in the classical model of risk-sharing and exchange under uncertainty. We prove three kinds of results. First, that in an equilibrium allocation, the scope for improving individual welfare by a given margin (an $\ve$-improvement) vanishes as the number of states increases. Second, that the scope for a change in aggregate resources that may be distributed to enhance individual welfare by a given margin also vanishes. Equivalently: in an inefficient allocation, for a given level of resource sub-optimality (as measured by the coefficient of resource under-utilization), the possibilities for enhancing welfare by perturbing aggregate resources decrease exponentially to zero with the number of states. Finally, we consider efficient risk-sharing in standard models of uncertainty aversion with multiple priors, and show that, in an inefficient allocation, certain sets of priors shrink with the size of the state space.
Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned extension operators.
We consider the capacitated clustering problem in general metric spaces where the goal is to identify $k$ clusters and minimize the sum of the radii of the clusters (we call this the Capacitated-$k$-sumRadii problem). We are interested in fixed-parameter tractable (FPT) approximation algorithms where the running time is of the form $f(k) \cdot \text{poly}(n)$, where $f(k)$ can be an exponential function of $k$ and $n$ is the number of points in the input. In the uniform capacity case, Bandyapadhyay et al. recently gave a $4$-approximation algorithm for this problem. Our first result improves this to an FPT $3$-approximation and extends to a constant factor approximation for any $L_p$ norm of the cluster radii. In the general capacities version, Bandyapadhyay et al. gave an FPT $15$-approximation algorithm. We extend their framework to give an FPT $(4 + \sqrt{13})$-approximation algorithm for this problem. Our framework relies on a novel idea of identifying approximations to optimal clusters by carefully pruning points from an initial candidate set of points. This is in contrast to prior results that rely on guessing suitable points and building balls of appropriate radii around them. On the hardness front, we show that assuming the Exponential Time Hypothesis, there is a constant $c > 1$ such that any $c$-approximation algorithm for the non-uniform capacity version of this problem requires running time $2^{\Omega \left(\frac{k}{polylog(k)} \right)}$.
A promising strategy to protect quantum information from noise-induced errors is to encode it into the low-energy states of a topological quantum memory device. However, readout errors from such memory under realistic settings is less understood. We study the problem of decoding quantum information encoded in the groundspaces of topological stabilizer Hamiltonians in the presence of generic perturbations, such as quenched disorder. We first prove that the standard stabilizer-based error correction and decoding schemes work adequately well in such perturbed quantum codes by showing that the decoding error diminishes exponentially in the distance of the underlying unperturbed code. We then prove that Quantum Neural Network (QNN) decoders provide an almost quadratic improvement on the readout error. Thus, we demonstrate provable advantage of using QNNs for decoding realistic quantum error-correcting codes, and our result enables the exploration of a wider range of non-stabilizer codes in the near-term laboratory settings.
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