Multivariate trace estimation using quantum state space linear algebra

In this paper, we present a quantum algorithm for approximating multivariate traces, i.e. the traces of matrix products. Our research is motivated by the extensive utility of multivariate traces in elucidating spectral characteristics of matrices, as well as by recent advancements in leveraging quantum computing for faster numerical linear algebra. Central to our approach is a direct translation of a multivariate trace formula into a quantum circuit, achieved through a sequence of low-level circuit construction operations. To facilitate this translation, we introduce \emph{quantum Matrix States Linear Algebra} (qMSLA), a framework tailored for the efficient generation of state preparation circuits via primitive matrix algebra operations. Our algorithm relies on sets of state preparation circuits for input matrices as its primary inputs and yields two state preparation circuits encoding the multivariate trace as output. These circuits are constructed utilizing qMSLA operations, which enact the aforementioned multivariate trace formula. We emphasize that our algorithm's inputs consist solely of state preparation circuits, eschewing harder to synthesize constructs such as Block Encodings. Furthermore, our approach operates independently of the availability of specialized hardware like QRAM, underscoring its versatility and practicality.

Capacity Analysis of Vector Symbolic Architectures

Hyperdimensional computing (HDC) is a biologically-inspired framework that uses high-dimensional vectors and various vector operations to represent and manipulate symbols. The ensemble of a particular vector space and two vector operations (one addition-like for "bundling" and one outer-product-like for "binding") form what is called a "vector symbolic architecture" (VSA). While VSAs have been employed in numerous applications and studied empirically, many theoretical questions about VSAs remain open. We provide theoretical analyses for the *representation capacities* of three popular VSAs: MAP-I, MAP-B, and Binary Sparse. Representation capacity here refers to upper bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks (such as testing for set membership $i \in S$ and estimating set intersection sizes $|S \cap T|$) to a given degree of accuracy. We also describe a relationship between the MAP-I VSA to Hopfield networks, which are simple models of associative memory, and analyze the ability of Hopfield networks to perform some of the same tasks that are typically asked of VSAs. Our analysis of MAP-I casts the VSA vectors as the outputs of *sketching* (dimensionality reduction) algorithms such as the Johnson-Lindenstrauss transform; this provides a clean, simple framework for obtaining bounds on MAP-I's representation capacity. We also provide, to our knowledge, the first analysis of testing set membership in a bundle of general pairwise bindings from MAP-I. Binary sparse VSAs are well-known to be related to Bloom filters; we give analyses of set intersection for Bloom and Counting Bloom filters. Our analysis of MAP-B and Binary Sparse bundling include new applications of several concentration inequalities.

Towards Quantum Advantage on Noisy Quantum Computers

Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical TDA algorithms are exorbitant, and quickly become impractical for high-order characteristics. Quantum computing promises exponential speedup for certain problems. Yet, many existing quantum algorithms with notable asymptotic speedups require a degree of fault tolerance that is currently unavailable. In this paper, we present NISQ-TDA, the first fully implemented end-to-end quantum machine learning algorithm needing only a linear circuit-depth, that is applicable to non-handcrafted high-dimensional classical data, with potential speedup under stringent conditions. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. Our approach includes three key innovations: (a) an efficient realization of the full boundary operator as a sum of Pauli operators; (b) a quantum rejection sampling and projection approach to restrict a uniform superposition to the simplices of the desired order in the complex; and (c) a stochastic rank estimation method to estimate the topological features in the form of approximate Betti numbers. We present theoretical results that establish additive error guarantees for NISQ-TDA, and the circuit and computational time and depth complexities for exponentially scaled output estimates, up to the error tolerance. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise.

PCENet: High Dimensional Surrogate Modeling for Learning Uncertainty

Learning data representations under uncertainty is an important task that emerges in numerous machine learning applications. However, uncertainty quantification (UQ) techniques are computationally intensive and become prohibitively expensive for high-dimensional data. In this paper, we present a novel surrogate model for representation learning and uncertainty quantification, which aims to deal with data of moderate to high dimensions. The proposed model combines a neural network approach for dimensionality reduction of the (potentially high-dimensional) data, with a surrogate model method for learning the data distribution. We first employ a variational autoencoder (VAE) to learn a low-dimensional representation of the data distribution. We then propose to harness polynomial chaos expansion (PCE) formulation to map this distribution to the output target. The coefficients of PCE are learned from the distribution representation of the training data using a maximum mean discrepancy (MMD) approach. Our model enables us to (a) learn a representation of the data, (b) estimate uncertainty in the high-dimensional data system, and (c) match high order moments of the output distribution; without any prior statistical assumptions on the data. Numerical experimental results are presented to illustrate the performance of the proposed method.

Efficient Scaling of Dynamic Graph Neural Networks

We present distributed algorithms for training dynamic Graph Neural Networks (GNN) on large scale graphs spanning multi-node, multi-GPU systems. To the best of our knowledge, this is the first scaling study on dynamic GNN. We devise mechanisms for reducing the GPU memory usage and identify two execution time bottlenecks: CPU-GPU data transfer; and communication volume. Exploiting properties of dynamic graphs, we design a graph difference-based strategy to significantly reduce the transfer time. We develop a simple, but effective data distribution technique under which the communication volume remains fixed and linear in the input size, for any number of GPUs. Our experiments using billion-size graphs on a system of 128 GPUs shows that: (i) the distribution scheme achieves up to 30x speedup on 128 GPUs; (ii) the graph-difference technique reduces the transfer time by a factor of up to 4.1x and the overall execution time by up to 40%

Quantum Topological Data Analysis with Linear Depth and Exponential Speedup

Quantum computing offers the potential of exponential speedups for certain classical computations. Over the last decade, many quantum machine learning (QML) algorithms have been proposed as candidates for such exponential improvements. However, two issues unravel the hope of exponential speedup for some of these QML algorithms: the data-loading problem and, more recently, the stunning dequantization results of Tang et al. A third issue, namely the fault-tolerance requirements of most QML algorithms, has further hindered their practical realization. The quantum topological data analysis (QTDA) algorithm of Lloyd, Garnerone and Zanardi was one of the first QML algorithms that convincingly offered an expected exponential speedup. From the outset, it did not suffer from the data-loading problem. A recent result has also shown that the generalized problem solved by this algorithm is likely classically intractable, and would therefore be immune to any dequantization efforts. However, the QTDA algorithm of Lloyd et~al. has a time complexity of $O(n^4/(\epsilon^2 \delta))$ (where $n$ is the number of data points, $\epsilon$ is the error tolerance, and $\delta$ is the smallest nonzero eigenvalue of the restricted Laplacian) and requires fault-tolerant quantum computing, which has not yet been achieved. In this paper, we completely overhaul the QTDA algorithm to achieve an improved exponential speedup and depth complexity of $O(n\log(1/(\delta\epsilon)))$. Our approach includes three key innovations: (a) an efficient realization of the combinatorial Laplacian as a sum of Pauli operators; (b) a quantum rejection sampling approach to restrict the superposition to the simplices in the complex; and (c) a stochastic rank estimation method to estimate the Betti numbers. We present a theoretical error analysis, and the circuit and computational time and depth complexities for Betti number estimation.

Projection techniques to update the truncated SVD of evolving matrices

This paper considers the problem of updating the rank-k truncated Singular Value Decomposition (SVD) of matrices subject to the addition of new rows and/or columns over time. Such matrix problems represent an important computational kernel in applications such as Latent Semantic Indexing and Recommender Systems. Nonetheless, the proposed framework is purely algebraic and targets general updating problems. The algorithm presented in this paper undertakes a projection view-point and focuses on building a pair of subspaces which approximate the linear span of the sought singular vectors of the updated matrix. We discuss and analyze two different choices to form the projection subspaces. Results on matrices from real applications suggest that the proposed algorithm can lead to higher accuracy, especially for the singular triplets associated with the largest modulus singular values. Several practical details and key differences with other approaches are also discussed.

Multilabel Classification by Hierarchical Partitioning and Data-dependent Grouping

In modern multilabel classification problems, each data instance belongs to a small number of classes from a large set of classes. In other words, these problems involve learning very sparse binary label vectors. Moreover, in large-scale problems, the labels typically have certain (unknown) hierarchy. In this paper we exploit the sparsity of label vectors and the hierarchical structure to embed them in low-dimensional space using label groupings. Consequently, we solve the classification problem in a much lower dimensional space and then obtain labels in the original space using an appropriately defined lifting. Our method builds on the work of (Ubaru & Mazumdar, 2017), where the idea of group testing was also explored for multilabel classification. We first present a novel data-dependent grouping approach, where we use a group construction based on a low-rank Nonnegative Matrix Factorization (NMF) of the label matrix of training instances. The construction also allows us, using recent results, to develop a fast prediction algorithm that has a logarithmic runtime in the number of labels. We then present a hierarchical partitioning approach that exploits the label hierarchy in large scale problems to divide up the large label space and create smaller sub-problems, which can then be solved independently via the grouping approach. Numerical results on many benchmark datasets illustrate that, compared to other popular methods, our proposed methods achieve competitive accuracy with significantly lower computational costs.

Tensor Graph Convolutional Networks for Prediction on Dynamic Graphs

Many irregular domains such as social networks, financial transactions, neuron connections, and natural language structures are represented as graphs. In recent years, a variety of graph neural networks (GNNs) have been successfully applied for representation learning and prediction on such graphs. However, in many of the applications, the underlying graph changes over time and existing GNNs are inadequate for handling such dynamic graphs. In this paper we propose a novel technique for learning embeddings of dynamic graphs based on a tensor algebra framework. Our method extends the popular graph convolutional network (GCN) for learning representations of dynamic graphs using the recently proposed tensor M-product technique. Theoretical results that establish the connection between the proposed tensor approach and spectral convolution of tensors are developed. Numerical experiments on real datasets demonstrate the usefulness of the proposed method for an edge classification task on dynamic graphs.

Find the dimension that counts: Fast dimension estimation and Krylov PCA

High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these covariance matrices and obtaining an approximation to the subspace. This problem arises in the popular principal component analysis (PCA), and in many applications of machine learning, data analysis, signal and image processing, and others. We first present a novel method for estimating the dimension of the principal subspace. We then show how this method can be coupled with a Krylov subspace method to simultaneously estimate the dimension and obtain an approximation to the subspace. The dimension estimation is achieved at no additional cost. The proposed method operates on a model selection framework, where the novel selection criterion is derived based on random matrix perturbation theory ideas. We present theoretical analyses which (a) show that the proposed method achieves strong consistency (i.e., yields optimal solution as the number of data-points $n\rightarrow \infty$), and (b) analyze conditions for exact dimension estimation in the finite $n$ case. Using recent results, we show that our algorithm also yields near optimal PCA. The proposed method avoids forming the sample covariance matrix (associated with the data) explicitly and computing the complete eigen-decomposition. Therefore, the method is inexpensive, which is particularly advantageous in modern data applications where the covariance matrices can be very large. Numerical experiments illustrate the performance of the proposed method in various applications.