A wide range of fundamental machine learning tasks that are addressed by the maximum a posteriori estimation can be reduced to a general minimum conical hull problem.The best-known solution to tackle general minimum conical hull problems is the divide-and-conquer anchoring learning scheme (DCA), whose runtime complexity is polynomial in size.However, big data are pushing these polynomial algorithms to their performance limits.In this paper, we propose a sublinear classical algorithm to tackle general minimum conical hull problems when the input has stored in a sample-based low-overhead data structure.The algorithm's runtime complexity is polynomial in the rank and polylogarithmic in size.The proposed algorithm achieves the exponential speedup over DCA and, therefore, provides advantages for high-dimensional problems.
Among matrix decomposition methods, the two factorized matrices obtained by Non-negative matrix factorization (NMF) generally yield a more natural and interpretable representations. It is computationally expensive to find the two factorized matrices W and H with a fixed rank r from a non-negative matrix X∈Rn⇥m by minimizing ||X-WH^Τ ||^2. Even if the proposed separability assumption enables polynomial separable NMF (SNMF) algorithms, the computational cost is not affordable for the continuously growing big data. As a powerful technique to deal with high dimensional data, quantum computing possesses quantum advantages for solving machine learning problems, at which the quantum advantages include both an exponential speedup and a more efficient learning ability with respect to the classical counterparts. In this research, based on the divide-and-conquer anchoring (DCA) scheme, quantum DCA (QDCA) is devised to solve SNMF problems. Benefiting from the linear algebraic nature of quantum mechanics and the convenience that statistical distribution is natively available in quantum mechanics with a superset, QDCA achieves an exponential speedup with respect to the classical counterpart. The runtime complexity is O(poly log(n + m)), where n x m is the size of the input matrix. In addition, the proposed QDCA can naturally generalize to solve near-separable NMF under a polynomial logarithmic runtime. Specifically, the time consuming divide step can be completed by a quantum algorithm for linear operations with an exponential speedup. In the conquer step, considering that reading out quantum data into classical forms is exponential expensive and attributing to that anchors can be sampled with a high probability from a probability distribution in the resulting quantum states, the proposed heuristic post-selection method enables QDCA to extract information from quantum states efficiently, which also maintains the quantum speedup after measurements. Under a plausible assumption, QDCA performs efficiently, achieves the exponential speedup, and is beneficial for high dimensional problems. The QDCA is a hybrid quantum and classical algorithm that maintains the quantum speedup after measurements, which provides a new way to develop quantum machine learning algorithms.
It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X-WH^Τ ||^2. Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems.
Non-negative Matrix Factorization (NMF) asks to decompose a (entry-wise) non-negative matrix into the product of two smaller-sized nonnegative matrices, which has been shown intractable in general. In order to overcome this issue, separability assumption is introduced which assumes all data points are in a conical hull. This assumption makes NMF tractable and widely used in text analysis and image processing, but still impractical for huge-scale datasets. In this paper, inspired by recent development on dequantizing techniques, we propose a new classical algorithm for separable NMF problem. Our new algorithm runs in polynomial time in the rank and logarithmic in the size of input matrices, which achieves an exponential speedup in the low-rank setting.
This paper presents a sublinear classical algorithm for principal component regression. The algorithm uses quantum-inspired linear algebra, an idea developed by Tang. Using this technique, her algorithm for recommendation systems achieved runtime only polynomially slower than its quantum counterpart. Her work was quickly adapted to solve many other problems in sublinear time complexity. In this work, we developed an algorithm for principal component regression that runs in time polylogarithmic to the number of data points, an exponential speed up over the state-of-the-art algorithm, under the mild assumption that the input is given in some data structure that supports a norm-based sampling procedure. This exponential speed up allows for potential applications in much larger data sets.
Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. Recently, quantum algorithms with superpolynomial speedups for solving SDPs have been proposed assuming access to its constraint matrices in quantum superposition. Mutually inspired by both classical and quantum SDP solvers, in this paper we present a sublinear classical algorithm for solving low-rank SDPs which is asymptotically as good as existing quantum algorithms. Specifically, given an SDP with
m constraint matrices, each of dimension
n and rank
poly(logn), our algorithm gives a succinct description and any entry of the solution matrix in time
O(m⋅poly(logn,1/ε)) given access to a sample-based low-overhead data structure of the constraint matrices, where
ε is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with both the SDP solvers based on the matrix multiplicative weight (MMW) framework and the recent studies of quantum-inspired machine learning algorithms. The cost of solving SDPs by MMW mainly comes from the exponentiation of Hermitian matrices, and we propose two new technical ingredients (compared to previous sample-based algorithms) for this task that may be of independent interest:
∙ Weighted sampling: assuming sampling access to each individual constraint matrix
A1,…,Aτ, we propose a procedure that gives a good approximation of
A=A1+⋯+Aτ.
∙ Symmetric approximation: we propose a sampling procedure that gives low-rank spectral decomposition of a Hermitian matrix
A. This improves upon previous sampling procedures that only give low-rank singular value decompositions, losing the signs of eigenvalues.
Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.