Efficient Scalable Accurate Regression Queries in In-DBMS Analytics

Recent trends aim to incorporate advanced data analytics capabilities within DBMSs. Linear regression queries are fundamental to exploratory analytics and predictive modeling. However, computing their exact answers leaves a lot to be desired in terms of efficiency and scalability. We contribute a novel predictive analytics model and associated regression query processing algorithms, which are efficient, scalable and accurate. We focus on predicting the answers to two key query types that reveal dependencies between the values of different attributes: (i) mean-value queries and (ii) multivariate linear regression queries, both within specific data subspaces defined based on the values of other attributes. Our algorithms achieve many orders of magnitude improvement in query processing efficiency and nearperfect approximations of the underlying relationships among data attributes

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.Comment: 67 pages, 1 figur

Postprocessing can speed up general quantum search algorithms

A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$. It uses a diffusion operator $D_{s}$ having source state as one of its eigenstates and $I_{t}$, where $I_{\psi}$ denotes the selective phase inversion of $|\psi\rangle$ state. It evolves $|s\rangle$ to a particular state $|w\rangle$, call it w-state, in $O(B/\alpha)$ time steps where $\alpha$ is $|\langle t|s\rangle|$ and $B$ is a characteristic of the diffusion operator. Measuring the w-state gives the target state with the success probability of $O(1/B^{2})$ and $O(B^{2})$ applications of the algorithm can boost it from $O(1/B^{2})$ to $O(1)$, making the total time complexity $O(B^{3}/\alpha)$. In the special case of Grover's algorithm, $D_{s}$ is $I_{s}$ and $B$ is very close to $1$. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement $I_{w}$. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using phase estimation algorithm. This algorithm is used to efficiently approximate $I_{w}$ which reduces the time complexity of general algorithm to $O(B/\alpha)$. Though $O(B/\alpha)$ algorithms are known to exist, our algorithm offers physical implementation advantages.Comment: Accepted for publication in Physical Review A. arXiv admin note: substantial text overlap with arXiv:1210.464

Large-scale predictive modeling and analytics through regression queries in data management systems

Regression analytics has been the standard approach to modeling the relationship between input and output variables, while recent trends aim to incorporate advanced regression analytics capabilities within data management systems (DMS). Linear regression queries are fundamental to exploratory analytics and predictive modeling. However, computing their exact answers leaves a lot to be desired in terms of efficiency and scalability. We contribute with a novel predictive analytics model and an associated statistical learning methodology, which are efficient, scalable and accurate in discovering piecewise linear dependencies among variables by observing only regression queries and their answers issued to a DMS. We focus on in-DMS piecewise linear regression and specifically in predicting the answers to mean-value aggregate queries, identifying and delivering the piecewise linear dependencies between variables to regression queries and predicting the data dependent variables within specific data subspaces defined by analysts and data scientists. Our goal is to discover a piecewise linear data function approximation over the underlying data only through query–answer pairs that is competitive with the best piecewise linear approximation to the ground truth. Our methodology is analyzed, evaluated and compared with exact solution and near-perfect approximations of the underlying relationships among variables achieving orders of magnitude improvement in analytics processing