Biblio

Kernel regression is an essential and ubiquitous tool for non-parametric data analysis, particularly popular among time series and spatial data. However, the central operation which is performed many times, evaluating a kernel on the data set, takes linear time. This is impractical for modern large data sets. In this paper we describe coresets for kernel regression: compressed data sets which can be used as proxy for the original data and have provably bounded worst case error. The size of the coresets are independent of the raw number of data points; rather they only depend on the error guarantee, and in some cases the size of domain and amount of smoothing. We evaluate our methods on very large time series and spatial data, and demonstrate that they incur negligible error, can be constructed extremely efficiently, and allow for great computational gains.

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. This is the quantum analogue of the problem of estimating a probability distribution given some number of samples. Previously, it was known only that estimating states to error є in trace distance required O(dr2/є2) copies for a d-dimensional density matrix of rank r. Here, we give a measurement scheme (POVM) that uses O( (dr/ δ ) ln(d/δ) ) copies to estimate ρ to error δ in infidelity. This implies O( (dr / є2)· ln(d/є) ) copies suffice to achieve error є in trace distance. For fixed d, our measurement can be implemented on a quantum computer in time polynomial in n. We also use the Holevo bound from quantum information theory to prove a lower bound of Ω(dr/є2)/ log(d/rє) copies needed to achieve error є in trace distance. This implies a lower bound Ω(dr/δ)/log(d/rδ) for the estimation error δ in infidelity. These match our upper bounds up to log factors. Our techniques can also show an Ω(r2d/δ) lower bound for measurement strategies in which each copy is measured individually and then the outcomes are classically post-processed to produce an estimate. This matches the known achievability results and proves for the first time that such “product” measurements have asymptotically suboptimal scaling with d and r.

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. This is the quantum analogue of the problem of estimating a probability distribution given some number of samples. Previously, it was known only that estimating states to error є in trace distance required O(dr2/є2) copies for a d-dimensional density matrix of rank r. Here, we give a measurement scheme (POVM) that uses O( (dr/ δ ) ln(d/δ) ) copies to estimate ρ to error δ in infidelity. This implies O( (dr / є2)· ln(d/є) ) copies suffice to achieve error є in trace distance. For fixed d, our measurement can be implemented on a quantum computer in time polynomial in n. We also use the Holevo bound from quantum information theory to prove a lower bound of Ω(dr/є2)/ log(d/rє) copies needed to achieve error є in trace distance. This implies a lower bound Ω(dr/δ)/log(d/rδ) for the estimation error δ in infidelity. These match our upper bounds up to log factors. Our techniques can also show an Ω(r2d/δ) lower bound for measurement strategies in which each copy is measured individually and then the outcomes are classically post-processed to produce an estimate. This matches the known achievability results and proves for the first time that such “product” measurements have asymptotically suboptimal scaling with d and r.