We study the problem of distribution to real-value regression, where one aims to regress a mapping @math that takes in a distribution input covariate @math (for a non-parametric family of distributions @math ) and outputs a real-valued response @math . This setting was recently studied, and a "Kernel-Kernel" estimator was introduced and shown to have a polynomial rate of convergence. However, evaluating a new prediction with the Kernel-Kernel estimator scales as @math . This causes the difficult situation where a large amount of data may be necessary for a low estimation risk, but the computation cost of estimation becomes infeasible when the data-set is too large. To this end, we propose the Double-Basis estimator, which looks to alleviate this big data problem in two ways: first, the Double-Basis estimator is shown to have a computation complexity that is independent of the number of of instances @math when evaluating new predictions after training; secondly, the Double-Basis estimator is shown to have a fast rate of convergence for a general class of mappings @math .
DRR is related to the functional analysis, where one regresses a mapping whose input domain are functions @cite_12 . However, the objects DRR works over--distributions and their pdfs--are inferred through sets of samples drawn from the objects, with finite sizes. In functional analysis, the functions are inferred through observations of @math pairs that are often taken to be an arbitrarily dense grid in the domain of the functions. For a comprehensive survey in functional analysis see @cite_12 @cite_11 . Also, recently @cite_15 studied the problem of distribution to distribution regression, where both input and output covariates are distributions.
@cite_15 @cite_12 @cite_11
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The vast majority of existing approaches to opinion feature extraction rely on mining patterns only from a single review corpus, ignoring the nontrivial disparities in word distributional characteristics of opinion features across different corpora. In this paper, we propose a novel method to identify opinion features from online reviews by exploiting the difference in opinion feature statistics across two corpora, one domain-specific corpus (i.e., the given review corpus) and one domain-independent corpus (i.e., the contrasting corpus). We capture this disparity via a measure called domain relevance (DR), which characterizes the relevance of a term to a text collection. We first extract a list of candidate opinion features from the domain review corpus by defining a set of syntactic dependence rules. For each extracted candidate feature, we then estimate its intrinsic-domain relevance (IDR) and extrinsic-domain relevance (EDR) scores on the domain-dependent and domain-independent corpora, respectively. Candidate features that are less generic (EDR score less than a threshold) and more domain-specific (IDR score greater than another threshold) are then confirmed as opinion features. We call this interval thresholding approach the intrinsic and extrinsic domain relevance (IEDR) criterion. Experimental results on two real-world review domains show the proposed IEDR approach to outperform several other well-established methods in identifying opinion features. Motivated by data-driven decision making and sampling problems, we investigate probabilistic interpretations of robust optimization (RO). We establish a connection between RO and distributionally robust stochastic programming (DRSP), showing that the solution to any RO problem is also a solution to a DRSP problem. Specifically, we consider the case where multiple uncertain parameters belong to the same fixed dimensional space and find the set of distributions of the equivalent DRSP problem. The equivalence we derive enables us to construct RO formulations for sampled problems (as in stochastic programming and machine learning) that are statistically consistent, even when the original sampled problem is not. In the process, this provides a systematic approach for tuning the uncertainty set. The equivalence further provides a probabilistic explanation for the common shrinkage heuristic, where the uncertainty set used in an RO problem is a shrunken version of the original uncertainty set. We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states ( annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios. Abstract The problem of reconstructing a function f from a set of non-uniformly distributed, weighted-average sampled values ∫ R d f(x)ψ x j (x) d x: j∈J is studied in the context of shift-invariant subspaces of L 2 ( R d ) . Necessary density conditions on the sampling set X= x j : j∈J for stable reconstruction are obtained, and fast iterative algorithms are described. The performance of the algorithms are analyzed when the data are corrupted by noise. Estimates are derived for the convergence rates of the algorithms in terms of the sampling density and the diameters of the sampling functionals ψ x j : x j ∈X . The results provide a mathematical framework for situations arising frequently in applications, e.g., when the sample values are not precise because they are gathered by real-world acquisition devices.