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Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of results provide tight bounds on the average errors between empirical and population statistics of eigenvectors, fewer results are tight for entrywise analyses, which are critical for a number of problems such as community detection and ranking.

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Following the recent trend in explicit neural memory structures, we present a new design of an external memory, wherein memories are stored in an Euclidean key space \mathbb R^ n . An LSTM controller performs read and write via specialized read and write heads. It can move a head by either providing a new address in the key space (aka random access) or moving from its previous position via a Lie group action (aka Lie access). In this way, the" L" and" R" instructions of a traditional Turing Machine are generalized to arbitrary elements of a fixed Lie group action. For this reason, we name this new model the Lie Access Neural Turing Machine, or LANTM.

In a Data-Generating Experiment, the observed sample, x, has intractable or unavailable c.d.f., F_θ, and θ’s statistical nature is unknown; θ is element of metric space (Θ, dΘ). Matching estimates of θ are introduced, learning from the “best” x-matches with samples X^∗ from F_{θ^∗}, θ^∗ ∈ Θ. Under mild conditions, these nonparametric estimates are uniformly consistent and the upper bounds on their rates of convergence in probability have the same rate and depend on the Kolmogorov entropies of an increasing sequence of sets covering Θ. When Θ ⊆ Rm and the observations are i.i.d. the upper bounds can be, \sqrt{log n}/\sqrt{n} when m is known, and \sqrt{m_n·log n}/\sqrt{n} when m is unknown; m ≥ 1, m_n ↑ ∞ at a desired rate. Upper bounds can also be obtained for dependent observations. These rates hold for observations in R^d, complementing recent results obtained for real, i.i.d. observations, under stronger assumptions and using weak probability distances; d ≥ 1. In simulations, the Matching estimates are successful for the mixture of 2 normals and for Tukey’s (a, b, g, h) and the (a, b, g, k) models. Computers’ evolution will allow for more and faster comparisons, resulting in improved Matching estimates for universal use in Machine Learning.