No abstract uploaded!

We study the problem of globally recovering a dictionary from a set of signals via ℓ1-minimization. We assume that the signals are generated as i.i.d. random linear combinations of the K atoms from a complete reference dictionary D∗∈RK×K, where the linear combination coefficients are from either a Bernoulli type model or exact sparse model. First, we obtain a necessary and sufficient norm condition for the reference dictionary D∗ to be a sharp local minimum of the expected ℓ1 objective function. Our result substantially extends that of Wu and Yu (2015) and allows the combination coefficient to be non-negative. Secondly, we obtain an explicit bound on the region within which the objective value of the reference dictionary is minimal. Thirdly, we show that the reference dictionary is the unique sharp local minimum, thus establishing the first known global property of ℓ1-minimization dictionary learning. Motivated by the theoretical results, we introduce a perturbation based test to determine whether a dictionary is a sharp local minimum of the objective function. In addition, we also propose a new dictionary learning algorithm based on Block Coordinate Descent, called DL-BCD, which is guaranteed to decrease the obective function monotonically. Simulation studies show that DL-BCD has competitive performance in terms of recovery rate compared to other state-of-the-art dictionary learning algorithms when the reference dictionary is generated from random Gaussian matrices.

We develop an auto-reservoir computing framework, Auto-Reservoir Neural Network (ARNN), to efficiently and accurately make multi-step-ahead predictions based on a short-term high-dimensional time series. Different from traditional reservoir computing whose reservoir is an external dynamical system irrelevant to the target system, ARNN directly transforms the observed high-dimensional dynamics as its reservoir, which maps the high-dimensional/spatial data to the future temporal values of a target variable based on our spatiotemporal information (STI) transformation. Thus, the multi-step prediction of the target variable is achieved in an accurate and computationally efficient manner. ARNN is successfully applied to both representative models and real-world datasets, all of which show satisfactory performance in the multi-step-ahead prediction, even when the data are peturbed by noise and when the system is time-varying. Actually, such ARNN transformation equivalently expands the sample size and thus has great potential in practical applications in artificial intelligence and machine learning.

In a Data-Generating Experiment (DGE), the data, X, is often obtained either from a Black-Box with inputs θ and Y, or from a Quantile function or a learning machine, f(Y, θ); θ is unknown, element of metric space (Θ, ρ), Y is random. If X has intractable or unknown c.d.f., Fθ, non-identifiability of θ cannot be confirmed and when present, among others, limits the predictive accuracy of the learned model, f(Y, \hat{θ}); \hat{θ} estimate of θ. In Machine Learning, non-identifiability of θ is ubiquitous and its extent is a criterion for selecting a learning machine. Empirical indices, EDI and PPVI, are introduced using P-values of Kolmogorov-Smirnov tests: i) to confirm almost surely, using generated data, the discrimination of θ from θ^∗, namely that the Kolmogorov distance, dK(Fθ, Fθ^∗), is positive, ii) to confirm identifiability of θ(∈ Θ) by repeating i) for θ^∗ in a sieve of Θ, since neighboring parameter values are in practice indistinguishable, and iii) most important, to compare EDI-graphs of DGEs, preferring more discrimination and less non-identifiability among parameters, and select one DGE to use. In applications, EDI-graphs confirm nonidentifiability in mixture models and in models parametrised with sums of parameters. EDI and PPVI explain why Tukey’s g-and-h model (DGE1) has better g-discrimination than the g-and-k model (DGE2), unless the sample size is extremely large; h_0 = k_0. EDIgraphs indicate that Normal learning machines have better parameter discrimination thanSigmoid learning machines and their parameters are non-identifiable.

When sample, X = x, is observed from intractable c.d.f. Fθ, or a Black-Box with input θ, an Approximate Bayesian Computation (ABC) method provides approximate posterior, π_ϵ. θ^∗ is included in the support of π_ϵ when the Matching distance ρ(S(x^∗), S(x)) ≤ ϵ; x^∗ is a sample drawn from F_{θ^∗}, θ^∗ is obtained from prior π, S is a summary statistic, ϵ > 0. ABC concerns include: the use of only one sample, x^∗, for each θ^∗; the choices of S, ρ and ϵ; π_ϵ(θ^∗), which is determined by arbitrary kernel, K(x, x^∗; ϵ), creating visual π_ϵ-artifacts. The concerns are accommodated with the introduced Fiducial(F)-ABC for all (θ^∗ drawn): M x^∗ are drawn from F_{θ^∗}; a universal S is used, the empirical measure indexed by sets which activate its sufficiency for exchangeable observations-vectors and have been neglected; a strong, probability distance ρ is used, inherently connected with S and Matching; light is thrown to ϵ’s nature and value; πϵ is obtained from the proportions of x∗ matching x. F-ABC for all posterior is closer to Bayesian philosophy, which does not use θ^∗-exclusions. Under few, mild assumptions, π_ϵ converges to the posterior, π(θ|x), when ϵ ↓ 0, and rates of concentration of T (F_{θ^∗}) to T (F_θ) are obtained when n ↑ ∞; T is a functional. Satisfactory F-ABC for all θ^∗ drawn posteriors are depicted for parametric and data generating models, including Tukey’s (a, b, g, h)-model, a 5-parameter normal mixture and a time series model. Various advantages of the F-ABC for all are presented over coarsened posteriors for observations in R^d, d ≥ 1.