arXiv subjects

Kweku Abraham

Publications and source records attributed to Kweku Abraham.

9 recordsLinked to original sources

arXiv2025

Sample Complexity Analysis of Multi-Target Detection via Markovian and Hard-Core Multi-Reference Alignment

Motivated by single-particle cryo-electron microscopy, we study the sample complexity of the multi-target detection (MTD) problem, in which an unknown signal appears multiple times at unknown locations within a long, noisy observation. We propose a patching scheme that reduces MTD to a non-i.i.d. multi-reference alignment (MRA) model. In the one-dimensional setting, the latent group elements form a Markov chain, and we show that the convergence rate of any estimator matches that of the corresponding i.i.d. MRA model, up to a logarithmic factor in the number of patches. Moreover, for estimators based on empirical averaging, such as the method of moments, the convergence rates are identical in both settings. We further establish an analogous result in two dimensions, where the latent structure arises from an exponentially mixing random field generated by a hard-core placement model. As a consequence, if the signal in the corresponding i.i.d. MRA model is determined by moments up to order $n_{\min}$, then in the low-SNR regime the number of patches required to estimate the signal in the MTD model scales as $\sigma^{2n_{\min}}$, where $\sigma^2$ denotes the noise variance.

eess.SP↗

arXiv2025

Frontiers to the learning of nonparametric hidden Markov models

Hidden Markov models (HMMs) are flexible tools for clustering dependent data coming from unknown populations, allowing nonparametric modelling of the population densities. Identifiability fails when the data is in fact independent and identically distributed (i.i.d.), and we study the frontier between learnable and unlearnable two-state nonparametric HMMs. Learning the parameters of the HMM requires solving a nonlinear inverse problem whose difficulty depends not only on the smoothnesses of the populations but also on the distance to the i.i.d. boundary of the parameter set. The latter difficulty is mostly ignored in the literature in favour of assumptions precluding nearly independent data. This is the first work conducting a precise nonasymptotic, nonparametric analysis of the minimax risk taking into account all aspects of the hardness of the problem, in the case of two populations. Our analysis reveals an unexpected interplay between the distance to the i.i.d. boundary and the relative smoothnesses of the two populations: a surprising and intriguing transition occurs in the rate when the two densities have differing smoothnesses. We obtain upper and lower bounds revealing that, close to the i.i.d. boundary, it is possible to "borrow strength" from the estimator of the smoother density to improve the risk of the other.

math.ST↗

arXiv2023

Deep Gaussian Process Priors for Bayesian Inference in Nonlinear Inverse Problems

We study the use of a deep Gaussian process (DGP) prior in a general nonlinear inverse problem satisfying certain regularity conditions. We prove that when the data arises from a true parameter $θ^*$ with a compositional structure, the posterior induced by the DGP prior concentrates around $θ^*$ as the number of observations increases. The DGP prior accounts for the unknown compositional structure through the use of a hierarchical structure prior. As examples, we show that our results apply to Darcy's problem of recovering the scalar diffusivity from a steady-state heat equation and the problem of determining the attenuation potential in a steady-state Schrödinger equation. We further provide a lower bound, proving in Darcy's problem that typical Gaussian priors based on Whittle-Matérn processes (which ignore compositional structure) contract at a polynomially slower rate than the DGP prior for certain diffusivities arising from a generalised additive model.

math.ST↗

arXiv2023

Sharp multiple testing boundary for sparse sequences

This work investigates multiple testing by considering minimax separation rates in the sparse sequence model, when the testing risk is measured as the sum FDR+FNR (False Discovery Rate plus False Negative Rate). First using the popular beta-min separation condition, with all nonzero signals separated from $0$ by at least some amount, we determine the sharp minimax testing risk asymptotically and thereby explicitly describe the transition from "achievable multiple testing with vanishing risk" to "impossible multiple testing". Adaptive multiple testing procedures achieving the corresponding optimal boundary are provided: the Benjamini--Hochberg procedure with a properly tuned level, and an empirical Bayes $\ell$-value (`local FDR') procedure. We prove that the FDR and FNR make non-symmetric contributions to the testing risk for most optimal procedures, the FNR part being dominant at the boundary. The multiple testing hardness is then investigated for classes of arbitrary sparse signals. A number of extensions, including results for classification losses and convergence rates in the case of large signals, are also investigated.

math.ST↗

arXiv2022

Fundamental limits for learning hidden Markov model parameters

We study the frontier between learnable and unlearnable hidden Markov models (HMMs). HMMs are flexible tools for clustering dependent data coming from unknown populations. The model parameters are known to be fully identifiable (up to label-switching) without any modeling assumption on the distributions of the populations as soon as the clusters are distinct and the hidden chain is ergodic with a full rank transition matrix. In the limit as any one of these conditions fails, it becomes impossible in general to identify parameters. For a chain with two hidden states we prove nonasymptotic minimax upper and lower bounds, matching up to constants, which exhibit thresholds at which the parameters become learnable. We also provide an upper bound on the relative entropy rate for parameters in a neighbourhood of the unlearnable region which may have interest in itself.

stat.ML↗

arXiv2022

Empirical Bayes cumulative $\ell$-value multiple testing procedure for sparse sequences

In the sparse sequence model, we consider a popular Bayesian multiple testing procedure and investigate for the first time its behaviour from the frequentist point of view. Given a spike-and-slab prior on the high-dimensional sparse unknown parameter, one can easily compute posterior probabilities of coming from the spike, which correspond to the well known local-fdr values, also called $\ell$-values. The spike-and-slab weight parameter is calibrated in an empirical Bayes fashion, using marginal maximum likelihood. The multiple testing procedure under study, called here the cumulative $\ell$-value procedure, ranks coordinates according to their empirical $\ell$-values and thresholds so that the cumulative ranked sum does not exceed a user-specified level $t$. We validate the use of this method from the multiple testing perspective: for alternatives of appropriately large signal strength, the false discovery rate (FDR) of the procedure is shown to converge to the target level $t$, while its false negative rate (FNR) goes to $0$. We complement this study by providing convergence rates for the method. Additionally, we prove that the $q$-value multiple testing procedure shares similar convergence rates in this model.

math.ST↗

arXiv2021

Multiple Testing in Nonparametric Hidden Markov Models: An Empirical Bayes Approach

Given a nonparametric Hidden Markov Model (HMM) with two states, the question of constructing efficient multiple testing procedures is considered, treating one of the states as an unknown null hypothesis. A procedure is introduced, based on nonparametric empirical Bayes ideas, that controls the False Discovery Rate (FDR) at a user--specified level. Guarantees on power are also provided, in the form of a control of the true positive rate. One of the key steps in the construction requires supremum--norm convergence of preliminary estimators of the emission densities of the HMM. We provide the existence of such estimators, with convergence at the optimal minimax rate, for the case of a HMM with $J\ge 2$ states, which is of independent interest.

math.ST↗

arXiv2020

On statistical Calderón problems

For $D$ a bounded domain in $\mathbb R^d, d \ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $γ$ determining solutions $u=u_{γ, f}$ of the partial differential equation \begin{equation*} \begin{split} \nabla \cdot(γ\nabla u)&=0 \quad \text{ in }D, \\ u&=f \quad \text { on } \partial D, \end{split} \end{equation*} from noisy observations $Y$ of the Dirichlet-to-Neumann map \[f \mapsto Λ_γ(f) = {γ\frac{\partial u_{γ,f}}{\partial ν}}\Big|_{\partial D},\] with $\partial/\partial ν$ denoting the outward normal derivative, is considered. The data $Y$ consists of $Λ_γ$ corrupted by additive Gaussian noise at noise level $\varepsilon>0$, and a statistical algorithm $\hat γ(Y)$ is constructed which is shown to recover $γ$ in supremum-norm loss at a statistical convergence rate of the order $\log(1/\varepsilon)^{-δ}$ as $\varepsilon \to 0$. It is further shown that this convergence rate is optimal, up to the precise value of the exponent $δ>0$, in an information theoretic sense. The estimator $\hat γ(Y)$ has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods.

math.ST↗

arXiv2018

Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data

We consider inference in the scalar diffusion model $dX_t=b(X_t)dt+σ(X_t)dW_t$ with discrete data $(X_{jΔ_n})_{0\leq j \leq n}$, $n\to \infty,~Δ_n\to 0$ and periodic coefficients. For $σ$ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in $L^2$-distance around the true drift function $b_0$ at the frequentist minimax rate (up to logarithmic factors) over Besov smoothness classes. We exhibit natural nonparametric priors which satisfy our conditions. Our results show that the Bayesian method adapts both to an unknown sampling regime and to unknown smoothness.

math.ST↗