This paper establishes minimax lower bounds for the amplitude-based phase retrieval model with Gaussian design measurements and additive Gaussian noise. We prove that for any estimator based on the measurement matrix and the observed vector, there exists a vector such that, with probability at least 15, the distance between the estimator and this vector is at least the critical radius. For k-sparse vectors, we derive an explicit minimax lower bound. This bound holds when the sample size is proportional to the product of the noise-to-signal ratio, the sparsity level k, and a logarithmic factor. Our proofs combine the strategy of Fano’s method with packing constructions. A key observation is that the Kullback–Leibler (KL) divergence between the distributions induced by two vectors is approximately proportional to the sample size and the squared distance between the two vectors, and inversely proportional to the noise variance. Our result provides a useful basis for assessing the statistical performance of various algorithms and estimators, and may serve as a reference when comparing different approaches.
The samples resulting from a time encoding machine (TEM) are intrinsically non-uniform and thus typical strategies for reconstruction rely on transforming the time encoded samples into an equivalent non-uniform sampling problem. In this work we take an alternative perspective and consider how integrate-and-fire time encoded samples represent uniform sampling on an equivalent ‘amplitude’-domain signal. From this perspective, existing reconstruction strategies are readily applicable but only for a limited class of signals. To remedy this, we propose a modification of the typical sampling hardware to create a new time-dilating integrate-and-fire sampler (TDIF) that can sample and reconstruct a broader range of signals. We show that, in amplitude-domain, dilation of time measurements is an almost equivalent operation to filtering in the time-domain. Thus signals can be sampled with arbitrary sampling kernels in amplitude-domain by dilating time measurements. We show that non-negative piecewise constant signals can be perfectly reconstructed from their sparse dilated time encodings.
We derive quantitative volume constraints for sampling measures mu t on the unit sphere Sd that satisfy Marcinkiewicz-Zygmund inequalities for polynomials of degree t. Using precise localization estimates for Jacobi polynomials, we obtain explicit upper and lower bounds on the ut-mass of geodesic balls at the natural scale t-1. Whereas constants are typically left implicit in the literature, we place special emphasis on fully explicit constants, so that the results are genuinely quantitative. Moreover, these bounds yield quantitative constraints for the s-dimensional Hausdorff volume of Marcinkiewicz-Zygmund sampling sets and, in particular, optimal lower bounds for the length of Marcinkiewicz-Zygmund curves.
In graph-based data analysis, k-nearest neighbor (kNN) graphs are widely used due to their adaptivity to local data densities. Allowing weighted edges in the graph, the kernelized graph affinity provides a more general type of kNN graph where the kNN distance is used to set the kernel bandwidth adaptively. In this work, we consider a general class of kNN graph where the graph affinity is W-ij = epsilon(-d/2) k(0)(parallel to x(t) - x(j)parallel to(2) / epsilon phi(rho(x(t)) (rho(x(j)))(2)), with rho(x) being the (rescaled) kNN distance at the point x, phi a symmetric bi-variate function, and k(0) a non-negative function on [0, infinity). Under the manifold data setting, where N i.i.d. samples x(t) are drawn from a density p on a d-dimensional unknown manifold embedded in a high dimensional Euclidean space, we prove the operator pointwise convergence of the kNN graph Laplacian to the limiting manifold operator (depending on p) at the rate of O(N-2/(d+6)), up to a log factor, when k(0) and phi have C-3 regularity and satisfy other technical conditions. This is obtained when epsilon similar to N-2/(d+6) and k similar to N6/(d+6), both at the optimal order to balance the theoretical bias and variance errors. Our improved convergence rate is based on a refined analysis of the kNN estimator, which can be of independent interest. We validate our theory by numerical experiments on simulated data.
