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.
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.
Nielsen and Zhou (Mean size of wavelet packets, ACHA 13 (2002), 22–34) conjectured that for every Daubechies filter of length at least four the associated wavelet packets and their integer shifts fail to be a Schauder basis of Lp(R) for every p ≠ 2; their own ℓ1/ℓ∞ estimate only reaches extreme exponents. For the length-four filter, we prove the Schauder-basis-failure assertion for every 1 < p < ∞ with p ≠ 2. The proof combines convexity of the wavelet packet pressure function with the uniqueness of equilibrium states for irreducible matrix families due to Feng and Käenmäki (2011) and an exact algebraic separation of two periodic spectral growth rates of the high-pass transition matrices.
In the finite-dimensional setting, it is known that STFT phase retrieval is always possible when the window's ambiguity function does not vanish. However, it is not known how many zeros are allowed in the ambiguity function for the window still to allow phase retrieval. In order to tackle this problem, we first consider a two-window approach where the second window equals the Fourier transform of the first window. This allows us to apply the uncertainty principle in order to obtain sufficient conditions for phase retrieval. Using the relation between STFT phase retrieval and ambiguity sampling, we can prove sufficient conditions for the single-window phase retrieval problem, showing that only approximately eight ninths of the entries of the window's ambiguity function (and only three quarters in prime dimensions) are required to be nonzero.
