Localization in 1D non-parametric latent space models from pairwise affinities

Abstract

We consider the problem of estimating latent positions in a one-dimensional torus from pairwise affinities. The observed affinity between a pair of items is modeled as a noisy observation of a function f (x * i , x * j ) of the latent positions x * i , x * j of the two items on the torus. The affinity function f is unknown, and it is only assumed to fulfill some shape constraints ensuring that f (x, y) is large when the distance between x and y is small, and vice-versa. This non-parametric modeling offers a good flexibility to fit data. We introduce an estimation procedure that provably localizes all the latent positions with a maximum error of the order of log(n)/n, with highprobability. This rate is proven to be minimax optimal. A computationally efficient variant of the procedure is also analyzed under some more restrictive assumptions. Our general results can be instantiated to the problem of statistical seriation, leading to new bounds for the maximum error in the ordering.

Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms

“…Alternatively, spectral approaches have been widely used for matrix reordering tasks. Among them, a spectral seriation method based on the Fiedler vector is particularly popular and has been extensively studied in the literature [5,2,27,60,61,31]. In this section, we show that, albeit the success of such a spectral seriation algorithm in many applications, it is nonetheless substantially suboptimal compared to the constrained LSE.…”

“…Many statistical seriation problems [6,33,44], that in one way or another aim to find an element in the discrete permutation set optimizing certain objective function, have been studied from various aspects under different settings. These include the well-known consecutive one's problem [29,41,42] that dates back to the 1960s; the feature matching problem [23,38,30] and the noisy ranking problem [11,39,54,20,63,51,55,22]; the matrix seriation problem for various shape-constrained matrices including the monotone or bi-monotone matrices [26,50,47,57], the Robinson matrices [2,27,61,1], and the Monge matrices [36]; and more recently, the seriation problem under the latent space models [31,37]. Many of the existing works have focused on recovering the underlying permutations, estimation of the (disordered) signal structures, or both.…”

Detection-Recovery Gap for Planted Dense Cycles

“…In the case where the bandwidth k is at most n o(1) , sharp characterizations of recovery conditions are given in [BDT + 20, DWXY20] under a more general model. Moreover, several other algorithms and analyses have been introduced for related models from the perspective of graphon estimation by [JS22,NS21,GIV21]. However, none of the previous works have shown computational lower bounds against a class of efficient algorithms.…”

A simple and optimal algorithm for strict circular seriation

“…For origins of circular seriation see the papers [10,12,11] and for recent applications of circular seriation in planar tomographic reconstruction, gene expression, DNA replication and 3D conformation, see the papers [7,17,16]. For a spectral approach to circular seriation, see the papers [8,9,21]. At the difference of the classical seriation, where the notion of Robinson dissimilarity is a well-established standard, in circular seriation there are several non-equivalent notions of circular Robinson dissimilarities.…”