6 research outputs found
Homomorphic Sensing of Subspace Arrangements
Homomorphic sensing is a recent algebraic-geometric framework that studies
the unique recovery of points in a linear subspace from their images under a
given collection of linear maps. It has been successful in interpreting such a
recovery in the case of permutations composed by coordinate projections, an
important instance in applications known as unlabeled sensing, which models
data that are out of order and have missing values. In this paper, we provide
tighter and simpler conditions that guarantee the unique recovery for the
single-subspace case, extend the result to the case of a subspace arrangement,
and show that the unique recovery in a single subspace is locally stable under
noise. We specialize our results to several examples of homomorphic sensing
such as real phase retrieval and unlabeled sensing. In so doing, in a unified
way, we obtain conditions that guarantee the unique recovery for those
examples, typically known via diverse techniques in the literature, as well as
novel conditions for sparse and unsigned versions of unlabeled sensing.
Similarly, our noise result also implies that the unique recovery in unlabeled
sensing is locally stable.Comment: 18 page
Accelerating Globally Optimal Consensus Maximization in Geometric Vision
Branch-and-bound-based consensus maximization stands out due to its important
ability of retrieving the globally optimal solution to outlier-affected
geometric problems. However, while the discovery of such solutions caries high
scientific value, its application in practical scenarios is often prohibited by
its computational complexity growing exponentially as a function of the
dimensionality of the problem at hand. In this work, we convey a novel, general
technique that allows us to branch over an dimensional space for an
n-dimensional problem. The remaining degree of freedom can be solved globally
optimally within each bound calculation by applying the efficient interval
stabbing technique. While each individual bound derivation is harder to compute
owing to the additional need for solving a sorting problem, the reduced number
of intervals and tighter bounds in practice lead to a significant reduction in
the overall number of required iterations. Besides an abstract introduction of
the approach, we present applications to three fundamental geometric computer
vision problems: camera resectioning, relative camera pose estimation, and
point set registration. Through our exhaustive tests, we demonstrate
significant speed-up factors at times exceeding two orders of magnitude,
thereby increasing the viability of globally optimal consensus maximizers in
online application scenarios
Unlabeled Principal Component Analysis
We consider the problem of principal component analysis from a data matrix
where the entries of each column have undergone some unknown permutation,
termed Unlabeled Principal Component Analysis (UPCA). Using algebraic geometry,
we establish that for generic enough data, and up to a permutation of the
coordinates of the ambient space, there is a unique subspace of minimal
dimension that explains the data. We show that a permutation-invariant system
of polynomial equations has finitely many solutions, with each solution
corresponding to a row permutation of the ground-truth data matrix. Allowing
for missing entries on top of permutations leads to the problem of unlabeled
matrix completion, for which we give theoretical results of similar flavor. We
also propose a two-stage algorithmic pipeline for UPCA suitable for the
practically relevant case where only a fraction of the data has been permuted.
Stage-I of this pipeline employs robust-PCA methods to estimate the
ground-truth column-space. Equipped with the column-space, stage-II applies
methods for linear regression without correspondences to restore the permuted
data. A computational study reveals encouraging findings, including the ability
of UPCA to handle face images from the Extended Yale-B database with
arbitrarily permuted patches of arbitrary size in seconds on a standard
desktop computer