6 research outputs found
Deterministic Conditions for Subspace Identifiability from Incomplete Sampling
Consider a generic -dimensional subspace of , , and
suppose that we are only given projections of this subspace onto small subsets
of the canonical coordinates. The paper establishes necessary and sufficient
deterministic conditions on the subsets for subspace identifiability.Comment: To appear in Proc. of IEEE ISIT, 201
A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion
Low-rank matrix completion (LRMC) problems arise in a wide variety of
applications. Previous theory mainly provides conditions for completion under
missing-at-random samplings. This paper studies deterministic conditions for
completion. An incomplete matrix is finitely rank- completable
if there are at most finitely many rank- matrices that agree with all its
observed entries. Finite completability is the tipping point in LRMC, as a few
additional samples of a finitely completable matrix guarantee its unique
completability. The main contribution of this paper is a deterministic sampling
condition for finite completability. We use this to also derive deterministic
sampling conditions for unique completability that can be efficiently verified.
We also show that under uniform random sampling schemes, these conditions are
satisfied with high probability if entries per column are
observed. These findings have several implications on LRMC regarding lower
bounds, sample and computational complexity, the role of coherence, adaptive
settings and the validation of any completion algorithm. We complement our
theoretical results with experiments that support our findings and motivate
future analysis of uncharted sampling regimes.Comment: This update corrects an error in version 2 of this paper, where we
erroneously assumed that columns with more than r+1 observed entries would
yield multiple independent constraint
Tensor Methods for Nonlinear Matrix Completion
In the low rank matrix completion (LRMC) problem, the low rank assumption
means that the columns (or rows) of the matrix to be completed are points on a
low-dimensional linear algebraic variety. This paper extends this thinking to
cases where the columns are points on a low-dimensional nonlinear algebraic
variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC).
Matrices whose columns belong to a union of subspaces (UoS) are an important
special case. We propose a LADMC algorithm that leverages existing LRMC methods
on a tensorized representation of the data. For example, a second-order
tensorization representation is formed by taking the outer product of each
column with itself, and we consider higher order tensorizations as well. This
approach will succeed in many cases where traditional LRMC is guaranteed to
fail because the data are low-rank in the tensorized representation but not in
the original representation. We also provide a formal mathematical
justification for the success of our method. In particular, we show bounds of
the rank of these data in the tensorized representation, and we prove sampling
requirements to guarantee uniqueness of the solution. Interestingly, the
sampling requirements of our LADMC algorithm nearly match the information
theoretic lower bounds for matrix completion under a UoS model. We also provide
experimental results showing that the new approach significantly outperforms
existing state-of-the-art methods for matrix completion in many situations