3 research outputs found
Using the Eigenvalue Relaxation for Binary Least-Squares Estimation Problems
The goal of this paper is to survey the properties of the eigenvalue
relaxation for least squares binary problems. This relaxation is a convex
program which is obtained as the Lagrangian dual of the original problem with
an implicit compact constraint and as such, is a convex problem with polynomial
time complexity. Moreover, as a main pratical advantage of this relaxation over
the standard Semi-Definite Programming approach, several efficient bundle
methods are available for this problem allowing to address problems of very
large dimension. The necessary tools from convex analysis are recalled and
shown at work for handling the problem of exactness of this relaxation. Two
applications are described. The first one is the problem of binary image
reconstruction and the second is the problem of multiuser detection in CDMA
systems
Learning with Semi-Definite Programming: new statistical bounds based on fixed point analysis and excess risk curvature
Many statistical learning problems have recently been shown to be amenable to
Semi-Definite Programming (SDP), with community detection and clustering in
Gaussian mixture models as the most striking instances [javanmard et al.,
2016]. Given the growing range of applications of SDP-based techniques to
machine learning problems, and the rapid progress in the design of efficient
algorithms for solving SDPs, an intriguing question is to understand how the
recent advances from empirical process theory can be put to work in order to
provide a precise statistical analysis of SDP estimators.
In the present paper, we borrow cutting edge techniques and concepts from the
learning theory literature, such as fixed point equations and excess risk
curvature arguments, which yield general estimation and prediction results for
a wide class of SDP estimators. From this perspective, we revisit some
classical results in community detection from [gu\'edon et al.,2016] and [chen
et al., 2016], and we obtain statistical guarantees for SDP estimators used in
signed clustering, group synchronization and MAXCUT