32 research outputs found
Least-Squares Covariance Matrix Adjustment
We consider the problem of finding the smallest adjustment to a given symmetric matrix, as measured by the Euclidean or Frobenius norm, so that it satisfies some given linear equalities and inequalities, and in addition is positive semidefinite. This least-squares covariance adjustment problem is a convex optimization problem, and can be efficiently solved using standard methods when the number of variables (i.e., entries in the matrix) is modest, say, under . Since the number of variables is , this corresponds to a limit around . Malick [{\it SIAM J. Matrix Anal.\ Appl.,} 26 (2005), pp. 272--284] studies a closely related problem and calls it the semidefinite least-squares problem. In this paper we formulate a dual problem that has no matrix inequality or matrix variables, and a number of (scalar) variables equal to the number of equality and inequality constraints in the original least-squares covariance adjustment problem. This dual problem allows us to solve far larger least-squares covariance adjustment problems than would be possible using standard methods. Assuming a modest number of constraints, problems with are readily solved by the dual method. The dual method coincides with the dual method proposed by Malick when there are no inequality constraints and can be obtained as an extension of his dual method when there are inequality constraints. Using the dual problem, we show that in many cases the optimal solution is a low rank update of the original matrix. When the original matrix has structure, such as sparsity, this observation allows us to solve very large least-squares covariance adjustment problems
An Efficient Dual Approach to Distance Metric Learning
Distance metric learning is of fundamental interest in machine learning
because the distance metric employed can significantly affect the performance
of many learning methods. Quadratic Mahalanobis metric learning is a popular
approach to the problem, but typically requires solving a semidefinite
programming (SDP) problem, which is computationally expensive. Standard
interior-point SDP solvers typically have a complexity of (with
the dimension of input data), and can thus only practically solve problems
exhibiting less than a few thousand variables. Since the number of variables is
, this implies a limit upon the size of problem that can
practically be solved of around a few hundred dimensions. The complexity of the
popular quadratic Mahalanobis metric learning approach thus limits the size of
problem to which metric learning can be applied. Here we propose a
significantly more efficient approach to the metric learning problem based on
the Lagrange dual formulation of the problem. The proposed formulation is much
simpler to implement, and therefore allows much larger Mahalanobis metric
learning problems to be solved. The time complexity of the proposed method is
, which is significantly lower than that of the SDP approach.
Experiments on a variety of datasets demonstrate that the proposed method
achieves an accuracy comparable to the state-of-the-art, but is applicable to
significantly larger problems. We also show that the proposed method can be
applied to solve more general Frobenius-norm regularized SDP problems
approximately
Worst-Case Linear Discriminant Analysis as Scalable Semidefinite Feasibility Problems
In this paper, we propose an efficient semidefinite programming (SDP)
approach to worst-case linear discriminant analysis (WLDA). Compared with the
traditional LDA, WLDA considers the dimensionality reduction problem from the
worst-case viewpoint, which is in general more robust for classification.
However, the original problem of WLDA is non-convex and difficult to optimize.
In this paper, we reformulate the optimization problem of WLDA into a sequence
of semidefinite feasibility problems. To efficiently solve the semidefinite
feasibility problems, we design a new scalable optimization method with
quasi-Newton methods and eigen-decomposition being the core components. The
proposed method is orders of magnitude faster than standard interior-point
based SDP solvers.
Experiments on a variety of classification problems demonstrate that our
approach achieves better performance than standard LDA. Our method is also much
faster and more scalable than standard interior-point SDP solvers based WLDA.
The computational complexity for an SDP with constraints and matrices of
size by is roughly reduced from to
( in our case).Comment: 14 page
Forecasting People Trajectories and Head Poses by Jointly Reasoning on Tracklets and Vislets
In this work, we explore the correlation between people trajectories and
their head orientations. We argue that people trajectory and head pose
forecasting can be modelled as a joint problem. Recent approaches on trajectory
forecasting leverage short-term trajectories (aka tracklets) of pedestrians to
predict their future paths. In addition, sociological cues, such as expected
destination or pedestrian interaction, are often combined with tracklets. In
this paper, we propose MiXing-LSTM (MX-LSTM) to capture the interplay between
positions and head orientations (vislets) thanks to a joint unconstrained
optimization of full covariance matrices during the LSTM backpropagation. We
additionally exploit the head orientations as a proxy for the visual attention,
when modeling social interactions. MX-LSTM predicts future pedestrians location
and head pose, increasing the standard capabilities of the current approaches
on long-term trajectory forecasting. Compared to the state-of-the-art, our
approach shows better performances on an extensive set of public benchmarks.
MX-LSTM is particularly effective when people move slowly, i.e. the most
challenging scenario for all other models. The proposed approach also allows
for accurate predictions on a longer time horizon.Comment: Accepted at IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE
INTELLIGENCE 2019. arXiv admin note: text overlap with arXiv:1805.0065
Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data
We study ridge estimation of the precision matrix in the high-dimensional
setting where the number of variables is large relative to the sample size. We
first review two archetypal ridge estimators and note that their utilized
penalties do not coincide with common ridge penalties. Subsequently, starting
from a common ridge penalty, analytic expressions are derived for two
alternative ridge estimators of the precision matrix. The alternative
estimators are compared to the archetypes with regard to eigenvalue shrinkage
and risk. The alternatives are also compared to the graphical lasso within the
context of graphical modeling. The comparisons may give reason to prefer the
proposed alternative estimators