In this paper, we design a Collaborative-Hierarchical Sparse and Low-Rank (C-HiSLR) model that is natural for recognizing human emotion in visual data. Previous attempts require explicit expression components, which are often unavailable and difficult to recover. Instead, our model exploits the low-rank property to subtract neutral faces from expressive facial frames as well as performs sparse representation on the expression components with group sparsity enforced. For the CK+ dataset, C-HiSLR on raw expressive faces performs as competitive as the Sparse Representation based Classification (SRC) applied on manually prepared emotions. Our C-HiSLR performs even better than SRC in terms of true positive rate.
In practice, we care more about how to recover @math @cite_20 . Enforcing sparsity is feasible since @math can be exactly recovered from @math under conditions for @math @cite_13 . However, finding the sparsest solution is NP-hard and difficult to solve exactly @cite_22 . But now, it is well-known that the @math norm is a good convex relaxation of sparsity -- -- minimizing the @math norm induces the sparsest solution under mild conditions @cite_21 . Exact recovery is also guaranteed by @math -minimization under suitable conditions @cite_31 . Typically, an iterative greedy algorithm is the Orthogonal Matching Pursuit (OMP) @cite_20 .
@cite_22 @cite_21 @cite_31 @cite_13 @cite_20
2075779886 2164452299 2129131372 2145096794 2127271355
We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, ⩾, > and ≠ are considered. While Min RVLS with equations was mentioned to be NP-hard in (Garey and Johnson, 1979), we established in (Amaldi; 1992; Amaldi and Kann, 1995) that min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in (, 1993). In this paper we determine strong bounds on the approximability of various variants of Min RVLS and min ULR, including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional. The various NP-hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor, unless P = NP. Particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning. In particular, we disprove a conjecture of van Horn and Martinez (1992) regarding the existence of a polynomial-time algorithm to design linear classifiers (or perceptrons) that involve a close-to-minimum number of features. Suppose we wish to recover a vector x0 ∈ R m (e.g., a digital signal or image) from incomplete and contaminated observations y = Ax0 + e; A is an n × m This paper considers a natural error correcting problem with real valued input output. We wish to recover an input vector f spl isin R sup n from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the spl lscr sub 1 -minimization problem ( spl par x spl par sub spl lscr 1 := spl Sigma sub i |x sub i |) min(g spl isin R sup n ) spl par y - Ag spl par sub spl lscr 1 provided that the support of the vector of errors is not too large, spl par e spl par sub spl lscr 0 :=| i:e sub i spl ne 0 | spl les spl rho spl middot m for some spl rho >0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of spl lscr sub 1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail. This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f spl isin C sup N and a randomly chosen set of frequencies spl Omega . Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set spl Omega ? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)= spl sigma sub spl tau spl isin T f( spl tau ) spl delta (t- spl tau ) obeying |T| spl les C sub M spl middot (log N) sup -1 spl middot | spl Omega | for some constant C sub M >0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N sup -M ), f can be reconstructed exactly as the solution to the spl lscr sub 1 minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C sub M which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T| spl middot logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N sup -M ) would in general require a number of frequency samples at least proportional to |T| spl middot logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f. This paper demonstrates theoretically and empirically that a greedy algorithm called orthogonal matching pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m2) measurements. The new results for OMP are comparable with recent results for another approach called basis pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.