Nonnegative least squares (NNLS) problems arise in models that rely on additive linear combinations. In particular, they are at the core of nonnegative matrix factorization (NMF) algorithms. The nonnegativity constraint is known to naturally favor sparsity, that is, solutions with few non-zero entries. However, it is often useful to further enhance this sparsity, as it improves the interpretability of the results and helps reducing noise. While the $\ell_0$-"norm", equal to the number of non-zeros entries in a vector, is a natural sparsity measure, its combinatorial nature makes it difficult to use in practical optimization schemes. Most existing approaches thus rely either on its convex surrogate, the $\ell_1$-norm, or on heuristics such as greedy algorithms. In the case of multiple right-hand sides NNLS (MNNLS), which are used within NMF algorithms, sparsity is often enforced column- or row-wise, and the fact that the solution is a matrix is not exploited. In this paper, we first introduce a novel formulation for sparse MNNLS, with a matrix-wise $\ell_0$ sparsity constraint. Then, we present a two-step algorithm to tackle this problem. The first step uses a homotopy algorithm to produce the whole regularization path for all the $\ell_1$-penalized NNLS problems arising in MNNLS, that is, to produce a set of solutions representing different tradeoffs between reconstruction error and sparsity. The second step selects solutions among these paths in order to build a sparsity-constrained matrix that minimizes the reconstruction error. We illustrate the advantages of our proposed algorithm for the unmixing of facial and hyperspectral images.

Disney already has humans performing stunts at places like Shanghai Disneyland. And of course, they already have animatronic figures, but those guys aren't exactly made to fly. Take the "Shaman" character in an "Avatar"-themed attraction in Florida, for example. "It's a very, very sophisticated, very, very expressive animatronic figure," says Tony Dohi, a principle R&D "imagineer" with Walt Disney Imagineering Research and Development. "Yet it's definitely not something we'd want to throw 90 feet across an outdoor space."

It starts without warning--or rather, the warnings are there, but your ability to detect them exists only in hindsight. First you're sitting in the car with your son, then he tells you: "I cannot find my old self again." You think, well, teenagers say dramatic stuff like this all the time. Then he's refusing to do his homework, he's writing suicidal messages on the wall in black magic marker, he's trying to cut himself with a razor blade. You sit down with him; you two have a long talk. A week later, he runs home from a nighttime gathering at his friend's apartment, he's bursting through the front door, shouting about how his friends are trying to kill him. He spends the night crouching in his mother's old room, clutching a stuffed animal to his chest. He's 17 years old at this point, and you are his father, Dick Russell, a traveler, a former staff reporter for Sports Illustrated, but a father first and foremost.

It starts without warning--or rather, the warnings are there, but your ability to detect them exists only in hindsight. First you're sitting in the car with your son, then he tells you: "I cannot find my old self again." You think, well, teenagers say dramatic stuff like this all the time. Then he's refusing to do his homework, he's writing suicidal messages on the wall in black magic marker, he's trying to cut himself with a razor blade. You sit down with him; you two have a long talk. A week later, he runs home from a nighttime gathering at his friend's apartment, he's bursting through the front door, shouting about how his friends are trying to kill him. He spends the night crouching in his mother's old room, clutching a stuffed animal to his chest. He's 17 years old at this point, and you are his father, Dick Russell, a traveler, a former staff reporter for Sports Illustrated, but a father first and foremost.