Analysis of the Optimization Landscapes for Overcomplete Representation Learning
This provides theoretical justification for unsupervised learning methods in high-dimensional data analysis, though it is incremental as it builds on existing empirical successes.
The paper tackles the problem of understanding why nonconvex algorithms work well for learning overcomplete representations, such as sparsely used and convolutional dictionaries, by showing that the optimization landscapes have benign geometric structures where local minimizers are close to target solutions and saddle points have negative curvature, enabling guaranteed global optimization with simple initializations.
We study nonconvex optimization landscapes for learning overcomplete representations, including learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries, where these unsupervised learning problems find many applications in high-dimensional data analysis. Despite the empirical success of simple nonconvex algorithms, theoretical justifications of why these methods work so well are far from satisfactory. In this work, we show these problems can be formulated as $\ell^4$-norm optimization problems with spherical constraint, and study the geometric properties of their nonconvex optimization landscapes. For both problems, we show the nonconvex objectives have benign (global) geometric structures, in the sense that every local minimizer is close to one of the target solutions and every saddle point exhibits negative curvature. This discovery enables the development of guaranteed global optimization methods using simple initializations. For both problems, we show the nonconvex objectives have benign geometric structures -- every local minimizer is close to one of the target solutions and every saddle point exhibits negative curvature -- either in the entire space or within a sufficiently large region. This discovery ensures local search algorithms (such as Riemannian gradient descent) with simple initializations approximately find the target solutions. Finally, numerical experiments justify our theoretical discoveries.