We consider variants of trust-region and adaptive cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under certain condition on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve ǫ-approximate second-order optimality which have been shown to be tight. Our Hessian approximation condition offers a range of advantages as compared with the prior works and allows for direct construction of the approximate Hessian with a priori guarantees through various techniques, including randomized sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding subsampled trust-region and adaptive cubic regularization methods.
We propose a deep hashing framework for sketch retrieval that, for the first time, works on a multi-million scale human sketch dataset. Leveraging on this large dataset, we explore a few sketch-specific traits that were otherwise under-studied in prior literature. Instead of following the conventional sketch recognition task, we introduce the novel problem of sketch hashing retrieval which is not only more challenging, but also offers a better testbed for large-scale sketch analysis, since: (i) more fine-grained sketch feature learning is required to accommodate the large variations in style and abstraction, and (ii) a compact binary code needs to be learned at the same time to enable efficient retrieval. Key to our network design is the embedding of unique characteristics of human sketch, where (i) a two-branch CNN-RNN architecture is adapted to explore the temporal ordering of strokes, and (ii) a novel hashing loss is specifically designed to accommodate both the temporal and abstract traits of sketches. By working with a 3.8M sketch dataset, we show that state-of-the-art hashing models specifically engineered for static images fail to perform well on temporal sketch data. Our network on the other hand not only offers the best retrieval performance on various code sizes, but also yields the best generalization performance under a zero-shot setting and when re-purposed for sketch recognition. Such superior performances effectively demonstrate the benefit of our sketch-specific design.
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