Efficient On-the-fly Category Retrieval using ConvNets and GPUs

We investigate the gains in precision and speed, that can be obtained by using Convolutional Networks (ConvNets) for on-the-fly retrieval - where classifiers are learnt at run time for a textual query from downloaded images, and used to rank large image or video datasets. We make three contributions: (i) we present an evaluation of state-of-the-art image representations for object category retrieval over standard benchmark datasets containing 1M+ images; (ii) we show that ConvNets can be used to obtain features which are incredibly performant, and yet much lower dimensional than previous state-of-the-art image representations, and that their dimensionality can be reduced further without loss in performance by compression using product quantization or binarization. Consequently, features with the state-of-the-art performance on large-scale datasets of millions of images can fit in the memory of even a commodity GPU card; (iii) we show that an SVM classifier can be learnt within a ConvNet framework on a GPU in parallel with downloading the new training images, allowing for a continuous refinement of the model as more images become available, and simultaneous training and ranking. The outcome is an on-the-fly system that significantly outperforms its predecessors in terms of: precision of retrieval, memory requirements, and speed, facilitating accurate on-the-fly learning and ranking in under a second on a single GPU.Comment: Published in proceedings of ACCV 201

Maximal Newton points and the quantum Bruhat graph

We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety.Comment: 39 pages, 4 figures best viewed in color; final version to appear in Michigan Math.

Virtual polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility

Virtual Polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided