Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations. We propose and study a novel framework to study sets of phylogenetic trees based on tropical geometry. In particular, we focus on characterizing our framework for statistical analyses of evolutionary biological processes represented by phylogenetic trees. Our setting exhibits analytic, geometric, and topological properties that are desirable for theoretical studies in probability and statistics, as well as increased computational efficiency over the current state-of-the-art. We demonstrate our approach on seasonal influenza data.Comment: 28 pages, 5 figures, 1 tabl

Classification of Occluded Objects using Fast Recurrent Processing

Recurrent neural networks are powerful tools for handling incomplete data problems in computer vision, thanks to their significant generative capabilities. However, the computational demand for these algorithms is too high to work in real time, without specialized hardware or software solutions. In this paper, we propose a framework for augmenting recurrent processing capabilities into a feedforward network without sacrificing much from computational efficiency. We assume a mixture model and generate samples of the last hidden layer according to the class decisions of the output layer, modify the hidden layer activity using the samples, and propagate to lower layers. For visual occlusion problem, the iterative procedure emulates feedforward-feedback loop, filling-in the missing hidden layer activity with meaningful representations. The proposed algorithm is tested on a widely used dataset, and shown to achieve 2$\times$ improvement in classification accuracy for occluded objects. When compared to Restricted Boltzmann Machines, our algorithm shows superior performance for occluded object classification.Comment: arXiv admin note: text overlap with arXiv:1409.8576 by other author

Learning Temporal Transformations From Time-Lapse Videos

Based on life-long observations of physical, chemical, and biologic phenomena in the natural world, humans can often easily picture in their minds what an object will look like in the future. But, what about computers? In this paper, we learn computational models of object transformations from time-lapse videos. In particular, we explore the use of generative models to create depictions of objects at future times. These models explore several different prediction tasks: generating a future state given a single depiction of an object, generating a future state given two depictions of an object at different times, and generating future states recursively in a recurrent framework. We provide both qualitative and quantitative evaluations of the generated results, and also conduct a human evaluation to compare variations of our models.Comment: ECCV201

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Disjunctive Normal Level Set: An Efficient Parametric Implicit Method

Level set methods are widely used for image segmentation because of their capability to handle topological changes. In this paper, we propose a novel parametric level set method called Disjunctive Normal Level Set (DNLS), and apply it to both two phase (single object) and multiphase (multi-object) image segmentations. The DNLS is formed by union of polytopes which themselves are formed by intersections of half-spaces. The proposed level set framework has the following major advantages compared to other level set methods available in the literature. First, segmentation using DNLS converges much faster. Second, the DNLS level set function remains regular throughout its evolution. Third, the proposed multiphase version of the DNLS is less sensitive to initialization, and its computational cost and memory requirement remains almost constant as the number of objects to be simultaneously segmented grows. The experimental results show the potential of the proposed method.Comment: 5 page