5,915 research outputs found

    On Single-Sequence and Multi-Sequence Factorizations

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    Subspace based factorization methods are commonly used for a variety of applications, such as 3D reconstruction, multi-body segmentation and optical flow estimation. These are usually applied to a single video sequence. In this paper we present an analysis of the multi-sequence case and place it under a single framework with the single sequence case. In particular, we start by analyzing the characteristics of subspace based spatial and temporal segmentation. We show that in many cases objects moving with different 3D motions will be captured as a single object using multi-body (spatial) factorization approaches. Similarly, frames viewing different shapes might be grouped as displaying the same shape in the temporal factorization framework. Temporal factorization provides temporal grouping of frames by employing a subspace based approach to capture non-rigid shape changes (Zelnik-Manor and Irani, 2004). We analyze what causes these degeneracies and show that in the case of multiple sequences these can be made useful and provide information for both temporal synchronization of sequences and spatial matching of points across sequences

    Unsupervised discovery of temporal sequences in high-dimensional datasets, with applications to neuroscience.

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    Identifying low-dimensional features that describe large-scale neural recordings is a major challenge in neuroscience. Repeated temporal patterns (sequences) are thought to be a salient feature of neural dynamics, but are not succinctly captured by traditional dimensionality reduction techniques. Here, we describe a software toolbox-called seqNMF-with new methods for extracting informative, non-redundant, sequences from high-dimensional neural data, testing the significance of these extracted patterns, and assessing the prevalence of sequential structure in data. We test these methods on simulated data under multiple noise conditions, and on several real neural and behavioral datas. In hippocampal data, seqNMF identifies neural sequences that match those calculated manually by reference to behavioral events. In songbird data, seqNMF discovers neural sequences in untutored birds that lack stereotyped songs. Thus, by identifying temporal structure directly from neural data, seqNMF enables dissection of complex neural circuits without relying on temporal references from stimuli or behavioral outputs

    D\'{e}vissage for Waldhausen K-theory

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    A d\'evissage-type theorem in algebraic K-theory is a statement that identifies the K-theory of a Waldhausen category C\mathscr{C} in terms of the K-theories of a collection of Waldhausen subcategories of C\mathscr{C} when a d\'evissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between d\'evissage theorems of single type and of multiple type depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original d\'evissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of d\'evissage-type theorems and prove a general d\'evissage theorem of single type and a general d\'evissage theorem of multiple type.Comment: 28 page
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