More Like this

1. Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons

   Anari, Nima ; Daskalakis, Constantinos ; Maass, Wolfgang ; Papadimitriou, Christos ; Vempala, Santosh ( January 2018 , 32nd Annual Conference on Neural Information Processing Systems)

   We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recovering assemblies of neurons. Assemblies are large sets of neurons representing specific memories or concepts. The size of the intersection of two assemblies has been shown in experiments to represent the extent to which these memories co-occur or these concepts are related; the phenomenon is called association of assemblies. This suggests that an animal's memory is a complex web of associations, and poses the problem of recovering this representation from cognitive data. Motivated by this problem, we study the following more general question: Can we reconstruct the Venn diagram of a family of sets, given the sizes of their ℓ-wise intersections? We show that as long as the family of sets is randomly perturbed, it is enough for the number of measurements to be polynomially larger than the number of nonempty regions of the Venn diagram to fully reconstruct the diagram. 

   more » « less
2. Brain Computation by Assemblies of Neurons

   Christos H. Papadimitriou, Santosh S. ( June 2020 , Proceedings of the National Academy of Sciences of the United States of America)

   Our expanding understanding of the brain at the level of neurons and synapses, and the level of cognitive phenomena such as language, leaves a formidable gap between these two scales. Here we introduce a computational system which promises to bridge this gap: the Assembly Calculus. It encompasses operations on assemblies of neurons, such as project, associate, and merge, which appear to be implicated in cognitive phenomena, and can be shown, analytically as well as through simulations, to be plausibly realizable at the level of neurons and synapses. We demonstrate the reach of this system by proposing a brain architecture for syntactic processing in the production of language, compatible with recent experimental results. Abstract Assemblies are large populations of neurons believed to imprint memories, concepts, words, and other cognitive information. We identify a repertoire of operations on assemblies. These operations correspond to properties of assemblies observed in experiments, and can be shown, analytically and through simulations, to be realizable by generic, randomly connected populations of neurons with Hebbian plasticity and inhibition. Assemblies and their operations constitute a computational model of the brain which we call the Assembly Calculus, occupying a level of detail intermediate between the level of spiking neurons and synapses and that of the whole brain. The resulting computational system can be shown, under assumptions, to be, in principle, capable of carrying out arbitrary computations. We hypothesize that something like it may underlie higher human cognitive functions such as reasoning, planning, and language. In particular, we propose a plausible brain architecture based on assemblies for implementing the syntactic processing of language in cortex, which is consistent with recent experimental results. 

   more » « less
3. Fast quasi-harmonic weights for geometric data interpolation

   https://doi.org/10.1145/3450626.3459801  

   Wang, Yu ; Solomon, Justin ( August 2021 , ACM Transactions on Graphics)

   We propose quasi-harmonic weights for interpolating geometric data, which are orders of magnitude faster to compute than state-of-the-art. Currently, interpolation (or, skinning) weights are obtained by solving large-scale constrained optimization problems with explicit constraints to suppress oscillative patterns, yielding smooth weights only after a substantial amount of computation time. As an alternative, our weights are obtained as minima of an unconstrained problem that can be optimized quickly using straightforward numerical techniques. We consider weights that can be obtained as solutions to a parameterized family of second-order elliptic partial differential equations. By leveraging the maximum principle and careful parameterization, we pose weight computation as an inverse problem of recovering optimal anisotropic diffusivity tensors. In addition, we provide a customized ADAM solver that significantly reduces the number of gradient steps; our solver only requires inverting tens of linear systems that share the same sparsity pattern. Overall, our approach achieves orders of magnitude acceleration compared to previous methods, allowing weight computation in near real-time. 

   more » « less
4. PLANC: Parallel Low-rank Approximation with Nonnegativity Constraints

   https://doi.org/10.1145/3432185  

   Eswar, Srinivas ; Hayashi, Koby ; Ballard, Grey ; Kannan, Ramakrishnan ; Matheson, Michael A. ; Park, Haesun ( June 2021 , ACM Transactions on Mathematical Software)

   null (Ed.)

   We consider the problem of low-rank approximation of massive dense nonnegative tensor data, for example, to discover latent patterns in video and imaging applications. As the size of data sets grows, single workstations are hitting bottlenecks in both computation time and available memory. We propose a distributed-memory parallel computing solution to handle massive data sets, loading the input data across the memories of multiple nodes, and performing efficient and scalable parallel algorithms to compute the low-rank approximation. We present a software package called Parallel Low-rank Approximation with Nonnegativity Constraints, which implements our solution and allows for extension in terms of data (dense or sparse, matrices or tensors of any order), algorithm (e.g., from multiplicative updating techniques to alternating direction method of multipliers), and architecture (we exploit GPUs to accelerate the computation in this work). We describe our parallel distributions and algorithms, which are careful to avoid unnecessary communication and computation, show how to extend the software to include new algorithms and/or constraints, and report efficiency and scalability results for both synthetic and real-world data sets. 

   more » « less
5. Adaptive granularity in tensors: A quest for interpretable structure

   https://doi.org/10.3389/fdata.2022.929511  

   Pasricha, Ravdeep S. ; Gujral, Ekta ; Papalexakis, Evangelos E. ( November 2022 , Frontiers in Big Data)

   Data collected at very frequent intervals is usually extremely sparse and has no structure that is exploitable by modern tensor decomposition algorithms. Thus, the utility of such tensors is low, in terms of the amount of interpretable and exploitable structure that one can extract from them. In this paper, we introduce the problem of finding a tensor of adaptive aggregated granularity that can be decomposed to reveal meaningful latent concepts (structures) from datasets that, in their original form, are not amenable to tensor analysis. Such datasets fall under the broad category of sparse point processes that evolve over space and/or time. To the best of our knowledge, this is the first work that explores adaptive granularity aggregation in tensors. Furthermore, we formally define the problem and discuss different definitions of “good structure” that are in practice and show that the optimal solution is of prohibitive combinatorial complexity. Subsequently, we propose an efficient and effective greedy algorithm called I CE B REAKER , which follows a number of intuitive decision criteria that locally maximize the “goodness of structure,” resulting in high-quality tensors. We evaluate our method on synthetic, semi-synthetic, and real datasets. In all the cases, our proposed method constructs tensors that have a very high structure quality. 

   more » « less