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1. Sampling Sparse Representations with Randomized Measurement Langevin Dynamics

   https://doi.org/10.1145/3427585  

   Wang, Kafeng ; Xiong, Haoyi ; Bian, Jiang ; Zhu, Zhanxing ; Gao, Qian ; Guo, Zhishan ; Xu, Cheng-Zhong ; Huan, Jun ; Dou, Dejing ( April 2021 , ACM Transactions on Knowledge Discovery from Data)

   Stochastic Gradient Langevin Dynamics (SGLD) have been widely used for Bayesian sampling from certain probability distributions, incorporating derivatives of the log-posterior. With the derivative evaluation of the log-posterior distribution, SGLD methods generate samples from the distribution through performing as a thermostats dynamics that traverses over gradient flows of the log-posterior with certainly controllable perturbation. Even when the density is not known, existing solutions still can first learn the kernel density models from the given datasets, then produce new samples using the SGLD over the kernel density derivatives. In this work, instead of exploring new samples from kernel spaces, a novelmore »SGLD sampler, namely, Randomized Measurement Langevin Dynamics (RMLD) is proposed to sample the high-dimensional sparse representations from the spectral domain of a given dataset. Specifically, given a random measurement matrix for sparse coding, RMLD first derives a novel likelihood evaluator of the probability distribution from the loss function of LASSO, then samples from the high-dimensional distribution using stochastic Langevin dynamics with derivatives of the logarithm likelihood and Metropolis–Hastings sampling. In addition, new samples in low-dimensional measuring spaces can be regenerated using the sampled high-dimensional vectors and the measurement matrix. The algorithm analysis shows that RMLD indeed projects a given dataset into a high-dimensional Gaussian distribution with Laplacian prior, then draw new sparse representation from the dataset through performing SGLD over the distribution. Extensive experiments have been conducted to evaluate the proposed algorithm using real-world datasets. The performance comparisons on three real-world applications demonstrate the superior performance of RMLD beyond baseline methods.« less
2. Extreme Compressed Sensing of Poisson Rates from Multiple Measurements

   Kota, Pavan K ; LeJeune, Daniel ; Drezek, Rebekah A ; Baraniuk, Richard G. ( March 2021 , ArXivorg)

   Compressed sensing (CS) is a signal processing technique that enables the efficient recovery of a sparse high-dimensional signal from low-dimensional measurements. In the multiple measurement vector (MMV) framework, a set of signals with the same support must be recovered from their corresponding measurements. Here, we present the first exploration of the MMV problem where signals are independently drawn from a sparse, multivariate Poisson distribution. We are primarily motivated by a suite of biosensing applications of microfluidics where analytes (such as whole cells or biomarkers) are captured in small volume partitions according to a Poisson distribution. We recover the sparse parametermore »vector of Poisson rates through maximum likelihood estimation with our novel Sparse Poisson Recovery (SPoRe) algorithm. SPoRe uses batch stochastic gradient ascent enabled by Monte Carlo approximations of otherwise intractable gradients. By uniquely leveraging the Poisson structure, SPoRe substantially outperforms a comprehensive set of existing and custom baseline CS algorithms. Notably, SPoRe can exhibit high performance even with one-dimensional measurements and high noise levels. This resource efficiency is not only unprecedented in the field of CS but is also particularly potent for applications in microfluidics in which the number of resolvable measurements per partition is often severely limited. We prove the identifiability property of the Poisson model under such lax conditions, analytically develop insights into system performance, and confirm these insights in simulated experiments. Our findings encourage a new approach to biosensing and are generalizable to other applications featuring spatial and temporal Poisson signals.« less
3. Scaling Up Sparse Support Vector Machine by Simultaneous Feature and Sample Reduction

   Zhang, W ; Hong, B ; Ye, J ; Cai, D ; He, X ; Wang, J. ( January 2017 , Proceedings of the 34th International Conference on Machine Learning)

   Sparse support vector machine (SVM) is a popular classification technique that can simultaneously learn a small set of the most interpretable features and identify the support vectors. It has achieved great successes in many real-world applications. However, for large-scale problems involving a huge number of samples and extremely high-dimensional features, solving sparse SVMs remains challenging. By noting that sparse SVMs induce sparsities in both feature and sample spaces, we propose a novel approach, which is based on accurate estimations of the primal and dual optima of sparse SVMs, to simultaneously identify the features and samples that are guaranteed to bemore »irrelevant to the outputs. Thus, we can remove the identified inactive samples and features from the training phase, leading to substantial savings in both the memory usage and computational cost without sacrificing accuracy. To the best of our knowledge, the proposed method is the first static feature and sample reduction method for sparse SVMs. Experiments on both synthetic and real datasets (e.g., the kddb dataset with about 20 million samples and 30 million features) demonstrate that our approach significantly outperforms state-of-the-art methods and the speedup gained by our approach can be orders of magnitude.« less
4. Low-Degree Hardness of Random Optimization Problems

   https://doi.org/10.1109/FOCS46700.2020.00021  

   Gamarnik, David ; Jagannath, Aukosh ; Wein, Alexander S ( November 2020 , 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS))

   We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Such problems arise widely in the theory of random graphs, theoretical computer science, and statistical physics. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model, and (b) finding a large independent set in a sparse Erdos-Renyi graph. Two families of algorithms are considered: (a) low-degree polynomials of the input-a general framework that captures methods such as approximate message passing and local algorithms on sparse graphs, among others; and (b) the Langevin dynamics algorithm, a canonicalmore »Monte Carlo analogue of the gradient descent algorithm (applicable only for the spherical p-spin glass Hamiltonian). We show that neither family of algorithms can produce nearly optimal solutions with high probability. Our proof uses the fact that both models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objective values are above a certain threshold are either close or far from each other. The crux of our proof is the stability of both algorithms: a small perturbation of the input induces a small perturbation of the output. By an interpolation argument, such a stable algorithm cannot overcome the OGP barrier. The stability of the Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials is established using concepts from Gaussian and Boolean Fourier analysis, including noise sensitivity, hypercontractivity, and total influence.« less
5. Algorithm and Hardware Co-Design for FPGA Acceleration of Hamiltonian Monte Carlo Based No-U-Turn Sampler

   Wang, Y. ; Li, P. ( July 2021 , The 32nd IEEE International Conference on Application-specific Systems, Architectures and Processors)

   Monte Carlo (MC) methods are widely used in many research areas such as physical simulation, statistical analysis, and machine learning. Application of MC methods requires drawing fast mixing samples from a given probability distribution. Among existing sampling methods, the Hamiltonian Monte Carlo (HMC) utilizes gradient information during Hamiltonian simulation and can produce fast mixing samples at the highest efficiency. However, without carefully chosen simulation parameters for a specific problem, HMC generally suffers from simulation locality and computation waste. As a result, the No-U-Turn Sampler (NUTS) has been proposed to automatically tune these parameters during simulation and is the current state-of-the-artmore »sampling algorithm. However, application of NUTS requires frequent gradient calculation of a given distribution and high-volume vector processing, especially for large-scale problems, leading to drawing an expensively large number of samples and a desire of hardware acceleration. While some hardware acceleration works have been proposed for traditional Markov Chain Monte Carlo (MCMC) and HMC methods, there is no existing work targeting hardware acceleration of the NUTS algorithm. In this paper, we present the first NUTS accelerator on FPGA while addressing the high complexity of this state-of-the-art algorithm. Our hardware and algorithm co-optimizations include an incremental resampling technique which leads to a more memory efficient architecture and pipeline optimization for multi-chain sampling to maximize the throughput. We also explore three levels of parallelism in the NUTS accelerator to further boost performance. Compared with optimized C++ NUTS package: RSTAN, our NUTS accelerator can reach a maximum speedup of 50.6X and an energy improvement of 189.7X.« less