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Iterative error correction of asymptotically large associative memories is equivalent to a one-step learning rule. This rule is the inverse of the activation function of the memory. Spectral representations of nonlinear activation functions are used to obtain the inverse in closed form for Sparse Distributed Memory, Selected-Coordinate Design, and Radial Basis Functions.
Here, we present a greedy algorithm for computing selected eigenpairs of a large sparse matrix $H$ that can exploit localization features of the eigenvector. When the eigenvector to be computed is localized, meaning only a small number of its components have large magnitudes, the proposed algorithm identifies the location of these components in a greedy manner, and obtains approximations to the desired eigenpairs of $H$ by computing eigenpairs of a submatrix extracted from the corresponding rows and columns of $H$. Even when the eigenvector is not completely localized, the approximate eigenvectors obtained by the greedy algorithm can be used as good starting guesses to accelerate the convergence of an iterative eigensolver applied to $H$. We discuss a few possibilities for selecting important rows and columns of $H$ and techniques for constructing good initial guesses for an iterative eigensolver using the approximate eigenvectors returned from the greedy algorithm. We demonstrate the effectiveness of this approach with examples from nuclear quantum many-body calculations and many-body localization studies of quantum spin chains.
Abstract The demographic history of a population, and the distribution of fitness effects (DFE) of newly arising mutations in functional genomic regions, are fundamental factors dictating both genetic variation and evolutionary trajectories. Although both demographic and DFE inference has been performed extensively in humans, these approaches have generally either been limited to simple demographic models involving a single population, or, where a complex population history has been inferred, without accounting for the potentially confounding effects of selection at linked sites. Taking advantage of the coding-sparse nature of the genome, we propose a 2-step approach in which coalescent simulations are first used to infer a complex multi-population demographic model, utilizing large non-functional regions that are likely free from the effects of background selection. We then use forward-in-time simulations to perform DFE inference in functional regions, conditional on the complex demography inferred and utilizing expected background selection effects in the estimation procedure. Throughout, recombination and mutation rate maps were used to account for the underlying empirical rate heterogeneity across the human genome. Importantly, within this framework it is possible to utilize and fit multiple aspects of the data, and this inference scheme represents a generalized approach for such large-scale inference in species with coding-sparse genomes.
The Sparse Matrix approach for obtaining lead crystallization conditions has proven to be very fruitful for the crystallization of proteins and nucleic acids. Here we report a Sparse Matrix developed specifically for the crystallization of protein-DNA complexes. This method is rapid and economical, typically requiring 2.5 mg of complex to test 48 conditions. The method was originally developed to crystallize basic fibroblast growth factor (bFGF) complexed with DNA sequences identified through in vitro selection, or SELEX, methods. Two DNA aptamers that bind with approximately nanomolar affinity and inhibit the angiogenic properties of bFGF were selected for co-crystallization. The Sparse Matrix produced lead crystallization conditions for both bFGF-DNA complexes.
We propose an effective and robust algorithm for identifying partial differential equations (PDEs) with space-time varying coefficients from the noisy observation of a single solution trajectory. Identifying unknown differential equations from noisy data is a difficult task, and it is even more challenging with space and time varying coefficients in the PDE. The proposed algorithm, GP-IDENT, has three ingredients: (i) we use B-spline bases to express the unknown space and time varying coefficients, (ii) we propose Group Projected Subspace Pursuit (GPSP) to find a sequence of candidate PDEs with various levels of complexity, and (iii) we propose a new criterion for model selection using the Reduction in Residual (RR) to choose an optimal one among a pool of candidates. The new GPSP considers group projected subspaces which is more robust than existing methods in distinguishing correlated group features. We test GP-IDENT on a variety of PDEs and PDE systems, and compare it with the state-of-the-art parametric PDE identification algorithms under different settings to illustrate its outstanding performance. Furthermore, our experiments show that GP-IDENT is effective in identifying the correct terms from a large dictionary, and our model selection scheme is robust to noise.
The increasing availability of electronic health record (EHR) systems has created enormous potential for translational research. However, it is difficult to know all the relevant codes related to a phenotype due to the large number of codes available. Traditional data mining approaches often require the use of patient-level data, which hinders the ability to share data across institutions. In this project, we demonstrate that multi-center large-scale code embeddings can be used to efficiently identify relevant features related to a disease of interest. We constructed large-scale code embeddings for a wide range of codified concepts from EHRs from two large medical centers. We developed knowledge extraction via sparse embedding regression (KESER) for feature selection and integrative network analysis. We evaluated the quality of the code embeddings and assessed the performance of KESER in feature selection for eight diseases. Besides, we developed an integrated clinical knowledge map combining embedding data from both institutions. The features selected by KESER were comprehensive compared to lists of codified data generated by domain experts. Features identified via KESER resulted in comparable performance to those built upon features selected manually or with patient-level data. The knowledge map created using an integrative analysis identified disease-disease and disease-drug pairs more accurately compared to those identified using single institution data. Analysis of code embeddings via KESER can effectively reveal clinical knowledge and infer relatedness among codified concepts. KESER bypasses the need for patient-level data in individual analyses providing a significant advance in enabling multi-center studies using EHR data.
The standard formulation of Kanerva's sparse distributed memory (SDM) involves the selection of a large number of data storage locations, followed by averaging the data contained in those locations to reconstruct the stored data. A variant of this model is discussed, in which the predominant pattern is the focus of reconstruction. First, one architecture is proposed which returns the predominant pattern rather than the average pattern. However, this model will require too much storage for most uses. Next, a hybrid model is proposed, called tagged SDM, which approximates the results of the predominant pattern machine, but is nearly as efficient as Kanerva's original formulation. Finally, some experimental results are shown which confirm that significant improvements in the recall capability of SDM can be achieved using the tagged architecture.
A new design for a Sparse Distributed Memory, called the selected-coordinate design, is described. As in the original design, there are a large number of memory locations, each of which may be activated by many different addresses (binary vectors) in a very large address space. Each memory location is defined by specifying ten selected coordinates (bit positions in the address vectors) and a set of corresponding assigned values, consisting of one bit for each selected coordinate. A memory location is activated by an address if, for all ten of the locations's selected coordinates, the corresponding bits in the address vector match the respective assigned value bits, regardless of the other bits in the address vector. Some comparative memory capacity and signal-to-noise ratio estimates for the both the new and original designs are given. A few possible hardware embodiments of the new design are described.
As the HVAC&R industry mobilizes to deploy more low-GWP refrigerants, relevant standards are being continually reviewed and updated. Those include the general safety standards ISO 5149 and ASHRAE 15, and the equipment standards IEC and UL. The standards systematically set the allowable maximum amount of refrigerant that should be used in different equipment types and different applications. To do so, they rely on predictions of how a leaked refrigerant mass will disperse into a space. Dispersion characteristics, such as total flammable volume and its residence time, determine the risk associated with the presence of the flammable refrigerant. The standards have included provisions for the use of flammable refrigerants for approximately two decades. They relied on limited analytical analyses and test cases in their development. Dispersion of a refrigerant into a space is complex. Computational fluid dynamics (CFD) are the most accurate in predicting a given problem. However, CFD is computationally expensive and requires specialized expertise and resources and is not suitable for use by standards development working group as prediction tool. This paper presents the development of a reduced order model (ROM) that predicts the key dispersion characteristics relevant to the dispersion of a leaked refrigerant into a space for any combination of input variables. The inputs are the refrigerant release height, the total released refrigerant mass and its release flow rate, the refrigerant molecular weight, the ventilation flow rate, the floor area and height of the space, recirculation air flow rate, and the tightness of the space. The outputs are histograms of volume fraction of the room in prescribed concentration bins and the total mass of the refrigerant in each bin normalized by the total refrigerant charge at 13 prescribed simulation time stamps between 1 and 900 seconds. The ROM is constructed from a set of CFD simulations with carefully chosen combinations of input parameters. The selection if done using a multidimensional sparse grid which is a generalization of the classical tensor approach but offers additional flexibility and thus can be more carefully tuned towards a specific model. The tuning is done to improve the accuracy, measured in the difference between the output values of the ROM and the CFD model, while minimizing the computational cost, measured in number of CFD simulations which is orders of magnitude more expensive than the processing the training data.
The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.
An algorithm that performs sparse linear discriminant analysis (Sparse-LDA) finds near-optimal solutions in far less time than the prior art when specialized to binary classification (of 2 classes). Sparse-LDA is a type of feature- or variable- selection problem with numerous applications in statistics, machine learning, computer vision, computational finance, operations research, and bio-informatics. Because of its combinatorial nature, feature- or variable-selection problems are NP-hard or computationally intractable in cases involving more than 30 variables or features. Therefore, one typically seeks approximate solutions by means of greedy search algorithms. The prior Sparse-LDA algorithm was a greedy algorithm that considered the best variable or feature to add/ delete to/ from its subsets in order to maximally discriminate between multiple classes of data. The present algorithm is designed for the special but prevalent case of 2-class or binary classification (e.g. 1 vs. 0, functioning vs. malfunctioning, or change versus no change). The present algorithm provides near-optimal solutions on large real-world datasets having hundreds or even thousands of variables or features (e.g. selecting the fewest wavelength bands in a hyperspectral sensor to do terrain classification) and does so in typical computation times of minutes as compared to days or weeks as taken by the prior art. Sparse LDA requires solving generalized eigenvalue problems for a large number of variable subsets (represented by the submatrices of the input within-class and between-class covariance matrices). In the general (fullrank) case, the amount of computation scales at least cubically with the number of variables and thus the size of the problems that can be solved is limited accordingly. However, in binary classification, the principal eigenvalues can be found using a special analytic formula, without resorting to costly iterative techniques. The present algorithm exploits this analytic form along with the inherent sequential nature of greedy search itself. Together this enables the use of highly-efficient partitioned-matrix-inverse techniques that result in large speedups of computation in both the forward-selection and backward-elimination stages of greedy algorithms in general.
This study examines orbit selection for small constellations at Mars that minimize time to achieve surface position accuracies at specified levels.
With increasing freshwater scarcity, advanced process design mechanisms such as Closed-Circuit Reverse Osmosis (CCRO) and Digital/Physical Twin systems are gaining traction in water treatment and reuse operations. While digital and physical twin models enable improved system insight and control, their development is often expensive and computationally intensive, requiring large volumes of synthetic or experimental data to characterize underlying process dynamics. This work introduces a sparse surrogate modeling framework to estimate power consumption from measured flow and pressure variables, along with their nonlinear polynomial and interaction expansions. To ensure model reliability and reduce overfitting, a two-stage pipeline is proposed. First, a dynamic data filtering algorithm is employed to remove uninformative observations and transient operational states. Second, a sparse penalized regression technique is applied to select a minimal set of parsimonious features. The proposed model achieves high sparsity, retaining only 7 out of 34 candidate features (≈79.41% sparsity) while delivering a root mean square error (RMSE) of 0.072 on the test dataset.
In computer simulation and optimal design, sequential batch sampling offers an appealing way to iteratively stipulate optimal sampling points based upon existing selections and efficiently construct surrogate modeling. Nonetheless, the issue of near duplicates poses tremendous quandary for sequential learning. It refers to the situation that selected critical points cluster together in each sampling batch, which are individually but not collectively informative towards the optimal design. Near duplicates severely diminish the computational efficiency as they barely contribute extra information towards update of the surrogate. To address this issue, we impose a dispersion criterion on concurrent selection of sampling points, which essentially forces a sparse distribution of critical points in each batch, and demonstrate the effectiveness of this approach in adaptive contour estimation. Specifically, we adopt Gaussian process surrogate to emulate the simulator, acquire variance reduction of the critical region from new sampling points as a dispersion criterion, and combine it with the modified expected improvement (EI) function for critical batch selection. The critical region here is the proximity of the contour of interest. This proposed approach is vindicated in numerical examples of a two-dimensional four-branch function, a four-dimensional function with a disjoint contour of interest and a time-delay dynamic system.
In this study, we have previously shown that augmenting orthogonal matching pursuit (OMP) with an additional step in the identification stage of each pursuit iteration yields improved $k$ -sparse reconstruction and denoising performance relative to baseline OMP. At each iteration a “path” or geodesic, is generated between the two dictionary atoms that are most correlated with the residual and from this path a new atom that has a greater correlation to the residual than either of the two bracketing atoms is selected. Here, we provide new computational results illustrating improvements in sparse coding and denoising on canonical datasets using both learned and structured dictionaries. The two methods of constructing a path are investigated for each dictionary type: the Euclidean geodesic formed by a linear combination of the two atoms and the 2-Wasserstein geodesic corresponding to the optimal transport map between the atoms. We prove here the existence of a higher-correlation atom in the Euclidean case under assumptions on the two bracketing atoms and introduce algorithmic modifications to improve the likelihood that the bracketing atoms meet those conditions. Although, we demonstrate our augmentation on OMP alone, in general it may be applied to any reconstruction algorithm that relies on the selection and sorting of high-similarity atoms during an analysis or identification phase.
Traditional probabilistic methods for the simulation of advection-diffusion equations (ADEs) often overlook the entropic contribution of the discretization, e.g., the number of particles, within associated numerical methods. Many times, the gain in accuracy of a highly discretized numerical model is outweighed by its associated computational costs or the noise within the data. Herein, we address the question of how many particles are needed in a simulation to best approximate and estimate parameters in one-dimensional advective-diffusive transport. To do so, we use the well-known Akaike Information Criterion (AIC) and a recently-developed correction called the Computational Information Criterion (COMIC) to guide the model selection process. Random-walk and mass-transfer particle tracking methods are employed to solve the model equations at various levels of discretization. Numerical results demonstrate that the COMIC provides an optimal number of particles that can describe a more efficient model in terms of parameter estimation and model prediction compared to the model selected by the AIC even when the data is sparse or noisy, the sampling volume is not uniform throughout the physical domain, or the error distribution of the data is non-IID Gaussian.
We extend the l0-norm "subspectral" algorithms for sparse-LDA [5] and sparse-PCA [6] to general quadratic costs such as MSE in linear (kernel) regression. The resulting "Sparse Least Squares" (SLS) problem is also NP-hard, by way of its equivalence to a rank-1 sparse eigenvalue problem (e.g., binary sparse-LDA [7]). Specifically, for a general quadratic cost we use a highly-efficient technique for direct eigenvalue computation using partitioned matrix inverses which leads to dramatic x103 speed-ups over standard eigenvalue decomposition. This increased efficiency mitigates the O(n4) scaling behaviour that up to now has limited the previous algorithms' utility for high-dimensional learning problems. Moreover, the new computation prioritizes the role of the less-myopic backward elimination stage which becomes more efficient than forward selection. Similarly, branch-and-bound search for Exact Sparse Least Squares (ESLS) also benefits from partitioned matrix inverse techniques. Our Greedy Sparse Least Squares (GSLS) generalizes Natarajan's algorithm [9] also known as Order-Recursive Matching Pursuit (ORMP). Specifically, the forward half of GSLS is exactly equivalent to ORMP but more efficient. By including the backward pass, which only doubles the computation, we can achieve lower MSE than ORMP. Experimental comparisons to the state-of-the-art LARS algorithm [3] show forward-GSLS is faster, more accurate and more flexible in terms of choice of regularization
The complex dynamics of the coronavirus disease 2019 (COVID-19) pandemic has made obtaining reliable long-term forecasts of the disease progression difficult. Simple mechanistic models with deterministic parameters are useful for short-term predictions but have ultimately been unsuccessful in extrapolating the trajectory of the pandemic because of unmodelled dynamics and the unrealistic level of certainty that is assumed in the predictions. We propose a 22-compartment epidemiological model that includes compartments not previously considered concurrently, to account for the effects of vaccination, asymptomatic individuals, inadequate access to hospital care, post-acute COVID-19 and recovery with long-term health complications. Additionally, new connections between compartments introduce new dynamics to the system and provide a framework to study the sensitivity of model outputs to several concurrent effects, including temporary immunity, vaccination rate and vaccine effectiveness. Subject to data availability for a given region, we discuss a means by which population demographics (age, comorbidity, socioeconomic status, sex and geographical location) and clinically relevant information (different variants, different vaccines) can be incorporated within the 22-compartment framework. Considering a probabilistic interpretation of the parameters allows the model's predictions to reflect the current state of uncertainty about the model parameters and model states. We propose the use of a sparse Bayesian learning algorithm for parameter calibration and model selection. This methodology considers a combination of prescribed parameter prior distributions for parameters that are known to be essential to the modelled dynamics and automatic relevance determination priors for parameters whose relevance is questionable. This is useful as it helps prevent overfitting the available epidemiological data when calibrating the parameters of the proposed model. Population-level administrative health data will serve as partial observations of the model states.