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We prove that the Schubert structure constants of the quantum K-theory rings of symplectic Grassmannians of lines have signs that alternate with codimension and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings, including the Seidel representation and a Chevalley formula.

Approximate nearest neighbor search (ANNS) plays an indispensable role in a wide variety of applications, including recommendation systems, information retrieval, and semantic search. Among the cutting-edge ANNS algorithms, graph-based approaches provide superior accuracy and scalability on massive datasets. However, the best-performing graph-based ANN search solutions incur tens of hundreds of memory footprints as well as costly distance computation, thus hindering their efficient deployment at scale. The 3D NAND flash is emerging as a promising device for data-intensive applications due to its high density and nonvolatility. In this work, we present the near-storage processing (NSP)-based ANNS solution Proxima, to accelerate graph-based ANNS with algorithm-hardware co-design in 3D NAND flash. Proxima significantly reduces the complexity of graph search by leveraging the distance approximation and early termination. On top of the algorithmic enhancement, we implement Proxima search algorithm in 3D NAND flash using the heterogeneous integration technique. To maximize 3D NAND's bandwidth utilization, we present customized dataflow and optimized data allocation scheme. Our evaluation results show that: compared to graph ANNS on CPU and GPU, Proxima achieves a magnitude improvement in throughput or energy efficiency. Proxima yields 7x to 13x speedup over existing ASIC designs. Furthermore, Proxima achieves a good balance between accuracy, efficiency and storage density compared to previous NSP-based accelerators.

Mass spectrometry-based proteomics is a key enabler for personalized healthcare, providing a deep dive into the complex protein compositions of biological systems. This technology has vast applications in biotechnology and biomedicine but faces significant computational bottlenecks. Current methodologies often require multiple hours or even days to process extensive datasets, particularly in the domain of spectral clustering. To tackle these inefficiencies, we introduce SpecHD, a hyperdimensional computing (HDC) framework supplemented by an FPGA-accelerated architecture with integrated near-storage preprocessing. Utilizing streamlined binary operations in an HDC environment, SpecHD capitalizes on the low-latency and parallel capabilities of FPGAs. This approach markedly improves clustering speed and efficiency, serving as a catalyst for real-time, high-throughput data analysis in future healthcare applications. Our evaluations demonstrate that SpecHD not only maintains but often surpasses existing clustering quality metrics while drastically cutting computational time. Specifically, it can cluster a large-scale human proteome dataset-comprising 25 million MS/MS spectra and 131 GB of MS data-in just 5 minutes. With energy efficiency exceeding 31x and a speedup factor that spans a range of 6x to 54x over existing state of-the-art solutions, SpecHD emerges as a promising solution for the rapid analysis of mass spectrometry data with great implications for personalized healthcare.

In a recent paper, we stated conjectural presentations for the equivariant quantum K ring of partial flag varieties, motivated by physics considerations. In this companion paper, we analyze these presentations mathematically. We prove that if the conjectured relations hold, then they must form a complete set of relations. Our main result is a proof of the conjectured presentation in the case of the incidence varieties. We also show that if a quantum K divisor axiom holds (as conjectured by Buch and Mihalcea), then the conjectured presentation also holds for the complete flag variety., Comment: 25 pages; revised Remark 5.9 and added Example 5.10

Publicly available collections of drug-like molecules have grown to comprise 10s of billions of possibilities in recent history due to advances in chemical synthesis. Traditional methods for identifying ``hit'' molecules from a large collection of potential drug-like candidates have relied on biophysical theory to compute approximations to the Gibbs free energy of the binding interaction between the drug to its protein target. A major drawback of the approaches is that they require exceptional computing capabilities to consider for even relatively small collections of molecules. Hyperdimensional Computing (HDC) is a recently proposed learning paradigm that is able to leverage low-precision binary vector arithmetic to build efficient representations of the data that can be obtained without the need for gradient-based optimization approaches that are required in many conventional machine learning and deep learning approaches. This algorithmic simplicity allows for acceleration in hardware that has been previously demonstrated for a range of application areas. We consider existing HDC approaches for molecular property classification and introduce two novel encoding algorithms that leverage the extended connectivity fingerprint (ECFP) algorithm. We show that HDC-based inference methods are as much as 90 times more efficient than more complex representative machine learning methods and achieve an acceleration of nearly 9 orders of magnitude as compared to inference with molecular docking. We demonstrate multiple approaches for the encoding of molecular data for HDC and examine their relative performance on a range of challenging molecular property prediction and drug-protein binding classification tasks. Our work thus motivates further investigation into molecular representation learning to develop ultra-efficient pre-screening tools.

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent., Comment: Final version, 13 pages, 2 tables, 2 figures

Mass spectrometry, commonly used for protein identification, generates a massive number of spectra that need to be matched against a large database. In reality, most of them remain unidentified or mismatched due to unexpected post-translational modifications. Open modification search (OMS) has been proposed as a strategy to improve the identification rate by considering every possible change in spectra, but it expands the search space exponentially. In this work, we propose HyperOMS, which redesigns OMS based on hyperdimensional computing to cope with such challenges. Unlike existing algorithms that represent spectral data with floating point numbers, HyperOMS encodes them with high dimensional binary vectors and performs the efficient OMS in high-dimensional space. With the massive parallelism and simple boolean operations, HyperOMS can be efficiently handled on parallel computing platforms. Experimental results show that HyperOMS on GPU is up to $17\times$ faster and $6.4\times$ more energy efficient than the state-of-the-art GPU-based OMS tool while providing comparable search quality to competing search tools., Comment: 6 pages, 7 figures, extension of PACT 2022 paper

Genome sequence alignment is the core of many biological applications. The advancement of sequencing technologies produces a tremendous amount of data, making sequence alignment a critical bottleneck in bioinformatics analysis. The existing hardware accelerators for alignment suffer from limited on-chip memory, costly data movement, and poorly optimized alignment algorithms. They cannot afford to concurrently process the massive amount of data generated by sequencing machines. In this paper, we propose a ReRAM-based accelerator, RAPIDx, using processing in-memory (PIM) for sequence alignment. RAPIDx achieves superior efficiency and performance via software-hardware co-design. First, we propose an adaptive banded parallelism alignment algorithm suitable for PIM architecture. Compared to the original dynamic programming-based alignment, the proposed algorithm significantly reduces the required complexity, data bit width, and memory footprint at the cost of negligible accuracy degradation. Then we propose the efficient PIM architecture that implements the proposed algorithm. The data flow in RAPIDx achieves four-level parallelism and we design an in-situ alignment computation flow in ReRAM, delivering $5.5$-$9.7\times$ efficiency and throughput improvements compared to our previous PIM design, RAPID. The proposed RAPIDx is reconfigurable to serve as a co-processor integrated into existing genome analysis pipeline to boost sequence alignment or edit distance calculation. On short-read alignment, RAPIDx delivers $131.1\times$ and $46.8\times$ throughput improvements over state-of-the-art CPU and GPU libraries, respectively. As compared to ASIC accelerators for long-read alignment, the performance of RAPIDx is $1.8$-$2.9\times$ higher.

We prove that Schubert varieties in potentially different Grassmannians are isomorphic as varieties if and only if their corresponding Young diagrams are identical up to a transposition. We also discuss a generalization of this result to Grassmannian Richardson varieties. In particular, we prove that Richardson varieties in potentially different Grassmannians are isomorphic as varieties if their corresponding skew diagrams are semi-isomorphic as posets, and we conjecture the converse. Here, two posets are said to be semi-isomorphic if there is a bijection between their sets of connected components such that the corresponding components are either isomorphic or opposite., Comment: 11 pages, 14 figures, final version published in Journal of Algebra

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n) and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov-Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov-Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov-Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive Littlewood-Richardson rule for the non-equivariant quantum K-theory ring of X. The Littlewood-Richardson rule in turn implies that non-empty Gromov-Witten varieties given by Schubert varieties in general position have arithmetic genus 0., Comment: Final version, 34 pages, 2 figures, 8 tables

We derive cancellation-free Chevalley-type multiplication formulas in the T-equivariant quantum K-theory of Grassmannians of type A and C, and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T-equivariant quantum K-theory of arbitrary flag manifolds G/B, which was derived earlier in terms of the quantum alcove model, by the last three authors., Comment: We added the proof of the positivity property for the Chevalley structure constants for full flag manifolds of arbitrary types, two-step flag manifolds of type A, and Grassmannians of type C. We also added Appendix A (a new proof of the existence of a multiplicative surjection due to Kato) and Appendix B (on the connection with a conjecture of Weihong Xu, who is a co-author of this Appendix)

We consider a conjecture that identifies two types of base point free divisors on $\bar{M}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to simple Lie algebras in type A. Here we reduce this conjecture on $\bar{M}_{0,n}$ to the same statement for $n=4$. A reinterpretation leads to a proof of the conjecture on $\bar{M}_{0,n}$ for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

In this paper, we present a sparse neural network decoder (SNND) of polar codes based on belief propagation (BP) and deep learning. At first, the conventional factor graph of polar BP decoding is converted to the bipartite Tanner graph similar to low-density parity-check (LDPC) codes. Then the Tanner graph is unfolded and translated into the graphical representation of deep neural network (DNN). The complex sum-product algorithm (SPA) is modified to min-sum (MS) approximation with low complexity. We dramatically reduce the number of weight by using single weight to parameterize the networks. Optimized by the training techniques of deep learning, proposed SNND achieves comparative decoding performance of SPA and obtains about $0.5$ dB gain over MS decoding on ($128,64$) and ($256,128$) codes. Moreover, $60 \%$ complexity reduction is achieved and the decoding latency is significantly lower than the conventional polar BP., Comment: submitted to Asilomar 2018

Recently, deep learning methods have shown significant improvements in communication systems. In this paper, we study the equalization problem over the nonlinear channel using neural networks. The joint equalizer and decoder based on neural networks are proposed to realize blind equalization and decoding process without the knowledge of channel state information (CSI). Different from previous methods, we use two neural networks instead of one. First, convolutional neural network (CNN) is used to adaptively recover the transmitted signal from channel impairment and nonlinear distortions. Then the deep neural network decoder (NND) decodes the detected signal from CNN equalizer. Under various channel conditions, the experiment results demonstrate that the proposed CNN equalizer achieves better performance than other solutions based on machine learning methods. The proposed model reduces about $2/3$ of the parameters compared to state-of-the-art counterparts. Besides, our model can be easily applied to long sequence with $\mathcal{O}(n)$ complexity.

In this paper, deep neural network (DNN) is utilized to improve the belief propagation (BP) detection for massive multiple-input multiple-output (MIMO) systems. A neural network architecture suitable for detection task is firstly introduced by unfolding BP algorithms. DNN MIMO detectors are then proposed based on two modified BP detectors, damped BP and max-sum BP. The correction factors in these algorithms are optimized through deep learning techniques, aiming at improved detection performance. Numerical results are presented to demonstrate the performance of the DNN detectors in comparison with various BP modifications. The neural network is trained once and can be used for multiple online detections. The results show that, compared to other state-of-the-art detectors, the DNN detectors can achieve lower bit error rate (BER) with improved robustness against various antenna configurations and channel conditions at the same level of complexity.