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The gravitational wave (GW) event S230518h is a potential binary neutron star-black hole merger (NSBH) event that was detected during engineering run 15 (ER15), which served as the commissioning period before the LIGO-Virgo-KAGRA (LVK) O4a observing run. Despite its low probability of producing detectable electromagnetic emissions, we performed extensive follow-up observations of this event using the GECKO telescopes in the southern hemisphere. Our observation covered 61.7\% of the 90\% credible region, a $\rm 284\:deg^2$ area accessible from the southern hemisphere, reaching a median limiting magnitude of $R=21.6$ mag. In these images, we conducted a systematic search for an optical counterpart of this event by combining a CNN-based classifier and human verification. We identified 128 transient candidates, but no significant optical counterpart was found that could have caused the GW signal. Furthermore, we provide feasible KN properties that are consistent with the upper limits of observation. Although no optical counterpart was found, our result demonstrates both GECKO's efficient wide-field follow-up capabilities and usefulness for constraining properties of kilonovae from NSBH mergers at distances of $\sim 200$ Mpc., Comment: 25 pages, 14 figures, Accepted for publication in ApJ

This work presents a design under uncertainty approach for a char combustion process in a limited-data setting, where simulations of the fluid-solid coupled system are computationally expensive. We integrate a polynomial dimensional decomposition (PDD) surrogate model into the design optimization and induce computational efficiency in three key areas. First, we transform the input random variables to have fixed probability measures, which eliminates the need to recalculate the PDD's basis functions associated with these probability quantities. Second, using the limited available data from a physics-based high-fidelity solver, we estimate the PDD coefficients via sparsity-promoting diffeomorphic modulation under observable response preserving homotopy regression. Third, we propose a single-pass surrogate model training process that avoids the need to generate new training data and update the PDD coefficients during the derivative-free optimization process. The results provide insights for optimizing process parameters to ensure consistently high energy production from char combustion., Comment: 19 pages, 6 figures

In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.

It is known that particles with exotic properties can emerge in systems made of simple constituents such as qubits, due to long-range quantum entanglement. In this paper, we provide quantitative characterizations of entanglement necessary for emergent anyons and fermions by using the geometric entanglement measure (GEM) which quantifies the maximal overlap between a given state and any short-range entangled states. For systems with emergent anyons, based on the braiding statistics, we show that the GEM scales linearly in the system size regardless of microscopic details. The phenomenon of emergent anyons can also be understood within the framework of quantum error correction (QEC). Specifically, we show that the GEM of any 2D stabilizer codes must be at least quadratic in the code distance. Our proof is based on a generic prescription for constructing string operators, establishing a rigorous and direct connection between emergent anyons and QEC. For systems with emergent fermions, despite that the ground state subspaces could be exponentially huge and their coding properties could be rather poor, we show that the GEM also scales linearly in the system size. Our results also establish an intriguing link between quantum anomaly and entanglement: a quantum state respecting anomalous $1$-form symmetries, be it pure or mixed, must be long-range entangled and have large GEM, offering a non-trivial class of intrinsically mixed state phases.

It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here we show that this belief may or may not be true depending on a particular code. To this end, we characterize a tradeoff between the code distance $d$ quantifying the number of correctable errors, and geometric entanglement of logical states quantifying their maximal overlap with product states or more general "topologically trivial" states. The maximum overlap is shown to be exponentially small in $d$ for three families of codes: (1) low-density parity check (LDPC) codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with $d$. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant $d$ and $k$ (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length.

Quantum measurement has conventionally been regarded as the final step in quantum information processing, which is essential for reading out the processed information but collapses the quantum state into a classical state. However, recent studies have shown that quantum measurement itself can induce novel quantum phenomena. One seminal example is a monitored random circuit, which can generate long-range entanglement faster than a random unitary circuit. Inspired by these results, in this paper, we address the following question: When quantum information is encoded in a quantum error-correcting code, how many physical qubits should be randomly measured to destroy the encoded information? We investigate this question for various quantum error-correcting codes and derive the necessary and sufficient conditions for destroying the information through measurements. In particular, we demonstrate that for a large class of quantum error-correcitng codes, it is impossible to destroy the encoded information through random single-qubit Pauli measurements when a tiny portion of physical qubits is still unmeasured. Our results not only reveal the extraordinary robustness of quantum codes under measurement decoherence, but also suggest potential applications in quantum information processing tasks., Comment: 30 pages

Real-world graphs are dynamic, constantly evolving with new interactions, such as financial transactions in financial networks. Temporal Graph Neural Networks (TGNNs) have been developed to effectively capture the evolving patterns in dynamic graphs. While these models have demonstrated their superiority, being widely adopted in various important fields, their vulnerabilities against adversarial attacks remain largely unexplored. In this paper, we propose T-SPEAR, a simple and effective adversarial attack method for link prediction on continuous-time dynamic graphs, focusing on investigating the vulnerabilities of TGNNs. Specifically, before the training procedure of a victim model, which is a TGNN for link prediction, we inject edge perturbations to the data that are unnoticeable in terms of the four constraints we propose, and yet effective enough to cause malfunction of the victim model. Moreover, we propose a robust training approach T-SHIELD to mitigate the impact of adversarial attacks. By using edge filtering and enforcing temporal smoothness to node embeddings, we enhance the robustness of the victim model. Our experimental study shows that T-SPEAR significantly degrades the victim model's performance on link prediction tasks, and even more, our attacks are transferable to other TGNNs, which differ from the victim model assumed by the attacker. Moreover, we demonstrate that T-SHIELD effectively filters out adversarial edges and exhibits robustness against adversarial attacks, surpassing the link prediction performance of the naive TGNN by up to 11.2% under T-SPEAR., Comment: Published at AAAI 2024

In uncertainty quantification, variance-based global sensitivity analysis quantitatively determines the effect of each input random variable on the output by partitioning the total output variance into contributions from each input. However, computing conditional expectations can be prohibitively costly when working with expensive-to-evaluate models. Surrogate models can accelerate this, yet their accuracy depends on the quality and quantity of training data, which is expensive to generate (experimentally or computationally) for complex engineering systems. Thus, methods that work with limited data are desirable. We propose a diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression to train a polynomial dimensional decomposition surrogate of the output that minimizes the number of training data. The new method first computes a sparse Lasso solution and uses it to define the cost function. A subsequent D-MORPH regression minimizes the difference between the D-MORPH and Lasso solution. The resulting D-MORPH based surrogate is more robust to input variations and more accurate with limited training data. We illustrate the accuracy and computational efficiency of the new surrogate for global sensitivity analysis using mathematical functions and an expensive-to-simulate model of char combustion. The new method is highly efficient, requiring only 15% of the training data compared to conventional regression., Comment: 26 pages, 11 figures

Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems., Comment: 41 pages, added additional numerical experiments

The aim of continual learning is to learn new tasks continuously (i.e., plasticity) without forgetting previously learned knowledge from old tasks (i.e., stability). In the scenario of online continual learning, wherein data comes strictly in a streaming manner, the plasticity of online continual learning is more vulnerable than offline continual learning because the training signal that can be obtained from a single data point is limited. To overcome the stability-plasticity dilemma in online continual learning, we propose an online continual learning framework named multi-scale feature adaptation network (MuFAN) that utilizes a richer context encoding extracted from different levels of a pre-trained network. Additionally, we introduce a novel structure-wise distillation loss and replace the commonly used batch normalization layer with a newly proposed stability-plasticity normalization module to train MuFAN that simultaneously maintains high plasticity and stability. MuFAN outperforms other state-of-the-art continual learning methods on the SVHN, CIFAR100, miniImageNet, and CORe50 datasets. Extensive experiments and ablation studies validate the significance and scalability of each proposed component: 1) multi-scale feature maps from a pre-trained encoder, 2) the structure-wise distillation loss, and 3) the stability-plasticity normalization module in MuFAN. Code is publicly available at https://github.com/whitesnowdrop/MuFAN., Comment: Accepted to ICLR2023

Graphs are widely used for modeling various types of interactions, such as email communications and online discussions. Many of such real-world graphs are temporal, and specifically, they grow over time with new nodes and edges. Counting the instances of each graphlet (i.e., an induced subgraph isomorphism class) has been successful in characterizing local structures of graphs, with many applications. While graphlets have been extended for temporal graphs, the extensions are designed for examining temporally-local subgraphs composed of edges with close arrival times, instead of long-term changes in local structures. In this paper, as a new lens for temporal graph analysis, we study the evolution of distributions of graphlet instances over time in real-world graphs at three different levels (graphs, nodes, and edges). At the graph level, we first discover that the evolution patterns are significantly different from those in random graphs. Then, we suggest a graphlet transition graph for measuring the similarity of the evolution patterns of graphs, and we find out a surprising similarity between the graphs from the same domain. At the node and edge levels, we demonstrate that the local structures around nodes and edges in their early stage provide a strong signal regarding their future importance. In particular, we significantly improve the predictability of the future importance of nodes and edges using the counts of the roles (a.k.a., orbits) that they take within graphlets., Comment: 13 pages, 7 figures

We propose novel methods for Conditional Value-at-Risk (CVaR) estimation for nonlinear systems under high-dimensional dependent random inputs. We develop a novel DD-GPCE-Kriging surrogate that merges dimensionally decomposed generalized polynomial chaos expansion and Kriging to accurately approximate nonlinear and nonsmooth random outputs. We use DD-GPCE-Kriging (1) for Monte Carlo simulation (MCS) and (2) within multifidelity importance sampling (MFIS). The MCS-based method samples from DD-GPCE-Kriging, which is efficient and accurate for high-dimensional dependent random inputs, yet introduces bias. Thus, we propose an MFIS-based method where DD-GPCE-Kriging determines the biasing density, from which we draw a few high-fidelity samples to provide an unbiased CVaR estimate. To accelerate the biasing density construction, we compute DD-GPCE-Kriging using a cheap-to-evaluate low-fidelity model. Numerical results for mathematical functions show that the MFIS-based method is more accurate than the MCS-based method when the output is nonsmooth. The scalability of the proposed methods and their applicability to complex engineering problems are demonstrated on a two-dimensional composite laminate with 28 (partly dependent) random inputs and a three-dimensional composite T-joint with 20 (partly dependent) random inputs. In the former, the proposed MFIS-based method achieves 104x speedup compared to standard MCS using the high-fidelity model, while accurately estimating CVaR with 1.15% error., Comment: 34 pages, 8 figures, research paper

Event cameras respond to brightness changes in the scene asynchronously and independently for every pixel. Due to the properties, these cameras have distinct features: high dynamic range (HDR), high temporal resolution, and low power consumption. However, the results of event cameras should be processed into an alternative representation for computer vision tasks. Also, they are usually noisy and cause poor performance in areas with few events. In recent years, numerous researchers have attempted to reconstruct videos from events. However, they do not provide good quality videos due to a lack of temporal information from irregular and discontinuous data. To overcome these difficulties, we introduce an E2V-SDE whose dynamics are governed in a latent space by Stochastic differential equations (SDE). Therefore, E2V-SDE can rapidly reconstruct images at arbitrary time steps and make realistic predictions on unseen data. In addition, we successfully adopted a variety of image composition techniques for improving image clarity and temporal consistency. By conducting extensive experiments on simulated and real-scene datasets, we verify that our model outperforms state-of-the-art approaches under various video reconstruction settings. In terms of image quality, the LPIPS score improves by up to 12% and the reconstruction speed is 87% higher than that of ET-Net., Comment: arXiv admin note: This submission has been withdrawn by arXiv administrators due to inappropriate text overlap with external sources. Additional information at https://doi.org/10.1109/CVPR52688.2022.01319

Although machine learning on hypergraphs has attracted considerable attention, most of the works have focused on (semi-)supervised learning, which may cause heavy labeling costs and poor generalization. Recently, contrastive learning has emerged as a successful unsupervised representation learning method. Despite the prosperous development of contrastive learning in other domains, contrastive learning on hypergraphs remains little explored. In this paper, we propose TriCL (Tri-directional Contrastive Learning), a general framework for contrastive learning on hypergraphs. Its main idea is tri-directional contrast, and specifically, it aims to maximize in two augmented views the agreement (a) between the same node, (b) between the same group of nodes, and (c) between each group and its members. Together with simple but surprisingly effective data augmentation and negative sampling schemes, these three forms of contrast enable TriCL to capture both microscopic and mesoscopic structural information in node embeddings. Our extensive experiments using 13 baseline approaches, five datasets, and two tasks demonstrate the effectiveness of TriCL, and most noticeably, TriCL consistently outperforms not just unsupervised competitors but also (semi-)supervised competitors mostly by significant margins for node classification. The code and datasets are available at https://github.com/wooner49/TriCL., Comment: Published at AAAI 2023

Digital twin models allow us to continuously assess the possible risk of damage and failure of a complex system. Yet high-fidelity digital twin models can be computationally expensive, making quick-turnaround assessment challenging. Towards this goal, this article proposes a novel bi-fidelity method for estimating the conditional value-at-risk (CVaR) for nonlinear systems subject to dependent and high-dimensional inputs. For models that can be evaluated fast, a method that integrates the dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) approximation with a standard sampling-based CVaR estimation is proposed. For expensive-to-evaluate models, a new bi-fidelity method is proposed that couples the DD-GPCE with a Fourier-polynomial expansion of the mapping between the stochastic low-fidelity and high-fidelity output data to ensure computational efficiency. The method employs measure-consistent orthonormal polynomials in the random variable of the low-fidelity output to approximate the high-fidelity output. Numerical results for a structural mechanics truss with 36-dimensional (dependent random variable) inputs indicate that the DD-GPCE method provides very accurate CVaR estimates that require much lower computational effort than standard GPCE approximations. A second example considers the realistic problem of estimating the risk of damage to a fiber-reinforced composite laminate. The high-fidelity model is a finite element simulation that is prohibitively expensive for risk analysis, such as CVaR computation. Here, the novel bi-fidelity method can accurately estimate CVaR as it includes low-fidelity models in the estimation procedure and uses only a few high-fidelity model evaluations to significantly increase accuracy.

Common AI music composition algorithms based on artificial neural networks are to train a machine by feeding a large number of music pieces and create artificial neural networks that can produce music similar to the input music data. This approach is a blackbox optimization, that is, the underlying composition algorithm is, in general, not known to users. In this paper, we present a way of machine composition that trains a machine the composition principle embedded in the given music data instead of directly feeding music pieces. We propose this approach by using the concept of {\color{black}{Overlap}} matrix proposed in \cite{TPJ}. In \cite{TPJ}, a type of Korean music, so-called the {\it Dodeuri} music such as Suyeonjangjigok has been analyzed using topological data analysis (TDA), particularly using persistent homology. As the raw music data is not suitable for TDA analysis, the music data is first reconstructed as a graph. The node of the graph is defined as a two-dimensional vector composed of the pitch and duration of each music note. The edge between two nodes is created when those nodes appear consecutively in the music flow. Distance is defined based on the frequency of such appearances. Through TDA on the constructed graph, a unique set of cycles is found for the given music. In \cite{TPJ}, the new concept of the {\it {\color{black}{Overlap}} matrix} has been proposed, which visualizes how those cycles are interconnected over the music flow, in a matrix form. In this paper, we explain how we use the {\color{black}{Overlap}} matrix for machine composition. The {\color{black}{Overlap}} matrix makes it possible to compose a new music piece algorithmically and also provide a seed music towards the desired artificial neural network. In this paper, we use the {\it Dodeuri} music and explain detailed steps.

Spiking neural networks (SNNs) that mimic information transmission in the brain can energy-efficiently process spatio-temporal information through discrete and sparse spikes, thereby receiving considerable attention. To improve accuracy and energy efficiency of SNNs, most previous studies have focused solely on training methods, and the effect of architecture has rarely been studied. We investigate the design choices used in the previous studies in terms of the accuracy and number of spikes and figure out that they are not best-suited for SNNs. To further improve the accuracy and reduce the spikes generated by SNNs, we propose a spike-aware neural architecture search framework called AutoSNN. We define a search space consisting of architectures without undesirable design choices. To enable the spike-aware architecture search, we introduce a fitness that considers both the accuracy and number of spikes. AutoSNN successfully searches for SNN architectures that outperform hand-crafted SNNs in accuracy and energy efficiency. We thoroughly demonstrate the effectiveness of AutoSNN on various datasets including neuromorphic datasets., Comment: Accepted in ICML22

Spiking neural networks (SNNs) have been gaining interest as energy-efficient alternatives of conventional artificial neural networks (ANNs) due to their event-driven computation. Considering the future deployment of SNN models to constrained neuromorphic devices, many studies have applied techniques originally used for ANN model compression, such as network quantization, pruning, and knowledge distillation, to SNNs. Among them, existing works on knowledge distillation reported accuracy improvements of student SNN model. However, analysis on energy efficiency, which is also an important feature of SNN, was absent. In this paper, we thoroughly analyze the performance of the distilled SNN model in terms of accuracy and energy efficiency. In the process, we observe a substantial increase in the number of spikes, leading to energy inefficiency, when using the conventional knowledge distillation methods. Based on this analysis, to achieve energy efficiency, we propose a novel knowledge distillation method with heterogeneous temperature parameters. We evaluate our method on two different datasets and show that the resulting SNN student satisfies both accuracy improvement and reduction of the number of spikes. On MNIST dataset, our proposed student SNN achieves up to 0.09% higher accuracy and produces 65% less spikes compared to the student SNN trained with conventional knowledge distillation method. We also compare the results with other SNN compression techniques and training methods., Comment: The manuscript was withdrawn because it contains inappropriate content for posting

Spiking neural networks (SNNs) have emerged as energy-efficient neural networks with temporal information. SNNs have shown a superior efficiency on neuromorphic devices, but the devices are susceptible to noise, which hinders them from being applied in real-world applications. Several studies have increased noise robustness, but most of them considered neither deep SNNs nor temporal information. In this paper, we investigate the effect of noise on deep SNNs with various neural coding methods and present a noise-robust deep SNN with temporal information. With the proposed methods, we have achieved a deep SNN that is efficient and robust to spike deletion and jitter., Comment: Accepted to DAC 2021

Consider traffic data (i.e., triplets in the form of source-destination-timestamp) that grow over time. Tensors (i.e., multi-dimensional arrays) with a time mode are widely used for modeling and analyzing such multi-aspect data streams. In such tensors, however, new entries are added only once per period, which is often an hour, a day, or even a year. This discreteness of tensors has limited their usage for real-time applications, where new data should be analyzed instantly as it arrives. How can we analyze time-evolving multi-aspect sparse data 'continuously' using tensors where time is'discrete'? We propose SLICENSTITCH for continuous CANDECOMP/PARAFAC (CP) decomposition, which has numerous time-critical applications, including anomaly detection, recommender systems, and stock market prediction. SLICENSTITCH changes the starting point of each period adaptively, based on the current time, and updates factor matrices (i.e., outputs of CP decomposition) instantly as new data arrives. We show, theoretically and experimentally, that SLICENSTITCH is (1) 'Any time': updating factor matrices immediately without having to wait until the current time period ends, (2) Fast: with constant-time updates up to 464x faster than online methods, and (3) Accurate: with fitness comparable (specifically, 72 ~ 100%) to offline methods., Comment: Updated Figures 4, 5, 6, 7, and 8 after fixing a bug in preprocessing the Divvy dataset. To appear at the 37th IEEE International Conference on Data Engineering (ICDE '21)

Consider multiple seasonal time series being collected in real-time, in the form of a tensor stream. Real-world tensor streams often include missing entries (e.g., due to network disconnection) and at the same time unexpected outliers (e.g., due to system errors). Given such a real-world tensor stream, how can we estimate missing entries and predict future evolution accurately in real-time? In this work, we answer this question by introducing SOFIA, a robust factorization method for real-world tensor streams. In a nutshell, SOFIA smoothly and tightly integrates tensor factorization, outlier removal, and temporal-pattern detection, which naturally reinforce each other. Moreover, SOFIA integrates them in linear time, in an online manner, despite the presence of missing entries. We experimentally show that SOFIA is (a) robust and accurate: yielding up to 76% lower imputation error and 71% lower forecasting error; (b) fast: up to 935X faster than the second-most accurate competitor; and (c) scalable: scaling linearly with the number of new entries per time step., Comment: Published at ICDE 2021

Photon-pair generation must satisfy both the energy conservation and phase-matching conditions with a specific pump wavelength and dispersion of nonlinear optical medium, but finding a photon-pair, which has a desired specific wavelength, generation medium is challenging. Here, we present a method to create photon pairs that functions efficiently even the pump wavelength is much larger than the zero GVD wavelength of medium. In this study, we employ short SMF-28 fibers having ~1310 nm zero GVD wavelength and C-band pump (1552.52 nm) to generate C-band photon pairs. The measured pair generation rate and coincidence-to-accidental ratios are comparable to those from a long dispersion-shifted fiber. Polarization-entangled states are prepared, and an S value of 2.659 +- 0.094 is achieved from Bell inequality measurements. Our results indicate that the use of a short SFWM medium yields adequate photon-pair generation rates regardless of its dispersion properties in almost any material at any pump wavelength., Comment: 11 pages, 7 figures

Predictive Maintenance (PdM) can only be implemented when the online knowledge of system condition is available, and this has become available with deployment of on-equipment sensors. To date, most studies on predicting the remaining useful lifetime of a system have been focusing on either single-component systems or systems with deterministic reliability structures. This assumption is not applicable on some realistic problems, where there exist uncertainties in reliability structures of complex systems. In this paper, a PdM scheme is developed by employing a Discrete Time Markov Chain (DTMC) for forecasting the health of monitored components and a Bayesian Network (BN) for modeling the multi-component system reliability. Therefore, probabilistic inferences on both the system and its components status can be made and PdM can be scheduled on both levels., Comment: 7 pages, This is a Conference paper submitted to Industrial and Systems Engineering Research Conference 2016 (ISERC)

We consider computing the $k$-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of $n\times n$ large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index $k.$ We present a three-stage algorithm for computing the $k$-th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the $k$-th eigenvalue $(1 \ll k \ll n)$ with a non-standard application of the Lanczos method. In the second stage, spectral bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the $k$-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the $k$-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm.