Chunhao Wang

Anahita Fathi Kazerooni, Nastaran Khalili, Xinyang Liu et al.

Pediatric central nervous system tumors are the leading cause of cancer-related deaths in children. The five-year survival rate for high-grade glioma in children is less than 20%. The development of new treatments is dependent upon multi-institutional collaborative clinical trials requiring reproducible and accurate centralized response assessment. We present the results of the BraTS-PEDs 2023 challenge, the first Brain Tumor Segmentation (BraTS) challenge focused on pediatric brain tumors. This challenge utilized data acquired from multiple international consortia dedicated to pediatric neuro-oncology and clinical trials. BraTS-PEDs 2023 aimed to evaluate volumetric segmentation algorithms for pediatric brain gliomas from magnetic resonance imaging using standardized quantitative performance evaluation metrics employed across the BraTS 2023 challenges. The top-performing AI approaches for pediatric tumor analysis included ensembles of nnU-Net and Swin UNETR, Auto3DSeg, or nnU-Net with a self-supervised framework. The BraTSPEDs 2023 challenge fostered collaboration between clinicians (neuro-oncologists, neuroradiologists) and AI/imaging scientists, promoting faster data sharing and the development of automated volumetric analysis techniques. These advancements could significantly benefit clinical trials and improve the care of children with brain tumors.

Zhenyu Yang, Zongsheng Hu, Hangjie Ji et al.

Purpose: To develop a neural ordinary differential equation (ODE) model for visualizing deep neural network (DNN) behavior during multi-parametric MRI (mp-MRI) based glioma segmentation as a method to enhance deep learning explainability. Methods: By hypothesizing that deep feature extraction can be modeled as a spatiotemporally continuous process, we designed a novel deep learning model, neural ODE, in which deep feature extraction was governed by an ODE without explicit expression. The dynamics of 1) MR images after interactions with DNN and 2) segmentation formation can be visualized after solving ODE. An accumulative contribution curve (ACC) was designed to quantitatively evaluate the utilization of each MRI by DNN towards the final segmentation results. The proposed neural ODE model was demonstrated using 369 glioma patients with a 4-modality mp-MRI protocol: T1, contrast-enhanced T1 (T1-Ce), T2, and FLAIR. Three neural ODE models were trained to segment enhancing tumor (ET), tumor core (TC), and whole tumor (WT). The key MR modalities with significant utilization by DNN were identified based on ACC analysis. Segmentation results by DNN using only the key MR modalities were compared to the ones using all 4 MR modalities. Results: All neural ODE models successfully illustrated image dynamics as expected. ACC analysis identified T1-Ce as the only key modality in ET and TC segmentations, while both FLAIR and T2 were key modalities in WT segmentation. Compared to the U-Net results using all 4 MR modalities, Dice coefficient of ET (0.784->0.775), TC (0.760->0.758), and WT (0.841->0.837) using the key modalities only had minimal differences without significance. Conclusion: The neural ODE model offers a new tool for optimizing the deep learning model inputs with enhanced explainability. The presented methodology can be generalized to other medical image-related deep learning applications.

Zhenyu Yang, Kyle Lafata, Eugene Vaios et al.

The projection of planar MRI data onto a spherical surface is equivalent to a nonlinear image transformation that retains global anatomical information. By incorporating this image transformation process in our proposed spherical projection-based U-Net (SPU-Net) segmentation model design, multiple independent segmentation predictions can be obtained from a single MRI. The final segmentation is the average of all available results, and the variation can be visualized as a pixel-wise uncertainty map. An uncertainty score was introduced to evaluate and compare the performance of uncertainty measurements. The proposed SPU-Net model was implemented on the basis of 369 glioma patients with MP-MRI scans (T1, T1-Ce, T2, and FLAIR). Three SPU-Net models were trained to segment enhancing tumor (ET), tumor core (TC), and whole tumor (WT), respectively. The SPU-Net model was compared with (1) the classic U-Net model with test-time augmentation (TTA) and (2) linear scaling-based U-Net (LSU-Net) segmentation models in terms of both segmentation accuracy (Dice coefficient, sensitivity, specificity, and accuracy) and segmentation uncertainty (uncertainty map and uncertainty score). The developed SPU-Net model successfully achieved low uncertainty for correct segmentation predictions (e.g., tumor interior or healthy tissue interior) and high uncertainty for incorrect results (e.g., tumor boundaries). This model could allow the identification of missed tumor targets or segmentation errors in U-Net. Quantitatively, the SPU-Net model achieved the highest uncertainty scores for three segmentation targets (ET/TC/WT): 0.826/0.848/0.936, compared to 0.784/0.643/0.872 using the U-Net with TTA and 0.743/0.702/0.876 with the LSU-Net (scaling factor = 2). The SPU-Net also achieved statistically significantly higher Dice coefficients, underscoring the improved segmentation accuracy.

Andrew M. Childs, Tongyang Li, Jin-Peng Liu et al.

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $κ$ and dimension $d$) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $ε$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(κ^{1/2}d)$ and $\widetilde{O}(κ^{1/2}d^{3/2}/ε)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $κ,d,ε$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/ε^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $ε$.

Yang Chen, Zhenyu Yang, Jingtong Zhao et al.

We developed a deep ensemble learning model with a radiomics spatial encoding execution for improved glioma segmentation accuracy using multi-parametric MRI (mp-MRI). This model was developed using 369 glioma patients with a 4-modality mp-MRI protocol: T1, contrast-enhanced T1 (T1-Ce), T2, and FLAIR. In each modality volume, a 3D sliding kernel was implemented across the brain to capture image heterogeneity: fifty-six radiomic features were extracted within the kernel, resulting in a 4th order tensor. Each radiomic feature can then be encoded as a 3D image volume, namely a radiomic feature map (RFM). PCA was employed for data dimension reduction and the first 4 PCs were selected. Four deep neural networks as sub-models following the U-Net architecture were trained for the segmenting of a region-of-interest (ROI): each sub-model utilizes the mp-MRI and 1 of the 4 PCs as a 5-channel input for a 2D execution. The 4 softmax probability results given by the U-net ensemble were superimposed and binarized by Otsu method as the segmentation result. Three ensemble models were trained to segment enhancing tumor (ET), tumor core (TC), and whole tumor (WT). The adopted radiomics spatial encoding execution enriches the image heterogeneity information that leads to the successful demonstration of the proposed deep ensemble model, which offers a new tool for mp-MRI based medical image segmentation.

Xiantao Li, Chunhao Wang

In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time $T$, where the system is governed by a time-dependent Schrödinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schrödinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.

Guneykan Ozgul, Xiantao Li, Mehrdad Mahdavi et al.

We present quantum algorithms for sampling from non-logconcave probability distributions in the form of $π(x) \propto \exp(-βf(x))$. Here, $f$ can be written as a finite sum $f(x):= \frac{1}{N}\sum_{k=1}^N f_k(x)$. Our approach is based on quantum simulated annealing on slowly varying Markov chains derived from unadjusted Langevin algorithms, removing the necessity for function evaluations which can be computationally expensive for large data sets in mixture modeling and multi-stable systems. We also incorporate a stochastic gradient oracle that implements the quantum walk operators inexactly by only using mini-batch gradients. As a result, our stochastic gradient based algorithm only accesses small subsets of data points in implementing the quantum walk. One challenge of quantizing the resulting Markov chains is that they do not satisfy the detailed balance condition in general. Consequently, the mixing time of the algorithm cannot be expressed in terms of the spectral gap of the transition density, making the quantum algorithms nontrivial to analyze. To overcome these challenges, we first build a hypothetical Markov chain that is reversible, and also converges to the target distribution. Then, we quantified the distance between our algorithm's output and the target distribution by using this hypothetical chain as a bridge to establish the total complexity. Our quantum algorithms exhibit polynomial speedups in terms of both dimension and precision dependencies when compared to the best-known classical algorithms.

Dominic LaBella, Ujjwal Baid, Omaditya Khanna et al.

We describe the design and results from the BraTS 2023 Intracranial Meningioma Segmentation Challenge. The BraTS Meningioma Challenge differed from prior BraTS Glioma challenges in that it focused on meningiomas, which are typically benign extra-axial tumors with diverse radiologic and anatomical presentation and a propensity for multiplicity. Nine participating teams each developed deep-learning automated segmentation models using image data from the largest multi-institutional systematically expert annotated multilabel multi-sequence meningioma MRI dataset to date, which included 1000 training set cases, 141 validation set cases, and 283 hidden test set cases. Each case included T2, FLAIR, T1, and T1Gd brain MRI sequences with associated tumor compartment labels delineating enhancing tumor, non-enhancing tumor, and surrounding non-enhancing FLAIR hyperintensity. Participant automated segmentation models were evaluated and ranked based on a scoring system evaluating lesion-wise metrics including dice similarity coefficient (DSC) and 95% Hausdorff Distance. The top ranked team had a lesion-wise median dice similarity coefficient (DSC) of 0.976, 0.976, and 0.964 for enhancing tumor, tumor core, and whole tumor, respectively and a corresponding average DSC of 0.899, 0.904, and 0.871, respectively. These results serve as state-of-the-art benchmarks for future pre-operative meningioma automated segmentation algorithms. Additionally, we found that 1286 of 1424 cases (90.3%) had at least 1 compartment voxel abutting the edge of the skull-stripped image edge, which requires further investigation into optimal pre-processing face anonymization steps.

Anahita Fathi Kazerooni, Nastaran Khalili, Xinyang Liu et al.

Pediatric tumors of the central nervous system are the most common cause of cancer-related death in children. The five-year survival rate for high-grade gliomas in children is less than 20%. Due to their rarity, the diagnosis of these entities is often delayed, their treatment is mainly based on historic treatment concepts, and clinical trials require multi-institutional collaborations. Here we present the CBTN-CONNECT-DIPGR-ASNR-MICCAI BraTS-PEDs challenge, focused on pediatric brain tumors with data acquired across multiple international consortia dedicated to pediatric neuro-oncology and clinical trials. The CBTN-CONNECT-DIPGR-ASNR-MICCAI BraTS-PEDs challenge brings together clinicians and AI/imaging scientists to lead to faster development of automated segmentation techniques that could benefit clinical trials, and ultimately the care of children with brain tumors.

Zhenyu Yang, Haiming Zhu, Rihui Zhang et al.

Background: Deep learning has significantly advanced medical image analysis, with Vision Transformers (ViTs) offering a powerful alternative to convolutional models by modeling long-range dependencies through self-attention. However, ViTs are inherently data-intensive and lack domain-specific inductive biases, limiting their applicability in medical imaging. In contrast, radiomics provides interpretable, handcrafted descriptors of tissue heterogeneity but suffers from limited scalability and integration into end-to-end learning frameworks. In this work, we propose the Radiomics-Embedded Vision Transformer (RE-ViT) that combines radiomic features with data-driven visual embeddings within a ViT backbone. Purpose: To develop a hybrid RE-ViT framework that integrates radiomics and patch-wise ViT embeddings through early fusion, enhancing robustness and performance in medical image classification. Methods: Following the standard ViT pipeline, images were divided into patches. For each patch, handcrafted radiomic features were extracted and fused with linearly projected pixel embeddings. The fused representations were normalized, positionally encoded, and passed to the ViT encoder. A learnable [CLS] token aggregated patch-level information for classification. We evaluated RE-ViT on three public datasets (including BUSI, ChestXray2017, and Retinal OCT) using accuracy, macro AUC, sensitivity, and specificity. RE-ViT was benchmarked against CNN-based (VGG-16, ResNet) and hybrid (TransMed) models. Results: RE-ViT achieved state-of-the-art results: on BUSI, AUC=0.950+/-0.011; on ChestXray2017, AUC=0.989+/-0.004; on Retinal OCT, AUC=0.986+/-0.001, which outperforms other comparison models. Conclusions: The RE-ViT framework effectively integrates radiomics with ViT architectures, demonstrating improved performance and generalizability across multimodal medical image classification tasks.

Guneykan Ozgul, Xiantao Li, Mehrdad Mahdavi et al.

We propose quantum algorithms that provide provable speedups for Markov Chain Monte Carlo (MCMC) methods commonly used for sampling from probability distributions of the form $π\propto e^{-f}$, where $f$ is a potential function. Our first approach considers Gibbs sampling for finite-sum potentials in the stochastic setting, employing an oracle that provides gradients of individual functions. In the second setting, we consider access only to a stochastic evaluation oracle, allowing simultaneous queries at two points of the potential function under the same stochastic parameter. By introducing novel techniques for stochastic gradient estimation, our algorithms improve the gradient and evaluation complexities of classical samplers, such as Hamiltonian Monte Carlo (HMC) and Langevin Monte Carlo (LMC) in terms of dimension, precision, and other problem-dependent parameters. Furthermore, we achieve quantum speedups in optimization, particularly for minimizing non-smooth and approximately convex functions that commonly appear in empirical risk minimization problems.

Xuri Ge, Chunhao Wang, Xindi Wang et al.

Composed Image Retrieval (CIR) aims to retrieve target images based on a reference image and modified texts. However, existing methods often struggle to extract the correct semantic cues from the reference image that best reflect the user's intent under textual modification prompts, resulting in interference from irrelevant visual noise. In this paper, we propose a novel Multi-level Vision Selection by Multi-modal Chain-of-Thought Reasoning (MCoT-MVS) for CIR, integrating attention-aware multi-level vision features guided by reasoning cues from a multi-modal large language model (MLLM). Specifically, we leverage an MLLM to perform chain-of-thought reasoning on the multimodal composed input, generating the retained, removed, and target-inferred texts. These textual cues subsequently guide two reference visual attention selection modules to selectively extract discriminative patch-level and instance-level semantics from the reference image. Finally, to effectively fuse these multi-granular visual cues with the modified text and the imagined target description, we design a weighted hierarchical combination module to align the composed query with target images in a unified embedding space. Extensive experiments on two CIR benchmarks, namely CIRR and FashionIQ, demonstrate that our approach consistently outperforms existing methods and achieves new state-of-the-art performance. Code and trained models are publicly released.

Rihui Zhang, Haiming Zhu, Jingtong Zhao et al.

Accurate evaluation of regional lung ventilation is essential for the management and treatment of lung cancer patients, supporting assessments of pulmonary function, optimization of therapeutic strategies, and monitoring of treatment response. Currently, ventilation scintigraphy using nuclear medicine techniques is widely employed in clinical practice; however, it is often time-consuming, costly, and entails additional radiation exposure. In this study, we propose an explainable neural radiomic sequence model to identify regions of compromised pulmonary ventilation based on four-dimensional computed tomography (4DCT). A cohort of 45 lung cancer patients from the VAMPIRE dataset was analyzed. For each patient, lung volumes were segmented from 4DCT, and voxel-wise radiomic features (56-dimensional) were extracted across the respiratory cycle to capture local intensity and texture dynamics, forming temporal radiomic sequences. Ground truth ventilation defects were delineated voxel-wise using Galligas-PET and DTPA-SPECT. To identify compromised regions, we developed a temporal saliency-enhanced explainable long short-term memory (LSTM) network trained on the radiomic sequences. Temporal saliency maps were generated to highlight key features contributing to the model's predictions. The proposed model demonstrated robust performance, achieving average (range) Dice similarity coefficients of 0.78 (0.74-0.79) for 25 PET cases and 0.78 (0.74-0.82) for 20 SPECT cases. The temporal saliency map explained three key radiomic sequences in ventilation quantification: during lung exhalation, compromised pulmonary function region typically exhibits (1) an increasing trend of intensity and (2) a decreasing trend of homogeneity, in contrast to healthy lung tissue.

Maruf Adewole, Jeffrey D. Rudie, Anu Gbadamosi et al.

Gliomas are the most common type of primary brain tumors. Although gliomas are relatively rare, they are among the deadliest types of cancer, with a survival rate of less than 2 years after diagnosis. Gliomas are challenging to diagnose, hard to treat and inherently resistant to conventional therapy. Years of extensive research to improve diagnosis and treatment of gliomas have decreased mortality rates across the Global North, while chances of survival among individuals in low- and middle-income countries (LMICs) remain unchanged and are significantly worse in Sub-Saharan Africa (SSA) populations. Long-term survival with glioma is associated with the identification of appropriate pathological features on brain MRI and confirmation by histopathology. Since 2012, the Brain Tumor Segmentation (BraTS) Challenge have evaluated state-of-the-art machine learning methods to detect, characterize, and classify gliomas. However, it is unclear if the state-of-the-art methods can be widely implemented in SSA given the extensive use of lower-quality MRI technology, which produces poor image contrast and resolution and more importantly, the propensity for late presentation of disease at advanced stages as well as the unique characteristics of gliomas in SSA (i.e., suspected higher rates of gliomatosis cerebri). Thus, the BraTS-Africa Challenge provides a unique opportunity to include brain MRI glioma cases from SSA in global efforts through the BraTS Challenge to develop and evaluate computer-aided-diagnostic (CAD) methods for the detection and characterization of glioma in resource-limited settings, where the potential for CAD tools to transform healthcare are more likely.

Anahita Fathi Kazerooni, Nastaran Khalili, Xinyang Liu et al.

Pediatric tumors of the central nervous system are the most common cause of cancer-related death in children. The five-year survival rate for high-grade gliomas in children is less than 20\%. Due to their rarity, the diagnosis of these entities is often delayed, their treatment is mainly based on historic treatment concepts, and clinical trials require multi-institutional collaborations. The MICCAI Brain Tumor Segmentation (BraTS) Challenge is a landmark community benchmark event with a successful history of 12 years of resource creation for the segmentation and analysis of adult glioma. Here we present the CBTN-CONNECT-DIPGR-ASNR-MICCAI BraTS-PEDs 2023 challenge, which represents the first BraTS challenge focused on pediatric brain tumors with data acquired across multiple international consortia dedicated to pediatric neuro-oncology and clinical trials. The BraTS-PEDs 2023 challenge focuses on benchmarking the development of volumentric segmentation algorithms for pediatric brain glioma through standardized quantitative performance evaluation metrics utilized across the BraTS 2023 cluster of challenges. Models gaining knowledge from the BraTS-PEDs multi-parametric structural MRI (mpMRI) training data will be evaluated on separate validation and unseen test mpMRI dataof high-grade pediatric glioma. The CBTN-CONNECT-DIPGR-ASNR-MICCAI BraTS-PEDs 2023 challenge brings together clinicians and AI/imaging scientists to lead to faster development of automated segmentation techniques that could benefit clinical trials, and ultimately the care of children with brain tumors.

Hongwei Bran Li, Gian Marco Conte, Qingqiao Hu et al.

Automated brain tumor segmentation methods have become well-established and reached performance levels offering clear clinical utility. These methods typically rely on four input magnetic resonance imaging (MRI) modalities: T1-weighted images with and without contrast enhancement, T2-weighted images, and FLAIR images. However, some sequences are often missing in clinical practice due to time constraints or image artifacts, such as patient motion. Consequently, the ability to substitute missing modalities and gain segmentation performance is highly desirable and necessary for the broader adoption of these algorithms in the clinical routine. In this work, we present the establishment of the Brain MR Image Synthesis Benchmark (BraSyn) in conjunction with the Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2023. The primary objective of this challenge is to evaluate image synthesis methods that can realistically generate missing MRI modalities when multiple available images are provided. The ultimate aim is to facilitate automated brain tumor segmentation pipelines. The image dataset used in the benchmark is diverse and multi-modal, created through collaboration with various hospitals and research institutions.

Florian Kofler, Felix Meissen, Felix Steinbauer et al.

A myriad of algorithms for the automatic analysis of brain MR images is available to support clinicians in their decision-making. For brain tumor patients, the image acquisition time series typically starts with an already pathological scan. This poses problems, as many algorithms are designed to analyze healthy brains and provide no guarantee for images featuring lesions. Examples include, but are not limited to, algorithms for brain anatomy parcellation, tissue segmentation, and brain extraction. To solve this dilemma, we introduce the BraTS inpainting challenge. Here, the participants explore inpainting techniques to synthesize healthy brain scans from lesioned ones. The following manuscript contains the task formulation, dataset, and submission procedure. Later, it will be updated to summarize the findings of the challenge. The challenge is organized as part of the ASNR-BraTS MICCAI challenge.

Dominic LaBella, Maruf Adewole, Michelle Alonso-Basanta et al.

Meningiomas are the most common primary intracranial tumor in adults and can be associated with significant morbidity and mortality. Radiologists, neurosurgeons, neuro-oncologists, and radiation oncologists rely on multiparametric MRI (mpMRI) for diagnosis, treatment planning, and longitudinal treatment monitoring; yet automated, objective, and quantitative tools for non-invasive assessment of meningiomas on mpMRI are lacking. The BraTS meningioma 2023 challenge will provide a community standard and benchmark for state-of-the-art automated intracranial meningioma segmentation models based on the largest expert annotated multilabel meningioma mpMRI dataset to date. Challenge competitors will develop automated segmentation models to predict three distinct meningioma sub-regions on MRI including enhancing tumor, non-enhancing tumor core, and surrounding nonenhancing T2/FLAIR hyperintensity. Models will be evaluated on separate validation and held-out test datasets using standardized metrics utilized across the BraTS 2023 series of challenges including the Dice similarity coefficient and Hausdorff distance. The models developed during the course of this challenge will aid in incorporation of automated meningioma MRI segmentation into clinical practice, which will ultimately improve care of patients with meningioma.

Hangjie Ji, Kyle Lafata, Yvonne Mowery et al.

This paper develops a method of biologically guided deep learning for post-radiation FDG-PET image outcome prediction based on pre-radiation images and radiotherapy dose information. Based on the classic reaction-diffusion mechanism, a novel biological model was proposed using a partial differential equation that incorporates spatial radiation dose distribution as a patient-specific treatment information variable. A 7-layer encoder-decoder-based convolutional neural network (CNN) was designed and trained to learn the proposed biological model. As such, the model could generate post-radiation FDG-PET image outcome predictions with possible time-series transition from pre-radiotherapy image states to post-radiotherapy states. The proposed method was developed using 64 oropharyngeal patients with paired FDG-PET studies before and after 20Gy delivery (2Gy/daily fraction) by IMRT. In a two-branch deep learning execution, the proposed CNN learns specific terms in the biological model from paired FDG-PET images and spatial dose distribution as in one branch, and the biological model generates post-20Gy FDG-PET image prediction in the other branch. The proposed method successfully generated post-20Gy FDG-PET image outcome prediction with breakdown illustrations of biological model components. Time-series FDG-PET image predictions were generated to demonstrate the feasibility of disease response rendering. The developed biologically guided deep learning method achieved post-20Gy FDG-PET image outcome predictions in good agreement with ground-truth results. With break-down biological modeling components, the outcome image predictions could be used in adaptive radiotherapy decision-making to optimize personalized plans for the best outcome in the future.

Tongyang Li, Chunhao Wang, Shouvanik Chakrabarti et al.

We investigate sublinear classical and quantum algorithms for matrix games, a fundamental problem in optimization and machine learning, with provable guarantees. Given a matrix $A\in\mathbb{R}^{n\times d}$, sublinear algorithms for the matrix game $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}} y^{\top} Ax$ were previously known only for two special cases: (1) $\mathcal{Y}$ being the $\ell_{1}$-norm unit ball, and (2) $\mathcal{X}$ being either the $\ell_{1}$- or the $\ell_{2}$-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed $q\in (1,2]$, we solve the matrix game where $\mathcal{X}$ is a $\ell_{q}$-norm unit ball within additive error $ε$ in time $\tilde{O}((n+d)/{ε^{2}})$. We also provide a corresponding sublinear quantum algorithm that solves the same task in time $\tilde{O}((\sqrt{n}+\sqrt{d})\textrm{poly}(1/ε))$ with a quadratic improvement in both $n$ and $d$. Both our classical and quantum algorithms are optimal in the dimension parameters $n$ and $d$ up to poly-logarithmic factors. Finally, we propose sublinear classical and quantum algorithms for the approximate Carathéodory problem and the $\ell_{q}$-margin support vector machines as applications.

Nai-Hui Chia, András Gilyén, Tongyang Li et al.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: $\ell^2$-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin et al.

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with $m$ constraint matrices, each of dimension $n$ and rank $r$, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time $O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))$ given access to a sampling-based low-overhead data structure for the constraint matrices, where $\varepsilon$ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: $\bullet$ Weighted sampling: assuming sampling access to each individual constraint matrix $A_{1},\ldots,A_τ$, we propose a procedure that gives a good approximation of $A=A_{1}+\cdots+A_τ$. $\bullet$ Symmetric approximation: we propose a sampling procedure that gives the \emph{spectral decomposition} of a low-rank Hermitian matrix $A$. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

Nai-Hui Chia, Han-Hsuan Lin, Chunhao Wang

We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A \in \mathbb{C}^{m \times n}$ be a rank-$k$ matrix, and $b \in \mathbb{C}^m$ be a vector. We present two algorithms: a "sampling" algorithm that provides a sample from $A^{-1}b$ and a "query" algorithm that outputs an estimate of an entry of $A^{-1}b$, where $A^{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $O(\mathrm{poly}(k, κ, \|A\|_F, 1/ε)\,\mathrm{polylog}(m, n))$, where $κ$ is the condition number of $A$ and $ε$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.