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We introduce a novel unit-time ordinary differential equation (ODE) flow called the preconditioned F\"{o}llmer flow, which efficiently transforms a Gaussian measure into a desired target measure at time 1. To discretize the flow, we apply Euler's method, where the velocity field is calculated either analytically or through Monte Carlo approximation using Gaussian samples. Under reasonable conditions, we derive a non-asymptotic error bound in the Wasserstein distance between the sampling distribution and the target distribution. Through numerical experiments on mixture distributions in 1D, 2D, and high-dimensional spaces, we demonstrate that the samples generated by our proposed flow exhibit higher quality compared to those obtained by several existing methods. Furthermore, we propose leveraging the F\"{o}llmer flow as a warmstart strategy for existing Markov Chain Monte Carlo (MCMC) methods, aiming to mitigate mode collapses and enhance their performance. Finally, thanks to the deterministic nature of the F\"{o}llmer flow, we can leverage deep neural networks to fit the trajectory of sample evaluations. This allows us to obtain a generator for one-step sampling as a result., Comment: 24 pages, 6 figures, 3 tables

Dislocation climb plays an important role in understanding plastic deformation of metallic materials at high temperature. In this paper, we present a continuum formulation for dislocation climb velocity based on densities of dislocations. The obtained continuum formulation is an accurate approximation of the Green's function based discrete dislocation dynamics method (Gu et al. J. Mech. Phys. Solids 83:319-337, 2015). The continuum dislocation climb formulation has the advantage of accounting for both the long-range effect of vacancy bulk diffusion and that of the Peach-Koehler climb force, and the two longrange effects are canceled into a short-range effect (integral with fast-decaying kernel) and in some special cases, a completely local effect. This significantly simplifies the calculation in the Green's function based discrete dislocation dynamics method, in which a linear system has to be solved over the entire system for the long-range effect of vacancy diffusion and the long-range Peach-Koehler climb force has to be calculated. This obtained continuum dislocation climb velocity can be applied in any available continuum dislocation dynamics frameworks. We also present numerical validations for this continuum climb velocity and simulation examples for implementation in continuum dislocation dynamics frameworks.

Disease heterogeneity has been a critical challenge for precision diagnosis and treatment, especially in neurologic and neuropsychiatric diseases. Many diseases can display multiple distinct brain phenotypes across individuals, potentially reflecting disease subtypes that can be captured using MRI and machine learning methods. However, biological interpretability and treatment relevance are limited if the derived subtypes are not associated with genetic drivers or susceptibility factors. Herein, we describe Gene-SGAN - a multi-view, weakly-supervised deep clustering method - which dissects disease heterogeneity by jointly considering phenotypic and genetic data, thereby conferring genetic correlations to the disease subtypes and associated endophenotypic signatures. We first validate the generalizability, interpretability, and robustness of Gene-SGAN in semi-synthetic experiments. We then demonstrate its application to real multi-site datasets from 28,858 individuals, deriving subtypes of Alzheimer's disease and brain endophenotypes associated with hypertension, from MRI and SNP data. Derived brain phenotypes displayed significant differences in neuroanatomical patterns, genetic determinants, biological and clinical biomarkers, indicating potentially distinct underlying neuropathologic processes, genetic drivers, and susceptibility factors. Overall, Gene-SGAN is broadly applicable to disease subtyping and endophenotype discovery, and is herein tested on disease-related, genetically-driven neuroimaging phenotypes.

In this paper, combined with the P-angle function of Banach spaces and the geometric constants that can characterize Hilbert spaces, the new angular geometric constant is defined. Firstly, this paper explores the basic properties of the new constant and obtains some inequalities with significant geometric constants. Then according to the derived inequalities, this paper studies the relationship between the new constant and the geometric properties of Banach spaces. Furthermore, the necessary and sufficient condition for uniform non-squareness, and the sufficient conditions for uniform convexity, the normal structure and the fixed point property will be established.

Generative adversarial networks (GANs) are one powerful type of deep learning models that have been successfully utilized in numerous fields. They belong to a broader family called generative methods, which generate new data with a probabilistic model by learning sample distribution from real examples. In the clinical context, GANs have shown enhanced capabilities in capturing spatially complex, nonlinear, and potentially subtle disease effects compared to traditional generative methods. This review appraises the existing literature on the applications of GANs in imaging studies of various neurological conditions, including Alzheimer's disease, brain tumors, brain aging, and multiple sclerosis. We provide an intuitive explanation of various GAN methods for each application and further discuss the main challenges, open questions, and promising future directions of leveraging GANs in neuroimaging. We aim to bridge the gap between advanced deep learning methods and neurology research by highlighting how GANs can be leveraged to support clinical decision making and contribute to a better understanding of the structural and functional patterns of brain diseases.

A plethora of machine learning methods have been applied to imaging data, enabling the construction of clinically relevant imaging signatures of neurological and neuropsychiatric diseases. Oftentimes, such methods don't explicitly model the heterogeneity of disease effects, or approach it via nonlinear models that are not interpretable. Moreover, unsupervised methods may parse heterogeneity that is driven by nuisance confounding factors that affect brain structure or function, rather than heterogeneity relevant to a pathology of interest. On the other hand, semi-supervised clustering methods seek to derive a dichotomous subtype membership, ignoring the truth that disease heterogeneity spatially and temporally extends along a continuum. To address the aforementioned limitations, herein, we propose a novel method, termed Surreal-GAN (Semi-SUpeRvised ReprEsentAtion Learning via GAN). Using cross-sectional imaging data, Surreal-GAN dissects underlying disease-related heterogeneity under the principle of semi-supervised clustering (cluster mappings from normal control to patient), proposes a continuously dimensional representation, and infers the disease severity of patients at individual level along each dimension. The model first learns a transformation function from normal control (CN) domain to the patient (PT) domain with latent variables controlling transformation directions. An inverse mapping function together with regularization on function continuity, pattern orthogonality and monotonicity was also imposed to make sure that the transformation function captures necessarily meaningful imaging patterns with clinical significance. We first validated the model through extensive semi-synthetic experiments, and then demonstrate its potential in capturing biologically plausible imaging patterns in Alzheimer's disease (AD).

The imaging community has increasingly adopted machine learning (ML) methods to provide individualized imaging signatures related to disease diagnosis, prognosis, and response to treatment. Clinical neuroscience and cancer imaging have been two areas in which ML has offered particular promise. However, many neurologic and neuropsychiatric diseases, as well as cancer, are often heterogeneous in terms of their clinical manifestations, neuroanatomical patterns or genetic underpinnings. Therefore, in such cases, seeking a single disease signature might be ineffectual in delivering individualized precision diagnostics. The current chapter focuses on ML methods, especially semi-supervised clustering, that seek disease subtypes using imaging data. Work from Alzheimer Disease and its prodromal stages, psychosis, depression, autism, and brain cancer are discussed. Our goal is to provide the readers with a broad overview in terms of methodology and clinical applications.

In this paper, we investigate the continuity of the attractors in time-dependent phase spaces. (i) We establish two abstract criteria on the upper semicontinuity and the residual continuity of the pullback $\mathscr D$-attractor with respect to the perturbations, and an equivalence criterion between their continuity and the pullback equi-attraction, which generalize the continuity theory of attractors developed recently in [27,28] to that in time-dependent spaces. (ii) We propose the notion of pullback $\mathscr D$-exponential attractor, which includes the notion of time-dependent exponential attractor [33] as its spacial case, and establish its existence and H\"{o}lder continuity criterion via quasi-stability method introduced originally by Chueshov and Lasiecka [12,13]. (iii) We apply above-mentioned criteria to the semilinear damped wave equations with perturbed time-dependent speed of propagation: $\e\rho(t) u_{tt}+\alpha u_t -\Delta u+f(u)=g$, with perturbation parameter $\e\in(0, 1]$, to realize above mentioned continuity of pullback $\mathscr D$ and $\mathscr D$-exponential attractors in time-dependent phase spaces, and the method developed here allows to overcome the difficulty of the hyperbolicity of the model. These results deepen and extend recent theory of attractors in time-dependent spaces in literatures [15,20,19].

Reconstructing the 3D pose of a person in metric scale from a single view image is a geometrically ill-posed problem. For example, we can not measure the exact distance of a person to the camera from a single view image without additional scene assumptions (e.g., known height). Existing learning based approaches circumvent this issue by reconstructing the 3D pose up to scale. However, there are many applications such as virtual telepresence, robotics, and augmented reality that require metric scale reconstruction. In this paper, we show that audio signals recorded along with an image, provide complementary information to reconstruct the metric 3D pose of the person. The key insight is that as the audio signals traverse across the 3D space, their interactions with the body provide metric information about the body's pose. Based on this insight, we introduce a time-invariant transfer function called pose kernel -- the impulse response of audio signals induced by the body pose. The main properties of the pose kernel are that (1) its envelope highly correlates with 3D pose, (2) the time response corresponds to arrival time, indicating the metric distance to the microphone, and (3) it is invariant to changes in the scene geometry configurations. Therefore, it is readily generalizable to unseen scenes. We design a multi-stage 3D CNN that fuses audio and visual signals and learns to reconstruct 3D pose in a metric scale. We show that our multi-modal method produces accurate metric reconstruction in real world scenes, which is not possible with state-of-the-art lifting approaches including parametric mesh regression and depth regression.

Based on the parallelogram law and isosceles orthogonality, we define a new orthogonal geometric constant. We first discuss some basic properties of this new constant. Next, we consider the relation between the constant and the uniformly non-square property. Moreover, a generalized constant is also introduced and some basic properties are presented. It is shown that, for a normed space, the constant value is equal to 1 if and only if the norm can be induced by the inner product. Finally, we verify that this constant is closely related to the well-known geometric constants through some inequalities.

We shall introduce a new geometric constant $L^{\prime}_{\mathrm{Y}}(\lambda,X)$ based on a generalization of the parallelogram law. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, $L^{\prime}_{\mathrm{Y}}(\lambda,X)$ becomes $1$ for some $\lambda_0\in (0,1)$ if and only if the norm is induced by an inner product. Moreover, some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

Late-life depression (LLD) is characterized by considerable heterogeneity in clinical manifestation. Unraveling such heterogeneity would aid in elucidating etiological mechanisms and pave the road to precision and individualized medicine. We sought to delineate, cross-sectionally and longitudinally, disease-related heterogeneity in LLD linked to neuroanatomy, cognitive functioning, clinical symptomatology, and genetic profiles. Multimodal data from a multicentre sample (N=996) were analyzed. A semi-supervised clustering method (HYDRA) was applied to regional grey matter (GM) brain volumes to derive dimensional representations. Two dimensions were identified, which accounted for the LLD-related heterogeneity in voxel-wise GM maps, white matter (WM) fractional anisotropy (FA), neurocognitive functioning, clinical phenotype, and genetics. Dimension one (Dim1) demonstrated relatively preserved brain anatomy without WM disruptions relative to healthy controls. In contrast, dimension two (Dim2) showed widespread brain atrophy and WM integrity disruptions, along with cognitive impairment and higher depression severity. Moreover, one de novo independent genetic variant (rs13120336) was significantly associated with Dim 1 but not with Dim 2. Notably, the two dimensions demonstrated significant SNP-based heritability of 18-27% within the general population (N=12,518 in UKBB). Lastly, in a subset of individuals having longitudinal measurements, Dim2 demonstrated a more rapid longitudinal decrease in GM and brain age, and was more likely to progress to Alzheimers disease, compared to Dim1 (N=1,413 participants and 7,225 scans from ADNI, BLSA, and BIOCARD datasets).

The paper is concerned with the exponential attractors for the viscoelastic wave model in $\Omega\subset \mathbb R^3$: $$u_{tt}-h_t(0)\Delta u-\int_0^\infty\partial_sh_t(s)\Delta u(t-s)\mathrm ds+f(u)=h,$$ with time-dependent memory kernel $h_t(\cdot)$ which is used to model aging phenomena of the material. Conti et al [Amer. J. Math., 2018] recently provided the correct mathematical setting for the model and a well-posedness result within the novel theory of dynamical systems acting on. time-dependent spaces, recently established by Conti, Pata and Temam [J. Differential Equations, 2013], and proved the existence and the regularity of the time-dependent global attractor. In this work, we further study the existence of the time-dependent exponential attractors as well as their regularity. We establish an abstract existence criterion via quasi-stability method introduced originally by Chueshov and Lasiecka [J. Dynam. Diff.Eqs.,2004], and on the basis of the theory and technique developed in [Amer. J. Math., 2018] we further provide a new method to overcome the difficulty of the lack of further regularity to show the existence of the time-dependent exponential attractor. And these techniques can be used to tackle other hyperbolic models.

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with $\mathrm{ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bounds on the approximation error of deep $\mathrm{ReLU}^2$ network in $H^1$ norm and on the Rademacher complexity of the non-Lipschitz composition of gradient norm and $\mathrm{ReLU}^2$ network, both of which are of independent interest.

Heterogeneity of brain diseases is a challenge for precision diagnosis/prognosis. We describe and validate Smile-GAN (SeMI-supervised cLustEring-Generative Adversarial Network), a novel semi-supervised deep-clustering method, which dissects neuroanatomical heterogeneity, enabling identification of disease subtypes via their imaging signatures relative to controls. When applied to MRIs (2 studies; 2,832 participants; 8,146 scans) including cognitively normal individuals and those with cognitive impairment and dementia, Smile-GAN identified 4 neurodegenerative patterns/axes: P1, normal anatomy and highest cognitive performance; P2, mild/diffuse atrophy and more prominent executive dysfunction; P3, focal medial temporal atrophy and relatively greater memory impairment; P4, advanced neurodegeneration. Further application to longitudinal data revealed two distinct progression pathways: P1$\rightarrow$P2$\rightarrow$P4 and P1$\rightarrow$P3$\rightarrow$P4. Baseline expression of these patterns predicted the pathway and rate of future neurodegeneration. Pattern expression offered better yet complementary performance in predicting clinical progression, compared to amyloid/tau. These deep-learning derived biomarkers offer promise for precision diagnostics and targeted clinical trial recruitment., Comment: 37 pages, 11 figures

We propose an Euler particle transport (EPT) approach for generative learning. The proposed approach is motivated by the problem of finding an optimal transport map from a reference distribution to a target distribution characterized by the Monge-Ampere equation. Interpreting the infinitesimal linearization of the Monge-Ampere equation from the perspective of gradient flows in measure spaces leads to a stochastic McKean-Vlasov equation. We use the forward Euler method to solve this equation. The resulting forward Euler map pushes forward a reference distribution to the target. This map is the composition of a sequence of simple residual maps, which are computationally stable and easy to train. The key task in training is the estimation of the density ratios or differences that determine the residual maps. We estimate the density ratios (differences) based on the Bregman divergence with a gradient penalty using deep density-ratio (difference) fitting. We show that the proposed density-ratio (difference) estimators do not suffer from the "curse of dimensionality" if data is supported on a lower-dimensional manifold. Numerical experiments with multi-mode synthetic datasets and comparisons with the existing methods on real benchmark datasets support our theoretical results and demonstrate the effectiveness of the proposed method., Comment: arXiv admin note: substantial text overlap with arXiv:2002.02862

Machine learning methods applied to complex biomedical data has enabled the construction of disease signatures of diagnostic/prognostic value. However, less attention has been given to understanding disease heterogeneity. Semi-supervised clustering methods can address this problem by estimating multiple transformations from a (e.g. healthy) control (CN) group to a patient (PT) group, seeking to capture the heterogeneity of underlying pathlogic processes. Herein, we propose a novel method, Smile-GANs (SeMi-supervIsed cLustEring via GANs), for semi-supervised clustering, and apply it to brain MRI scans. Smile-GANs first learns multiple distinct mappings by generating PT from CN, with each mapping characterizing one relatively distinct pathological pattern. Moreover, a clustering model is trained interactively with mapping functions to assign PT into corresponding subtype memberships. Using relaxed assumptions on PT/CN data distribution and imposing mapping non-linearity, Smile-GANs captures heterogeneous differences in distribution between the CN and PT domains. We first validate Smile-GANs using simulated data, subsequently on real data, by demonstrating its potential in characterizing heterogeneity in Alzheimer's Disease (AD) and its prodromal phases. The model was first trained using baseline MRIs from the ADNI2 database and then applied to longitudinal data from ADNI1 and BLSA. Four robust subtypes with distinct neuroanatomical patterns were discovered: 1) normal brain, 2) diffuse atrophy atypical of AD, 3) focal medial temporal lobe atrophy, 4) typical-AD. Further longitudinal analyses discover two distinct progressive pathways from prodromal to full AD: i) subtypes 1 - 2 - 4, and ii) subtypes 1 - 3 - 4. Although demonstrated on an important biomedical problem, Smile-GANs is general and can find application in many biomedical and other domains.

The main goal of 1-bit compressive sampling is to decode $n$ dimensional signals with sparsity level $s$ from $m$ binary measurements. This is a challenging task due to the presence of nonlinearity, noises and sign flips. In this paper, the cardinality constraint least square is proposed as a desired decoder. We prove that, up to a constant $c$, with high probability, the proposed decoder achieves a minimax estimation error as long as $m \geq \mathcal{O}( s\log n)$. Computationally, we utilize a generalized Newton algorithm (GNA) to solve the cardinality constraint minimization problem with the cost of solving a least squares problem with small size at each iteration. We prove that, with high probability, the $\ell_{\infty}$ norm of the estimation error between the output of GNA and the underlying target decays to $\mathcal{O}(\sqrt{\frac{\log n }{m}}) $ after at most $\mathcal{O}(\log s)$ iterations. Moreover, the underlying support can be recovered with high probability in $\mathcal{O}(\log s)$ steps provided that the target signal is detectable. Extensive numerical simulations and comparisons with state-of-the-art methods are presented to illustrate the robustness of our proposed decoder and the efficiency of the GNA algorithm., Comment: arXiv admin note: text overlap with arXiv:1711.01206

Thanks to advancements in diagnosis and treatment, prostate cancer patients have high long-term survival rates. Currently, an important goal is to preserve quality-of-life during and after treatment. The relationship between the radiation a patient receives and the subsequent side effects he experiences is complex and difficult to model or predict. Here, we use machine learning algorithms and statistical models to explore the connection between radiation treatment and post-treatment gastro-urinary function. Since only a limited number of patient datasets are currently available, we used image flipping and curvature-based interpolation methods to generate more data in order to leverage transfer learning. Using interpolated and augmented data, we trained a convolutional autoencoder network to obtain near-optimal starting points for the weights. A convolutional neural network then analyzed the relationship between patient-reported quality-of-life and radiation. We also used analysis of variance and logistic regression to explore organ sensitivity to radiation and develop dosage thresholds for each organ region. Our findings show no connection between the bladder and quality-of-life scores. However, we found a connection between radiation applied to posterior and anterior rectal regions to changes in quality-of-life. Finally, we estimated radiation therapy dosage thresholds for each organ. Our analysis connects machine learning methods with organ sensitivity, thus providing a framework for informing cancer patient care using patient reported quality-of-life metrics.

The paper investigates the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity: $u_{tt}-\Delta u_{t}-(1+\epsilon\|\nabla u\|^{2})\Delta u+f(u)=g(x,t)$, where $\epsilon\in [0,1]$ is a perturbed parameter. It shows that when the nonlinearity $f(u)$ is of supercritical growth $p: \frac{N+2}{N-2}=p^*