An algorithm for determining copositive matrices

In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of $\hat{T}^{-}=\{y\in \Delta_{m}| \beta^Ty\leq 0\}$ on the standard simplex $\Delta_m$, where each component of the vector $\beta$ is -1, 0 or 1.Comment: 15 page

Learning prediction function of prior measures for statistical inverse problems of partial differential equations

In this paper, we view the statistical inverse problems of partial differential equations (PDEs) as PDE-constrained regression and focus on learning the prediction function of the prior probability measures. From this perspective, we propose general generalization bounds for learning infinite-dimensionally defined prior measures in the style of the probability approximately correct Bayesian learning theory. The theoretical framework is rigorously defined on infinite-dimensional separable function space, which makes the theories intimately connected to the usual infinite-dimensional Bayesian inverse approach. Inspired by the concept of $\alpha$-differential privacy, a generalized condition (containing the usual Gaussian measures employed widely in the statistical inverse problems of PDEs) has been proposed, which allows the learned prior measures to depend on the measured data (the prediction function with measured data as input and the prior measure as output can be introduced). After illustrating the general theories, the specific settings of linear and nonlinear problems have been given and can be easily casted into our general theories to obtain concrete generalization bounds. Based on the obtained generalization bounds, infinite-dimensionally well-defined practical algorithms are formulated. Finally, numerical examples of the backward diffusion and Darcy flow problems are provided to demonstrate the potential applications of the proposed approach in learning the prediction function of the prior probability measures.Comment: 57 page

The anti-tumor histone deacetylase inhibitor SAHA and the natural flavonoid curcumin exhibit synergistic neuroprotection against amyloid-beta toxicity

With the trend of an increasing aged population worldwide, Alzheimer’s disease (AD), an age-related neurodegenerative disorder, as one of the major causes of dementia in elderly people is of growing concern. Despite the many hard efforts attempted during the past several decades in trying to elucidate the pathological mechanisms underlying AD and putting forward potential therapeutic strategies, there is still a lack of effective treatments for AD. The efficacy of many potential therapeutic drugs for AD is of main concern in clinical practice. For example, large bodies of evidence show that the antitumor histone deacetylase (HDAC) inhibitor, suberoylanilidehydroxamic acid (SAHA), may be of benefit for the treatment of AD; however, its extensive inhibition of HDACs makes it a poor therapeutic. Moreover, the natural flavonoid, curcumin, may also have a potential therapeutic benefit against AD; however, it is plagued by low bioavailability. Therefore, the integrative effects of SAHA and curcumin were investigated as a protection against amyloid-beta neurotoxicity in vitro. We hypothesized that at low doses their synergistic effect would improve therapeutic selectivity, based on experiments that showed that at low concentrations SAHA and curcumin could provide comprehensive protection against Ab25–35-induced neuronal damage in PC12 cells, strongly implying potent synergism. Furthermore, network analysis suggested that the possible mechanism underlying their synergistic action might be derived from restoration of the damaged functional link between Akt and the CBP/p300 pathway, which plays a crucial role in the pathological development of AD. Thus, our findings provided a feasible avenue for the application of a synergistic drug combination, SAHA and curcumin, in the treatment of AD

Text-driven Visual Synthesis with Latent Diffusion Prior

There has been tremendous progress in large-scale text-to-image synthesis driven by diffusion models enabling versatile downstream applications such as 3D object synthesis from texts, image editing, and customized generation. We present a generic approach using latent diffusion models as powerful image priors for various visual synthesis tasks. Existing methods that utilize such priors fail to use these models' full capabilities. To improve this, our core ideas are 1) a feature matching loss between features from different layers of the decoder to provide detailed guidance and 2) a KL divergence loss to regularize the predicted latent features and stabilize the training. We demonstrate the efficacy of our approach on three different applications, text-to-3D, StyleGAN adaptation, and layered image editing. Extensive results show our method compares favorably against baselines.Comment: Project website: https://latent-diffusion-prior.github.io