Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions

In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic geometry on the regression. Specifically, we analyze the impact of the manifold's curvatures (or higher order nonlinearity in the parameterization when the curvatures are locally zero) on the uniqueness of the regression solution. Our findings suggest that the corresponding linear regression does not have a unique solution when the embedded submanifold is flat in some dimensions. Otherwise, the manifold's curvature (or higher order nonlinearity in the embedding) may contribute significantly, particularly in the solution associated with the normal directions of the manifold. Our findings thus reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.Comment: 13 pages, 6 figures, accepted to TAGML23 workshop of ICML2023, to be published in PML

MgNO: Efficient Parameterization of Linear Operators via Multigrid

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a nonlinear operator layer is defined by $\mathcal O_i(u) = \sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$. Here, $\mathcal W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $\mathcal B_{ij}$ takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs)

Rescue of Retinal Degeneration in rd1 Mice by Intravitreally Injected Metformin

Retinitis pigmentosa (RP) is a progressive hereditary retinal degenerative disease in which photoreceptor cells undergo degeneration and apoptosis, eventually resulting in irreversible loss of visual function. Currently, no effective treatment exists for this disease. Neuroprotection and inflammation suppression have been reported to delay the development of RP. Metformin is a well-tested drug used to treat type 2 diabetes, and it has been reported to exert beneficial effects in neurodegenerative diseases, such as Parkinson’s disease and Alzheimer’s disease. In the present study, we used immunofluorescence staining, electroretinogram (ERG) recordings and RNA-Seq to explore the effects of metformin on photoreceptor degeneration and its mechanism in rd1 mice. We found that metformin significantly reduced apoptosis in photoreceptors and delayed the degeneration of photoreceptors and rod bipolar cells in rd1 mice, thus markedly improving the visual function of rd1 mice at P14, P18, and P22 when tested with a light/dark transition test and ERG. Microglial activation in the outer nuclear layer (ONL) of the retina of rd1 mice was significantly suppressed by metformin. RNA-Seq showed that metformin markedly downregulated inflammatory genes and upregulated the expression of crystallin proteins, which have been demonstrated to be important neuroprotective molecules in the retina, revealing the therapeutic potential of metformin for RP treatment. αA-crystallin proteins were further confirmed to be involved in the neuroprotective effects of metformin in a Ca2+ ionophore-damaged 661W photoreceptor-like cell line. These data suggest that metformin exerts a protective effect in rd1 mice via both immunoregulatory and new neuroprotective mechanisms

Microglia Mediate Synaptic Material Clearance at the Early Stage of Rats With Retinitis Pigmentosa

Resident microglia are the main immune cells in the retina and play a key role in the pathogenesis of retinitis pigmentosa (RP). Many previous studies on the roles of microglia mainly focused on the neurotoxicity or neuroprotection of photoreceptors, while their contributions to synaptic remodeling of neuronal circuits in the retina of early RP remained unclarified. In the present study, we used Royal College of Surgeons (RCS) rats, a classic RP model characterized by progressive microglia activation and synapse loss, to investigate the constitutive effects of microglia on the synaptic lesions and ectopic neuritogenesis. Rod degeneration resulted in synapse disruption and loss in the outer plexiform layer (OPL) at the early stage of RP. Coincidentally, the resident microglia in the OPL increased phagocytosis and mainly engaged in phagocytic engulfment of postsynaptic mGluR6 of rod bipolar cells (RBCs). Complement pathway might be involved in clearance of postsynaptic elements of RBCs by microglia. We pharmacologically deleted microglia using a CSF1 receptor (CSF1R) inhibitor to confirm this finding, and found that it caused the accumulation of postsynaptic mGluR6 levels and increased the number and length of ectopic dendrites in the RBCs. Interestingly, the numbers of presynaptic sites expressing CtBP2 and colocalized puncta in the OPL of RCS rats were not affected by microglia elimination. However, sustained microglial depletion led to progressive functional deterioration in the retinal responses to light in RCS rats. Based on our results, microglia mediated the remodeling of RBCs by phagocytosing postsynaptic materials and inhibiting ectopic neuritogenesis, contributing to delay the deterioration of vision at the early stage of RP