We study exact Kullback--Leibler (KL) projection for low-rank factorizations whose two nonnegative factors have prescribed row marginals and a shared, learned column marginal. For arbitrary positive row marginals of equal total mass, the joint KL projection reduces exactly to a strictly convex gauge-fixed dual with only $r-1$ effective variables; its Hessian is a sum of categorical covariance terms and admits $O((n+m)r)$ matrix-free Hessian--vector products. The projection theorem is objective-independent. We then specialize this geometry to doubly stochastic (DS) graph learning through $W=U\operatorname{Diag}(g)^{-1}V^\top$, where row-simplex factors with a common column mass induce an exactly DS graph without materializing an $n\times n$ optimization variable. Combined with observed-edge sparse fitting, a stochastic anchor-reduced manifold regularizer, and Bregman backtracking, the resulting mirror-descent method preserves exact feasibility at every accepted step. Under a nonvanishing latent-mass condition, it satisfies sufficient decrease and an $O(1/N)$ mirror-stationarity bound, while strictly positive accumulation points are KKT stationary. Matched clustering experiments show competitive accuracy, feasibility residuals near numerical precision, and favorable anytime behavior without a dense learned graph.
Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.
Many researchers investigated neural networks with some of their weights fixed to values randomly drawn from a given distribution, e.g., $N(0, I)$. Our proposed HalfNet draws random weights from $N(0, Σ)$, where $Σ$, which defines the geometry of the distribution, has a low-rank factorization that we learn from data. Experiments on MNIST and CIFAR-10 demonstrate that HalfNet can match the performance of fully trained multilayer perceptrons while using substantially fewer parameters. Spectral analysis indicates that much of the predictive power of neural networks lies in the geometry of their weight space rather than in the precise values of individual parameters, and we observe that accuracy scales smoothly with rank. HalfNet is not a neural architecture trick for low-rank structure; it implements a data-dependent random embedding that can also be interpreted through supervised metric learning, or random-feature and kernel perspectives.
Visual Prompting (VP) has emerged as an efficient paradigm for adapting large-scale pre-trained vision models to downstream tasks by incorporating learnable prompts at the input level. However, existing VP methods typically employ dense pixel-level prompts, which often suffer from redundant perturbations, limited generalization and energy inefficiency. To overcome these limitations, we propose to integrate brain-inspired spiking learning into visual prompt learning tasks. As we know that spiking neuron can perform inexpensive information processing by transmitting the input data into discrete spike trains and return sparse outputs. Inspired by this, we propose \textbf{Lo}w-\textbf{R}ank visual \textbf{S}pike \textbf{P}rompting (LoRSP), a novel framework that learns dynamic low-rank sparse visual prompts naturally via a Spiking neuron learning mechanism. The core idea of LoRSP is to exploit the brain-inspired sparse firing mechanism of spiking neurons to generate pixel-level sparse prompt for each instance. To be specific, we first construct a series of prompt factors via low-rank factorization to capture distinct prompt subspaces. These prompt factors are then fed into an SNN architecture, which performs the integrate-and-fire process to emit spikes. As a result, our LoRSP generates a \emph{sparse} visual prompt while maintaining the low-rank constraint. This design enables instance-specific selective prompting, leading to more compact and robust adaptation across diverse downstream tasks. Extensive experiments on five heterogeneous vision backbones and multiple benchmarks demonstrate that LoRSP achieves competitive performance while requiring fewer tunable parameters compared to existing VP methods.