We propose a deterministic simulated-oracle-direction mechanism for escaping spurious local minima in low-rank matrix sensing without explicitly lifting the problem. The method projects escape directions from an overparametrized landscape back into the original parameter space and guarantees objective decrease from existing local minima. Numerical experiments show reliable escapes with low computational overhead.

We study how loss-function choice affects robustness and optimization landscapes in noisy matrix sensing. A kernel-based residual loss matches mean-squared error under Gaussian noise while remaining stable for heavy-tailed and heterogeneous corruption. Theoretical and empirical results show improved robustness and convergence across diverse noise distributions.

We study matrix sensing through the convex matrix-lasso relaxation, where the nuclear norm replaces rank. We show that the relaxation can certifiably recover a low-rank ground-truth matrix and establish a connection to its Burer–Monteiro factorization. The guarantees extend to finite-variance noise and clarify the relationship between convex and non-convex formulations.

We introduce VAMO, a variance-reduced mixed-gradient optimizer that combines full-batch zeroth-order gradients with mini-batch stochastic updates. Its hybrid estimator achieves a dimension-agnostic convergence rate while using less dynamic memory than many first-order baselines. Neural-network training and LLM fine-tuning experiments show strong performance with a smaller memory footprint.

We study certifying ReLU neural networks against adversarial input perturbations by partitioning the input uncertainty set and solving relaxations on each part. We show how intelligently designed branching can eliminate relaxation error for linear programs and reduce worst-case error for semidefinite programs. Experiments on MNIST, CIFAR-10, and Wisconsin breast-cancer classifiers demonstrate substantially higher certification rates.

We introduce Topology-Aware 3D Gaussian Splatting, combining local persistent Voronoi interpolation with a persistent-homology regularizer. The method improves geometric coverage and preserves topological structure during optimization. Experiments across novel-view-synthesis benchmarks improve PSNR, SSIM, and LPIPS while maintaining efficient memory use.

We analyze matrix-sensing landscapes far from the ground truth, where spurious solutions can obstruct local optimization. Under suitable conditions, distant critical points are strict saddles rather than local minima, and higher-order losses amplify the negative curvature around them. This provides a theoretical alternative to increasing model complexity through overparametrization.

We examine how gradient descent induces implicit regularization in tensor optimization through a lifted matrix-sensing framework. With sufficiently small initialization, gradient descent produces approximately rank-one tensors and critical points with escape directions. The results highlight how tensor parametrization and first-order methods can work together to reach global optima.

We study over-parametrization for low-rank matrix sensing by constructing an infinite hierarchy of lifted non-convex problems through tensor lifting and Burer–Monteiro factorization. Spurious solutions remain stationary through the hierarchy but become strict saddle points under technical conditions, allowing local search methods to escape them. We also derive a bound on how much over-parametrization is needed to eliminate spurious solutions.

We compare semidefinite programming and Burer–Monteiro factorization for matrix sensing. We identify structured instances where one method succeeds while the other fails, and show that the success of semidefinite programming depends on the rank of the unknown solution whereas Burer–Monteiro performance is rank-independent. The results clarify the distinct strengths of the two approaches.

We develop a mathematical framework for low-rank matrix optimization with noisy data under a less restrictive restricted-isometry condition than prior work. When the noiseless RIP constant is below one third, spurious local solutions of the noisy problem must remain close to the ground truth, while the strict-saddle analysis gives polynomial-time search guarantees. The results characterize local-minimum geometry across both favorable and difficult regimes.

We study low-rank matrix recovery with noisy linear measurements and ask when local search can find the ground truth with small error. We derive sharp guarantees for local minimizers under a restricted-isometry assumption, show how the guarantee improves as noise decreases, and prove a strict-saddle property that supports polynomial-time convergence of perturbed gradient descent.

We study noisy matrix sensing in both exact-parametrization and overparametrization regimes. The analysis quantifies how the restricted-isometry property shapes the non-convex landscape, bounds the distance of local minimizers from the ground truth, and proves a strict-saddle property that yields global convergence of perturbed gradient descent.

We develop a sequential semidefinite-programming framework for neural-network robustness certification. By adding non-convex cuts through disjunctive programming, the method shrinks the relaxation gap toward zero. We analyze the approach theoretically and empirically as the number of cuts increases.

We study certification of neural-network robustness to perturbed and adversarial inputs. A partition-based procedure exploits the structure of ReLU networks to tighten existing linear-programming relaxations, reaching zero relaxation error with a finite partition. Experiments show that the method can issue robustness certificates where prior approaches fail.

We present a certifiably globally optimal algorithm for determining the extrinsic calibration between two sensors that can produce independent egomotion estimates. The problem is formulated as a quadratically constrained quadratic program and solved through a semidefinite-programming relaxation with a global optimality certificate. The method is fast, robust to noise, and does not require a prior estimate or operator intervention.