When Your Manifold Learning Metric Ignores the Underlying Geometry
You spent hours tuning perplexity, learning rate, and min_dist. Your t-SNE or UMAP plot looks clean—clusters separated, colors matching labels. But wh...
Navigating the Frontier of Mathematical Modeling and Nonlinear Dynamics for the Advanced Practitioner
Explore rigorous expositions on asymptotic analysis, optimization theory, and computational methods — crafted for seasoned mathematicians seeking deeper structural insights.
Articles — Page 4
Computational Topology for Data
When Wasserstein Distances Deceive: Landscape Mismatches in Topological Summaries
I watched a staff burn two months chasing a false signal. They had computed persistence landscape from lone-cell RNA data, then used Wasserstein dista...
Computational Topology for Data
Choosing a Filtration That Doesn't Collapse Your Meaningful Loops
Persistent homology is a beautiful lens for data—but only if the filtra you choose doesn't crush the very structure you're after. Pick off, and your p...
Stochastic Control & Filtering
Why Your Particle Filter Collapses in High Dimensions (And What to Reach For Instead)
So your particle filter is dead. You threw 10,000 particles at a 20-dimensional state space, watched the weights collapse to one particle after three ...
Stochastic Control & Filtering
What to Fix First in Nonlinear Filtering: Model Error vs. Sampling Degeneracy
Nonlinear filtered is a messy business. You inherit a model from the literature, throw in a particle filter, and watch it diverge after twenty steps. ...
Inverse Problems & Regularization
When Your Prior Dominates the Data: Bayesian Regularization in Low-SNR Regimes
You spent months designing a prior that encodes physics, smoothness, or sparsity. Then the data arrive—noisy, sparse, maybe corrupted. The posterior l...
Inverse Problems & Regularization
Choosing the Regularization Parameter When the L-Curve Lies
You spend hours computing the L-curve. The corner looks clear—perfect trade-off between residual and solual norm. You pick that lambda. But the recons...
Inverse Problems & Regularization
Why Tikhonov Fails for Discontinuous Solutions (And What to Reach For Instead)
Smoothness assumptions are baked into classic regularization. Tikhonov penalizes large derivatives quadratically, which forces solutions to be everywh...
Spectral Optimization
When Spectral Convergence Fails for Smooth Data: The Hidden Role of Boundary Layers
You have smooth data — infinitely differentiable, even. The kind of function spectral methods were built for. Yet your Chebyshev expansion, that suppo...
Spectral Optimization
When Tiny Eigenvalue Shifts Crash Your Solver
Imagine you are solving a substantial eigenvalue issue for a bridge block. The solver converges— then a load shift by 0.1%. Your eigenpairs jump, and ...
Spectral Optimization
When Your Spectrum Lies: Aliasing in Non-Uniform Spectral Methods
If you have ever tried to compute a spectrum from unevenly spaced data—say, stock prices logged at random times, or astronomical observations interrup...
Choosing the Right Norm for an Ill-Posed Inverse Problem
You have a blurry image of a star field and a known point-spread function. You want the sharp original. But the inverse problem is ill-posed—tiny nois...