Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2104.03448

Visual Diagnostics for Constrained Optimisation with Application to Guided Tours

H. Sherry Zhang·Dianne Cook·Ursula Laa·Nicolas Langrené·Patricia Menéndez
View original source ↗Read source document ↗Export metadata ↓

Abstract

A guided tour helps to visualise high-dimensional data by showing low-dimensional projections along a projection pursuit optimisation path. Projection pursuit is a generalisation of principal component analysis, in the sense that different indexes are used to define the interestingness of the projected data. While much work has been done in developing new indexes in the literature, less has been done on understanding the optimisation. Index functions can be noisy, might have multiple local maxima as well as an optimal maximum, and are constrained to generate orthonormal projection frames, which complicates the optimization. In addition, projection pursuit is primarily used for exploratory data analysis, and finding the local maxima is also useful. The guided tour is especially useful for exploration, because it conducts geodesic interpolation connecting steps in the optimisation and shows how the projected data changes as a maxima is approached. This work provides new visual diagnostics for examining a choice of optimisation procedure, based on the provision of a new data object which collects information throughout the optimisation. It has helped to diagnose and fix several problems with projection pursuit guided tour. This work might be useful more broadly for diagnosing optimisers, and comparing their performance. The diagnostics are implemented in the R package, ferrn.

Explore related subjects

stat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

H. Sherry Zhang, Dianne Cook, Ursula Laa, Nicolas Langrené, Patricia Menéndez. 2021-04-08. Visual Diagnostics for Constrained Optimisation with Application to Guided Tours. https://doi.org/10.32614/rj-2021-105

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fast inversion of the generalized Fisher transformation of correlation matrices

The generalized Fisher transformation maps a non-singular correlation matrix to an unconstrained real vector through the off-diagonal elements of its matrix logarithm. Evaluating its inverse is a computational bottleneck in dynamic correlation and multivariate volatility models. We develop a fast inversion algorithm by characterizing the unknown diagonal as the minimizer of a smooth, strictly convex, and coercive objective. An explicit Hessian and global spectral bounds identify the standard fixed-point iteration as a quasi-Newton method and explain why it can converge slowly near singularity. Every fixed-point step decreases the objective, and the iteration converges from every starting point. These results motivate GFT-FP+N, a hybrid of fixed-point and matrix-free Newton steps that never forms the Jacobian. In benchmarks with up to 1,000 replications per design and dimensions up to 800, GFT-FP+N reduces computation time by up to a factor of forty-five relative to the fixed-point iteration and converged in every replication, including on designs where Broyden's method almost always fails. Julia and R packages are provided.

stat.CO↗
arXiv2026

Exact Simulation of Diffusions via Brownian Bridge Range Reconstruction

We develop an exact simulation algorithm for scalar diffusion paths and diffusion bridges when the Poisson potential is unbounded in both tails. The method reconstructs the realized range of a Brownian bridge proposal by sampling its maximum and location, together with the maxima and locations of the two adjacent restricted Brownian meanders. Conditional on this finite information, the remaining path decomposes into four conditionally independent interval-constrained Brownian bridges, which can be sampled exactly at the Poisson times required by the rejection test. In contrast to constructions based on an enclosing range layer, the proposed representation retains the exact extrema and their locations. Our algorithm returns an exact finite-dimensional skeleton without time-discretization error and permits exact post-acceptance refinement at arbitrary finite collections of times. Numerical experiments validate the resulting finite-dimensional laws and identify the restricted-meander extremum simulation as the principal computational cost in the nonlinear example.

stat.CO↗
arXiv2026

Wasserstein mixing of a systematic-scan random rotation sampler

We study the mixing time of a systematic-scan analogue of Kac's walk that was proposed as a fast surrogate for Haar-distributed orthogonal matrices in randomized high-dimensional algorithms and was conjectured to approach Haar measure after only logarithmically many sweeps. We show that this conjectured speed-up does not occur for convergence of the full matrix law to Haar measure in Frobenius Wasserstein distance. At fixed normalized accuracy, the mixing time lies between order $n/\log n$ and order $n$ sweeps; at fixed absolute Frobenius accuracy, the corresponding bounds are between order $n$ and order $n\log n$. More strongly, below the scale $n/\log n$, the normalized Wasserstein distance remains asymptotically at its extremal value. We also show that the output law is singular with respect to Haar measure for fewer than $n/2$ sweeps. Thus the sampler may provide effective application-specific randomization without exhibiting the much faster full-Haar mixing.

stat.CO↗