Eigenvalue-Decomposition Cost Denoising as an Alternative to Predict-then-Optimize for Shortest-Path Problems

Henry Aldridge-Krawciw; Irene Aldridge

econ.EMSep 14, 2026

Eigenvalue-Decomposition Cost Denoising as an Alternative to Predict-then-Optimize for Shortest-Path Problems

Authors: Henry Aldridge-Krawciw, Irene Aldridge

Abstract

Predict-then-optimize methods such as Smart "Predict, then Optimize" (SPO+) of Elmachtoub and Grigas (2022) learn a mapping from contextual features to unknown edge costs and then solve the induced combinatorial problem on the predicted costs. This approach is powerful but relies on the predictive model being well specified: when the true cost-generating process is nonlinear in the features and the predictor is linear, SPO+'s performance degrades as the misspecification grows. We propose and evaluate a structurally different remedy for a specific but common setting: when the decision-maker observes many noisy realizations of the same underlying cost process, the realized cost vectors themselves can be treated as a noisy signal and denoised directly, via eigenvalue decomposition (equivalently, Principal Component Analysis) of their covariance matrix, before ever invoking a predictive model. We instantiate this idea on the $5\times5$ grid shortest-path benchmark introduced by Elmachtoub and Grigas (2022), retaining only the top- $k$ eigenvectors of the training cost covariance matrix and projecting new noisy cost observations onto that subspace prior to solving with Dijkstra's (1959) algorithm. We find that the choice of $k$ is decisive: keeping only $k{=}2$ eigenvectors discards real signal and underperforms even the naive noisy-cost baseline, while setting $k{=}5$ to match the true latent feature dimension makes eigenvalue-denoised Dijkstra the best-performing method at every misspecification level tested, outperforming SPO+ by a wide margin under high misspecification.

Explore similar work

Apr 30, 2026cs.LG

A Short Note on Batch-efficient Divide-and-Conquer Algorithm for EigenDecomposition

EigenDecomposition (ED) is at the heart of many computer vision algorithms and applications. One crucial bottleneck limiting its usage is the expensive computation cost, particularly for a mini-batch of matrices in deep neural networks. Our previous work proposed a dedicated QR-based ED algorithm for batched small matrices (dim

{<}32

). This short paper targets the limitation and proposes a batch-efficient Divide-and-Conquer based ED algorithm for larger matrices. The numerical test shows that for a mini-batch of matrices whose dimensions are smaller than

64

, our method can be much faster than the Pytorch SVD function.

Yue Song

Singular Value Decomposition Largest Eigenvalue Computer Vision

Ask PDF

Sep 17, 2026cs.AI

Customizable and Jointly Optimized Route Planning: A Deep Architecture Enabling Differentiable Shortest-Path Search

With the widespread use of online navigation and ride-hailing services, achieving optimal route planning for diverse user preferences has recently attracted increasing attention. Classic graph algorithms for pathfinding use heuristic cost functions to define edge weight, thus providing no optimality guarantee of route quality. Prior data-driven approaches equating ground truth of the optimal route with user trajectory, which is however moderately influenced by the navigation service, suffers from the feedback loop problem. To address these issues, we propose a deep architecture that is able to jointly optimize cost functions and route-ranking model towards any route preference. First, we run a multi-objective Dijkstra algorithm offline to collect the set of Pareto optimal routes, deeming it as the complete candidate set. Exploiting the property of such a set, we design a neural network structure that emulates shortest-path search and route ranking in an end-to-end differentiable manner. Second, we define route preference as a task of constrained optimization of route attributes, and propose a novel loss function that optimizes a single-objective variable, with other variables strictly under constraints. We conduct extensive experiments on real-world datasets. The results show that our architecture significantly outperforms state-of-the-art methods in route quality and customizability.

Rui Zhao, Chao Chen, Longfei Xu +4

Shortest Paths Expert Routing

Ask PDF

Jun 30, 2026math.OC

Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics

Low-rank matrix optimization is often carried out via the Burer-Monteiro (BM) formulation, but choosing the factorization rank

r

is delicate and can substantially slow optimization. We propose a unified framework, termed direction-magnitude decomposition (DMD), that decomposes the optimization variable to improve optimization efficiency even when the target rank is unknown. We develop two DMD-based approaches and establish their theoretical advantages on the canonical problem of matrix factorization. The first, overparameterized DMD, uses a rank

r

larger than necessary and enjoys faster convergence as

r

increases. The second, recursive DMD, is motivated by the incremental eigenpair learning, or saddle-to-saddle, behavior of overparameterized DMD. It achieves lower memory and computational costs, complementing overparameterized DMD. Both approaches are exponentially faster than gradient descent applied to the BM formulation. Numerical experiments on matrix factorization, sensing, and completion corroborate our theoretical findings and demonstrate the practical effectiveness of DMD.

Yudong Wei, Liang Zhang, Bingcong Li +1

Low-Rank Factorization Universal Factorization Problem

Ask PDF