{"paper":{"title":"Stochastic global optimization of continuous functions via random walks on Grassmannians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Random walks on Grassmannians converge to global minima of continuous functions at a rate set by a geometric gap parameter.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Kartik Gupta, Pradeep Ravikumar, Ramarathnam Venkatesan, Stephen D. Miller","submitted_at":"2026-05-13T22:06:50Z","abstract_excerpt":"We introduce a stochastic global optimization method based on random walks on Grassmannian manifolds. To minimize a continuous objective $\\ell:\\mathbb{R}^d\\rightarrow\\mathbb{R}$, the method repeatedly samples random $k$-dimensional linear subspaces (with $k\\ll d$), solves the resulting low-dimensional restrictions of these problems to these subspaces using an arbitrary black-box optimizer, and updates the iterate (which monotonically improves upon the previous iterate). Unlike classical optimization analyses that rely on convexity, smoothness, Lipschitz bounds, or Polyak-Lojasiewicz-type condi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We identify a gap parameter -- an analogue of a spectral gap for random walks -- that controls the rate at which the iterates approach the global minimum value. Finally, we argue that the same analysis yields a blind-spot robustness property: sufficiently narrow, deep dips of the loss function have limited influence on the algorithm's trajectory.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The convergence analysis rests on the existence and positivity of the gap parameter defined from the geometric distribution of restricted minima across random k-dimensional subspaces; the abstract provides no explicit construction, bound, or verification procedure for this quantity on arbitrary continuous functions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A stochastic global optimizer samples random k-dimensional subspaces, solves the restricted problem on each, and moves to the improved point, with rate controlled by a gap parameter on the distribution of restricted minima.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Random walks on Grassmannians converge to global minima of continuous functions at a rate set by a geometric gap parameter.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"840268697d2a67ba14d3f35bd924de5f570f0e60ed7ab68c4bd91567121219dd"},"source":{"id":"2605.14151","kind":"arxiv","version":1},"verdict":{"id":"b06bf1d4-917d-4925-bee9-5e8a14fd18c6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:02:11.639255Z","strongest_claim":"We identify a gap parameter -- an analogue of a spectral gap for random walks -- that controls the rate at which the iterates approach the global minimum value. Finally, we argue that the same analysis yields a blind-spot robustness property: sufficiently narrow, deep dips of the loss function have limited influence on the algorithm's trajectory.","one_line_summary":"A stochastic global optimizer samples random k-dimensional subspaces, solves the restricted problem on each, and moves to the improved point, with rate controlled by a gap parameter on the distribution of restricted minima.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The convergence analysis rests on the existence and positivity of the gap parameter defined from the geometric distribution of restricted minima across random k-dimensional subspaces; the abstract provides no explicit construction, bound, or verification procedure for this quantity on arbitrary continuous functions.","pith_extraction_headline":"Random walks on Grassmannians converge to global minima of continuous functions at a rate set by a geometric gap parameter."},"references":{"count":24,"sample":[{"doi":"10.1080/10586458.2009.10129052","year":2009,"title":"Ballinger, B., Blekherman, G., Cohn, H., Giansiracusa, N., Kelly, E., & Schürmann, A. (2009). Experimental Study of Energy-Minimizing Point Configurations on Spheres. Experimental Mathematics, 18(3), ","work_id":"04ffccf2-3cc9-4716-b2db-cde09d87eecb","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Point configurations minimizing harmonic energy on spheres","work_id":"f4f62a8c-cd8c-424a-a138-ecc1dfc31033","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"B. Bekka, P. de la Harpe, and A. Valette. Kazhdan's Property (T) . New Mathematical Monographs. Cambridge University Press, 2008","work_id":"4a0d1141-a5c6-4ee6-aa99-ffc0e1775f43","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Thomas Bendokat, Ralf Zimmermann, and P.-A. Absil. A Grassmann manifold handbook: basic geometry and computational aspects. Advances in Computational Mathematics , 50(1), January 2024","work_id":"d2329719-4b7a-414b-be59-b21cef6b14f8","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Moser, Alina Oprea, Battista Biggio, Marcello Pelillo, and Fabio Roli","work_id":"ed33d0c1-4453-4efb-8be7-be6a2f23906c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"2c15d2e59f863b31a95f5c3095109ed658d612b396985e58d799f8bf9d39092a","internal_anchors":1},"formal_canon":{"evidence_count":1,"snapshot_sha256":"9a99ff3b68afa39a4c063c4452c9565be8bf309044da66dd2cd202f3b4f987ef"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}