Pith Number

pith:ZISAT6AI

pith:2025:ZISAT6AID6BXC5V5LBOOEQ3WWR

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Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings

Daniel Levy, Efstratios Tsoukanis, Matthias Wellershoff, Nadav Dym, Radu Balan

Sorting-based embeddings of point sets have bi-Lipschitz distortion quadratic in the number of points and independent of dimension.

arxiv:2510.22186 v2 · 2025-10-25 · cs.LG · cs.IT · math.FA · math.IT · math.MG

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\usepackage{pith}
\pithnumber{ZISAT6AID6BXC5V5LBOOEQ3WWR}

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We construct matrices of projection vectors so that the bi-Lipschitz distortion of the mapping depends quadratically on the number of points n, and is completely independent of the dimension d. We also show that for any choice of projection vectors, the distortion of the mapping will never be better than a bound proportional to the square root of n.

C2weakest assumption

The analysis assumes projections are in general position (as stated in prior work) and that point sets lie in Euclidean space where generic linear projections produce distinct values with probability 1; the bi-Lipschitz claims further rely on the specific matrix constructions provided in the paper.

C3one line summary

Improved bounds on injectivity dimension D for sorting-based permutation-invariant embeddings of n-point sets, with bi-Lipschitz distortion quadratic in n and independent of ambient dimension d, plus a matching sqrt(n) lower bound on distortion for any choice of projections.

Receipt and verification

First computed	2026-05-26T02:03:58.613415Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ca2409f8081f837176bd585ce24376b4489d42402d1b0c61af73b1075c576bfd

Aliases

arxiv: 2510.22186 · arxiv_version: 2510.22186v2 · doi: 10.48550/arxiv.2510.22186 · pith_short_12: ZISAT6AID6BX · pith_short_16: ZISAT6AID6BXC5V5 · pith_short_8: ZISAT6AI

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZISAT6AID6BXC5V5LBOOEQ3WWR \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: ca2409f8081f837176bd585ce24376b4489d42402d1b0c61af73b1075c576bfd

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "30c685dbdfbf666dccbebdd19b7f969e609201b73830079d37d7fa6d6d566b32",
    "cross_cats_sorted": [
      "cs.IT",
      "math.FA",
      "math.IT",
      "math.MG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2025-10-25T06:44:08Z",
    "title_canon_sha256": "400f1d2225a76cf93b24990ab698cc083bd7ff88d9bc68142aeeaa121d9f2bdc"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2510.22186",
    "kind": "arxiv",
    "version": 2
  }
}