Paper Citation Record · LEDGER

Understanding Mode Connectivity via Parameter Space Symmetry

As of 22 August 2026, this Paper Citation Record lists 21 of 21 outbound references and 0 inbound Pith citation observations for arXiv:2505.23681.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2505.23681 v1

Coverage vector

measured 21 of 21 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T12:51:54.083693Z

measured 21 of 21 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-22T06:32:14.747728+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

21 of 21 outbound references displayed

verified exact0
verified fuzzy14
unresolved7
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation ddd0e29f-da12-48a8-9f94-309827f7f50d · outbound

This paper cites an unresolved cited work.

Understanding Mode Connectivity via Parameter Space Symmetry Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:51:57.832234Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation b4ad161d-d249-4a1b-9bf0-8c6be15c86d9 · outbound

This paper cites an unresolved cited work.

Understanding Mode Connectivity via Parameter Space Symmetry Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:51:57.611267Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 6b4e625c-02de-4858-b5c1-a137e834ceef · outbound

This paper cites For any g ∈ GL(h) such that det(g) < 0, g · (W1, W2) and (W ′ 1, W′.

Understanding Mode Connectivity via Parameter Space Symmetry For any g ∈ GL(h) such that det(g) < 0, g · (W1, W2) and (W ′ 1, W′

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:57.376976Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 571a76dd-6500-4c6c-ab24-70028680e546 · outbound

This paper cites an unresolved cited work.

Understanding Mode Connectivity via Parameter Space Symmetry Unresolved cited work

Reference 6

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:51:57.176093Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 271d8241-fbdf-48e6-80be-3aa60b242d3c · outbound

This paper cites Equivalently, det(gg ′) < 0.

Understanding Mode Connectivity via Parameter Space Symmetry Equivalently, det(gg ′) < 0

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:56.992325Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 86f4cde0-1e88-4edd-a49c-c12637dd56e9 · outbound

This paper cites an unresolved cited work.

Understanding Mode Connectivity via Parameter Space Symmetry Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:51:56.780838Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation b7bc7e22-193d-4ce9-9a56-f9b9499592c5 · outbound

This paper cites By Lemma 5.1, any g ∈ GL(h) with det(g) < 0 can bring (W1, W2) and (W ′ 1, W′.

Understanding Mode Connectivity via Parameter Space Symmetry By Lemma 5.1, any g ∈ GL(h) with det(g) < 0 can bring (W1, W2) and (W ′ 1, W′

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:56.600100Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 8ccbf496-c9f8-4e61-806b-90e98512179c · outbound

This paper cites Let g be the permutation matrix 0 1 1 0.

Understanding Mode Connectivity via Parameter Space Symmetry Let g be the permutation matrix 0 1 1 0

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:56.450488Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 672719e0-5f78-41ce-bce8-1ff1a294dc52 · outbound

This paper cites Then W, W′ belong to the same connected component of L−1(0), connected by curve γ : R → Param, γ(t) = ((1− t)Wl + tWlm−k, (1 − t)Wl−1 + tmWl−1, Wl−2, ..., W1).

Understanding Mode Connectivity via Parameter Space Symmetry Then W, W′ belong to the same connected component of L−1(0), connected by curve γ : R → Param, γ(t) = ((1− t)Wl + tWlm−k, (1 − t)Wl−1 + tmWl−1, Wl−2, ..., W1)

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:56.274995Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation efa16d2f-00af-4ef3-9659-dd16c885d4ad · outbound

This paper cites Then L ((1 − α)W + αW ′) = 1 − 1 2 + 1 2 m−k 1 2 + 1 2 m k!2 ||Y ||2 2 = 1 − 2−(k+1)(1 +m−k)(1 +m)k 2 ||Y ||2 2 (15) Let m = 2k+1 √ b ||Y ||2 + 1 − 1 k.

Understanding Mode Connectivity via Parameter Space Symmetry Then L ((1 − α)W + αW ′) = 1 − 1 2 + 1 2 m−k 1 2 + 1 2 m k!2 ||Y ||2 2 = 1 − 2−(k+1)(1 +m−k)(1 +m)k 2 ||Y ||2 2 (15) Let m = 2k+1 √ b ||Y ||2 + 1 − 1 k

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:56.144745Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 74d28a4a-e160-435a-91ed-b699e11e4112 · outbound

This paper cites At large m, the two minima are farther apart, and the loss evaluated at the middle point of their linear interpolation grows unboundedly as predicted by Proposition 5.3.

Understanding Mode Connectivity via Parameter Space Symmetry At large m, the two minima are farther apart, and the loss evaluated at the middle point of their linear interpolation grows unboundedly as predicted by Proposition 5.3

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:55.958870Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation c5a2e286-9545-439e-9cce-17542dc965b1 · outbound

This paper cites Since W ∈ L−1(0), we have Wlσ [Wl−1f (Wl−2, ..., W1, X)] =Y.

Understanding Mode Connectivity via Parameter Space Symmetry Since W ∈ L−1(0), we have Wlσ [Wl−1f (Wl−2, ..., W1, X)] =Y

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:55.792841Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 9cdf196e-4244-4fa4-9e0a-eda162c8a8b1 · outbound

This paper cites Then L ((1 − α)W + αW ′) =||Y − 1 4 Wl(I + m−kP )σ (I + mP −1)Wl−1f (Wl−2, ..., W1, X) ||2.

Understanding Mode Connectivity via Parameter Space Symmetry Then L ((1 − α)W + αW ′) =||Y − 1 4 Wl(I + m−kP )σ (I + mP −1)Wl−1f (Wl−2, ..., W1, X) ||2

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:55.572217Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 3eef6d25-a3e3-46c5-bc5f-4b04aedeab76 · outbound

This paper cites an unresolved cited work.

Understanding Mode Connectivity via Parameter Space Symmetry Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:51:55.383579Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 1e1afdde-d229-4518-aa53-139b77a07bcc · outbound

This paper cites Therefore, L ((1 − α)W + αW ′) is unbounded for any P.

Understanding Mode Connectivity via Parameter Space Symmetry Therefore, L ((1 − α)W + αW ′) is unbounded for any P

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:55.198525Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 66c91ef4-c6a7-4df4-b05c-176515dacd0b · outbound

This paper cites an unresolved cited work.

Understanding Mode Connectivity via Parameter Space Symmetry Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:51:54.995591Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 74de1ee3-7aff-4d80-8127-ba0509a6a1bd · outbound

This paper cites Let α = 1.

Understanding Mode Connectivity via Parameter Space Symmetry Let α = 1

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:54.820965Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 0582a42b-45bf-4300-866b-274e26447dd9 · outbound

This paper cites (23) As β → ∞, g and g−1 cannot approach β+β−1 2 I simultaneously.

Understanding Mode Connectivity via Parameter Space Symmetry (23) As β → ∞, g and g−1 cannot approach β+β−1 2 I simultaneously

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:54.589548Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 681d458a-a0f9-4f43-83c2-9ac5e34b260d · outbound

This paper cites The connectedness results derived from symmetry raise several interesting questions about mode connectivity.

Understanding Mode Connectivity via Parameter Space Symmetry The connectedness results derived from symmetry raise several interesting questions about mode connectivity

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:54.334587Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation e1c06f88-fb73-434e-9d12-b1dbd40f7808 · outbound

This paper cites Freeman, C.

Understanding Mode Connectivity via Parameter Space Symmetry Freeman, C

Reference 3269

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:51:58.011853Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation fa0d2c83-da96-4350-8ef7-d9971ddf499d · outbound

This paper cites A Note on Connectivity of Sublevel Sets in Deep Learning.

Understanding Mode Connectivity via Parameter Space Symmetry A Note on Connectivity of Sublevel Sets in Deep Learning

Reference 4799

Resolution: unresolved; no resolver link, observed 2026-08-07T12:51:52.006681Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.