Paper Citation Record · LEDGER

Training Deep Morphological Neural Networks as Universal Approximators

As of 19 August 2026, this Paper Citation Record lists 17 of 17 outbound references and 1 inbound Pith citation observation for arXiv:2505.09710.

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

pith.paper-citation-record.v1
2505.09710 v4

Coverage vector

measured 17 of 17 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-15T21:38:03.720582Z

measured 18 of 18 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-19T06:32:44.657259+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-30T13:43:22.153045Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-06-30T13:44:40.608583Z

Reference resolution

17 of 17 outbound references displayed

verified exact0
verified fuzzy13
unresolved1
parse uncertain1
malformed identifier2
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation d25f631f-dc8e-4517-9c24-e126c3ace724 · outbound

This paper cites slack" and continue the path from this argument. We continue this process until either 1) we reach a.

Training Deep Morphological Neural Networks as Universal Approximators slack" and continue the path from this argument. We continue this process until either 1) we reach a

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:04.061371Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 04f2673f-1981-4380-b0a1-9c4ff0dca53b · outbound

This paper cites Consider 3 samples (−1.7,1; 2.3),(5,−2.2; 3.7),(1,1; 4.7).

Training Deep Morphological Neural Networks as Universal Approximators Consider 3 samples (−1.7,1; 2.3),(5,−2.2; 3.7),(1,1; 4.7)

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:04.043964Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation d3b6dcf8-ff1f-47a8-8589-686318e0cb39 · outbound

This paper cites maximum of Gaussians.

Training Deep Morphological Neural Networks as Universal Approximators maximum of Gaussians

Reference 3

Resolution: malformed identifier; raw_fallback, observed 2026-08-15T21:38:04.026273Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation bacc3577-04cc-4d9d-a733-ae84732b92fa · outbound

This paper cites After extensive trial and error, we found that initializing the weights with a mean of−5/3 and a standard deviation of 3 yielded the best results.

Training Deep Morphological Neural Networks as Universal Approximators After extensive trial and error, we found that initializing the weights with a mean of−5/3 and a standard deviation of 3 yielded the best results

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:04.009166Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation e4c1195c-f0b0-4bf0-aa3f-1ff34a6e51b4 · outbound

This paper cites As a result, we used the same initialization as for DEP networks withλ= 1/2.

Training Deep Morphological Neural Networks as Universal Approximators As a result, we used the same initialization as for DEP networks withλ= 1/2

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.991108Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 7dd05e59-49a3-43f5-97dd-9ed6f7f9f86b · outbound

This paper cites Given this, the ideal weight initialization follows the same approach as DEP networks withλ= 1/2.

Training Deep Morphological Neural Networks as Universal Approximators Given this, the ideal weight initialization follows the same approach as DEP networks withλ= 1/2

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.974007Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation d6fb82cf-9da1-4869-9dd0-cb09c7714da8 · outbound

This paper cites finalMNNs*.ipynb.

Training Deep Morphological Neural Networks as Universal Approximators finalMNNs*.ipynb

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.956406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation c764cc9b-ac1b-438e-b029-5ef7b27d9ba5 · outbound

This paper cites an unresolved cited work.

Training Deep Morphological Neural Networks as Universal Approximators Unresolved cited work

Reference 9

Resolution: parse uncertain; raw_fallback, observed 2026-08-15T21:38:03.920447Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation a03c15e6-8695-4069-8539-60e6d6af57b8 · outbound

This paper cites Increasing.

Training Deep Morphological Neural Networks as Universal Approximators Increasing

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.900920Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation d74d1185-6165-49e4-9cac-df9b23675b1a · outbound

This paper cites This is a condition for the proof to work.

Training Deep Morphological Neural Networks as Universal Approximators This is a condition for the proof to work

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.882257Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 21830970-9061-4515-800c-1cc51a8d4f4f · outbound

This paper cites the weights of the supremum is not inside the previous infimum like they are in the idenities obtained from the Representation Theorem.

Training Deep Morphological Neural Networks as Universal Approximators the weights of the supremum is not inside the previous infimum like they are in the idenities obtained from the Representation Theorem

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.862675Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 76f6d580-fdcd-4dd9-9311-fcb85f0fda4f · outbound

This paper cites linear" activations. We perform regression of simple single-variate single-output functions sampled with zero-mean i.i.d. gaussian noise. To ablate the effect of our.

Training Deep Morphological Neural Networks as Universal Approximators linear" activations. We perform regression of simple single-variate single-output functions sampled with zero-mean i.i.d. gaussian noise. To ablate the effect of our

Reference 13

Resolution: malformed identifier; raw_fallback, observed 2026-08-15T21:38:03.939507Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation e427c7b2-7f27-4fb3-bdf7-b4bc370c5710 · outbound

This paper cites First, we see how having suprema and infima over finite domains is implicitly used in the proof.

Training Deep Morphological Neural Networks as Universal Approximators First, we see how having suprema and infima over finite domains is implicitly used in the proof

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.841249Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 95b2f106-44a5-4f80-af6a-c1112ff789c8 · outbound

This paper cites an unresolved cited work.

Training Deep Morphological Neural Networks as Universal Approximators Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-15T21:38:03.822060Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation a954cc10-8860-4d33-8b86-40bfd4f6fd54 · outbound

This paper cites Let us see the above in more detail.

Training Deep Morphological Neural Networks as Universal Approximators Let us see the above in more detail

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.801406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation c3f9a9dd-d199-4c80-8e19-bd4795280f06 · outbound

This paper cites Sinceyn is Lipschitz, so isgn.

Training Deep Morphological Neural Networks as Universal Approximators Sinceyn is Lipschitz, so isgn

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.783033Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 3960052c-c4ca-4ccc-96c5-f9d1765067f5 · outbound

This paper cites 35 Definegn(t) =y n(x+t1)andg(t) =y(x+t1).

Training Deep Morphological Neural Networks as Universal Approximators 35 Definegn(t) =y n(x+t1)andg(t) =y(x+t1)

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T21:38:03.763263Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Pith citing papers

Observation a3bb27e2-4177-437f-bb0b-edee05d6b438 · inbound

Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology cites this paper.

Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology Training Deep Morphological Neural Networks as Universal Approximators

Reference 16

Resolution: verified exact; arxiv_id, observed 2026-08-04T02:29:27.318617Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.