https://github.com/appleweiping/inr-aliasing-limits

Learned Nyquist: fundamental aliasing limits of implicit neural representations (IEEE ICASSP SPTM). Achievability + converse, Monte-Carlo-validated, real data both sides.
https://github.com/appleweiping/inr-aliasing-limits

aliasing fourier-features icassp implicit-neural-representation neural-fields reproducible-research sampling-theory signal-processing

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Learned Nyquist: fundamental aliasing limits of implicit neural representations (IEEE ICASSP SPTM). Achievability + converse, Monte-Carlo-validated, real data both sides.

• Host: GitHub
• URL: https://github.com/appleweiping/inr-aliasing-limits
• Owner: appleweiping
• License: mit
• Created: 2026-07-10T18:19:33.000Z (14 days ago)
• Default Branch: main
• Last Pushed: 2026-07-12T19:36:40.000Z (12 days ago)
• Last Synced: 2026-07-12T21:11:12.257Z (12 days ago)
• Topics: aliasing, fourier-features, icassp, implicit-neural-representation, neural-fields, reproducible-research, sampling-theory, signal-processing
• Language: Python
• Size: 13.5 MB
• Stars: 0
• Watchers: 0
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# Silent Aliasing and Certified Sampling Design for Fixed Fourier-Feature Models

Identifiability analysis and a **constructive, certified anti-aliasing sampling design**

for coordinate models built on a **fixed finite set of Fourier features** — the linear

core of Fourier-feature implicit neural representations. Target venue: IEEE ICASSP 2027

(Signal Processing Theory & Methods).

**Scope, stated up front.** Every theorem is about the fixed-feature, linear-coefficient

model (least squares on an exponential dictionary Λ). Trained nonlinear networks

(Fourier-feature MLPs, SIRENs) are an **explicitly labeled empirical extension** — and the

extension **fails for SIRENs**, which we report. Adaptive / learned frequency sets are

**outside** the theory. This is not a theory of "all INRs".

## The three-act contribution

1. **Structured indistinguishability (T1).** A visibility/aliasability calculus shows that

   *arbitrary nonuniform subsets of a rate-Q sampling grid inherit the grid's aliasing

   equivalence classes*: an integer tone with ν ≡ ω (mod Q) is **exactly** indistinguishable

   from the model atom ω — zero training residual, full-energy reconstruction failure, and

   (two-point argument) no estimator can avoid error ≥ |a|/√2. **Irregular-on-a-grid

   sampling does not break exact aliasing** — the paper's central corrective.

2. **Randomization breaks it, at a rate (T3).** Off-grid jitter destroys the fold with

   visibility equal to the jitter characteristic function (Gaussian: `v ≈ 2π|r|Q·σ_t`);

   i.i.d. sampling gives a finite-N, dictionary-referenced concentration bound on

   worst-case aliasability over a finite candidate set.

3. **Certified design (`design.py`).** The anti-aliasing objective is a **Ds-optimal

   (nuisance-parameter) / LCMV null-steering** criterion on sample times — *we credit that

   classical principle and do not claim it as new*. Our additions: the identifiability

   reading (which cross-terms to null, and why grid designs can't — T1), a fast joint

   surrogate that an **ablation shows is necessary** (coherence-only wrecks conditioning;

   condition-only leaves aliasing), and — the genuinely new tool — a **continuum

   aliasability certificate**: because `a_T(ν)²` is a real trig polynomial in ν, a

   Bernstein/Lipschitz bound certifies the worst case over an *entire band* from the

   samples alone, which the finite-candidate bound and random designs cannot.

**Honest novelty position** (full audit in [`docs/novelty-matrix.md`](docs/novelty-matrix.md)

and the claim ledger [`docs/claim-ledger.md`](docs/claim-ledger.md)): T2 (Riesz

bookkeeping) and T4 (noncentral-χ² detection) are standard and live in the supplement; the

defensible, non-derivable pieces are the **T1 grid-inheritance corrective**, the **T3

finite-N dictionary-referenced quantification**, and the **continuum certificate** plus its

identifiability-driven design use. Full proofs: [`paper/supplement.pdf`](paper/supplement.tex).

## Experiments (theory ↔ experiment closed loop)

| # | Script | What it validates |

|---|---|---|

| E1 | `run_synthetic_matrix.py` | T1 grid persistence vs i.i.d. (fixed & adversarial tones), T3 jitter law + concentration bound, error decomposition |

| E2 | `run_diagnostic_roc.py` | detection: coherent fold vs its in-band twin sits at chance (all 5 detectors, CI covers 0.5); visible tones follow the exact noncentral-χ² power curve; calibration/test separated |

| E3 | `run_real_signal.py` | sample-only bandwidth selection vs an evaluation-only oracle; anti-aliased decimation; 9 speech segments; paired CIs |

| **E6** | **`run_aliasguard.py`** | **certified design: ablation (joint objective needed), held-out & misspecified generalization, budget sweep, signal-domain payoff, continuum certificate, 1-D & 2-D** |

| E4 | `run_nonlinear.py` | trained-net extension: 20 seeds × 3 archs, NTK-prediction, **SIREN failure reported** |

| E5 | `run_image2d.py` | 2-D exact fold (linear) + lattice-vs-random masks (trained), all PSNR variants separate |

Headline E6 numbers (from `results/aliasguard.json`, regenerated by the script; the paper

pulls them via `paper/sync_macros.py`, no hand-entered numbers): designed on a focused

concern set and evaluated on **held-out unseen frequencies**, worst-case aliasability drops

to ≈0.19 vs ≈0.63 (random jitter) / ≈0.59 (E-optimal), holds under candidate-set

misspecification, and is **certified ≤ ≈0.29 over an entire continuous band** vs ≈0.57 for

random; the 2-D design gives ≈0.18 vs ≈0.68.

## Reproduction

```bash

uv venv --python 3.11 && source .venv/bin/activate   # Windows: .venv/Scripts/activate

uv pip install -r requirements.lock && uv pip install -e . --no-deps

python -m pytest -q                     # 39 theorem/design tests (CPU)

python experiments/run_all.py           # CPU reproduction (E1-E3, E6, E5 part A)

python experiments/run_all.py --full    # + trained-network studies (torch + GPU)

```

`requirements.lock` pins the CPU stack. Torch experiments were run with torch 2.8.0+cu128

on an RTX 4090 (recorded per-JSON in `_meta`); GPU training uses

`torch.use_deterministic_algorithms(True, warn_only=True)`, but bitwise reproducibility

across CUDA versions is not guaranteed. The CPU pipeline is deterministic (fixed seeds).

Build the paper + supplement:

```bash

cd paper && python sync_macros.py       # regenerate macros_ag.tex from results/aliasguard.json

pdflatex main && bibtex main && pdflatex main && pdflatex main

pdflatex supplement && pdflatex supplement

```

## Data provenance & licenses

| Data | Source | Terms | Processing |

|---|---|---|---|

| Speech (3 recordings) | Open Speech Repository, Harvard sentences | Freely available per OSR (see source page); **not** covered by this repo's MIT license | SHA-256 in `data/speech_provenance.json`; 2-s segments; `resample_poly` anti-aliasing |

| Mauna Loa CO₂ | NOAA GML | Public (NOAA disclaimer) | monthly means, full record, linear detrend |

| Sunspots (daily + smoothed) | SILSO, Royal Observatory of Belgium | CC BY-NC 4.0 | smoothed series is an intentionally low-pass **proxy**, labeled everywhere |

MIT license covers **code only**; third-party data remain under their own terms.

## Repository layout

```

src/inralias/  identifiability.py (T1-T4 objects)  design.py (AliasGuard + certificate)

               sampling.py  limits.py (background)  inr.py  diagnostics.py  signals.py

experiments/   run_synthetic_matrix  run_diagnostic_roc  run_real_signal  run_aliasguard

               run_nonlinear  run_image2d  run_all

tests/         test_identifiability  test_theory_vs_sim  test_sampling  test_diagnostics_inr

               test_design           (39 tests; GitHub Actions on every push)

paper/         main.tex  supplement.tex (full proofs)  sync_macros.py  figures  refs.bib

docs/          novelty-matrix.md  claim-ledger.md  CHANGELOG-major-revision.md

results/       *.json (with _meta provenance) + figures/

submission/    ICASSP 2027 submission package (see submission/README.md)

```

## Honesty notes

* AliasGuard is a Ds-optimal / null-steering design — **classical principle, credited**;

  the new pieces are the identifiability use and the continuum certificate.

* AliasGuard needs a **focused** concern set (K = O(N)); for a broad band with K ≫ N no

  design beats random much, and a badly misspecified set loses the edge — reported.

* The concentration bound is a loose union bound, plotted against the empirics.

* Trained-net extension **fails for SIRENs**; all seed-level failures are released.

* CV bandwidth selection is reported even where it underperforms.

* Detection thresholds come only from `null_calibrated_threshold` (calibration null).

* "Monte-Carlo-validated" = closed forms asserted against independent simulation in

  `tests/`; not a substitute for the proofs (supplement).

## Review process (ICASSP 2027)

ICASSP 2027 uses **single-anonymous** review (reviewers see author names); no manuscript

anonymization is required, so this public repository and the author name on the PDF are

compliant. The submission package is in [`submission/`](submission/).

## AI assistance disclosure

Substantial portions of the code, experiments, and manuscript were produced with AI

assistance (Anthropic Claude) under the author's direction; all results were generated by

the released, reproducible code. AI is not an author. Git history records this via

`Co-Authored-By` trailers and is not rewritten. This is disclosed in the paper's

Compliance-with-Ethical-Standards statement.

## License

MIT (code only; see `LICENSE`). Third-party data under their own terms (table above).