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".
- 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.
- 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. - 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: becausea_T(ν)²is 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
and the claim ledger 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.
| # | 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.
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:
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 | 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.
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)
- 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).
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/.
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.
MIT (code only; see LICENSE). Third-party data under their own terms (table above).