FM-FINER: Nyquist-Informed INR

• FM-FINER is an implicit neural representation that uses Nyquist-informed, neuron-specific frequency multipliers in chirp-based activations.
• It reduces hidden feature redundancy by replacing a single global multiplier with deterministic per-neuron scaling, achieving a more distinct spectral allocation.
• The method maintains FINER’s architecture with no extra parameters while improving performance in audio, image, 3D shape, and NeRF tasks.

FM-FINER is a variant of the FINER implicit neural representation (INR) that replaces FINER’s single, global frequency multiplier with Nyquist-informed, neuron-specific multipliers in periodic activations. It was introduced in the context of periodic activation-based multilayer perceptrons for continuous signal representation, where hidden feature redundancy limits expressive capacity even when sine-based or chirp-like activations are used. The method retains FINER’s chirp-style activation, changes only the frequency construction inside each neuron, introduces no new learnable parameters, and is evaluated on 1D audio, 2D image, 3D shape, and neural radiance field tasks (Alsakabi et al., 27 Sep 2025).

1. Definition and relation to periodic-activation INRs

FM-FINER belongs to the class of implicit neural representations that model a signal as a continuous function $f_\theta : \mathbf{x} \mapsto \mathbf{y}$ implemented by an MLP. In this setting, a coordinate $\mathbf{x}$ is mapped directly to a signal value such as audio amplitude, image RGB, occupancy, or radiance. The relevant baseline family is periodic-activation INR, where high-frequency structure is represented through sinusoidal nonlinearities rather than ReLU or $\tanh$.

The point of departure is the contrast between SIREN and FINER. SIREN uses a fixed frequency multiplier $\omega_0$ in activations of the form

$\sigma(x) = \sin(\omega_0 x),$

with layer activations

$$\mathbf{h} = \sin(\omega_0 W \mathbf{x} + \mathbf{b}).$$

FINER modifies this with a frequency-variable sinusoid,

$\sigma(x) = \sin\big(\omega_0 (|x| + 1) x\big),$

which behaves like a chirp because the local effective frequency depends on the pre-activation $x$ (Alsakabi et al., 27 Sep 2025).

FM-FINER preserves FINER’s chirp-like functional form but removes the assumption that all neurons in a layer should share the same frequency scale. Instead, each neuron receives its own fixed multiplier derived from the Nyquist frequency and the neuron index. The result is a deterministic frequency bank inside each hidden layer rather than a layer-wide shared multiplier.

2. Motivation: hidden feature redundancy and the Nyquist view

The central motivation is hidden feature redundancy. In SIREN and FINER, every neuron in a layer uses the same global multiplier $\omega_0$, so differences between neurons arise only through learned weights and biases. The paper argues that this causes many neurons to capture overlapping parts of the spectrum, producing strongly correlated hidden features rather than a diverse set of frequency responses (Alsakabi et al., 27 Sep 2025).

Redundancy is quantified by collecting each neuron’s output over 10,000 randomly sampled coordinates, forming a feature vector per neuron, computing the covariance matrix of these neuron-wise feature vectors, and then taking the Frobenius norm of that covariance matrix. On the image reconstruction network used in the paper’s first figure, FM-FINER reduces this covariance norm by 50.43% relative to FINER, while FM-SIREN reduces it by 49.92% relative to SIREN (Alsakabi et al., 27 Sep 2025).

The theoretical analogy is to classical transforms such as the Discrete Sine Transform. In that view, a signal approximation is written as

$f^*(x) = \sum_{m=0}^{M-1} c_m \phi_m(x),$

where each $\mathbf{x}$0 corresponds to a distinct frequency. The paper’s claim is not that FM-FINER creates an orthogonal basis in the strict DST sense, but that assigning structured, neuron-specific frequencies encourages a layer to behave more like a non-redundant frequency decomposition. This suggests that the principal benefit is not merely “more frequency,” but more even spectral allocation across neurons.

The Nyquist connection is explicit. Given sampling frequency $\mathbf{x}$1, the Nyquist limit is

$\mathbf{x}$2

FM-FINER uses this quantity to set the maximum usable frequency scale, so the hidden layer is organized according to the signal’s representable band rather than an arbitrary global multiplier (Alsakabi et al., 27 Sep 2025).

3. Mathematical formulation and implementation

For a layer with $\mathbf{x}$3 neurons, FM-FINER defines per-neuron basis frequencies as

$\mathbf{x}$4

The method then modifies FINER’s activation by replacing the global $\mathbf{x}$5 with a neuron-specific multiplier and an additional scaling factor: $\mathbf{x}$6 where

$\mathbf{x}$7

The paper states that the factor $\mathbf{x}$8 is empirically chosen via ablation to reduce aliasing, because FINER’s chirp activation already spans a broader effective band and full Nyquist frequencies would push it beyond the safe range (Alsakabi et al., 27 Sep 2025).

In practical implementation, the paper keeps FINER’s recommended $\mathbf{x}$9 values as scales and treats the effective per-neuron frequency multiplier as

$\tanh$0

with NyquistFactor fixed at $\tanh$1 for FM-FINER. This means that architecture, initialization philosophy, and training procedure remain aligned with FINER; only the multiplier schedule changes.

The reported architectures are unchanged from the baseline FINER settings. For audio and images, the paper uses a 2-layer MLP with 256 neurons per hidden layer. For 3D shapes, it uses a 3-layer MLP with 256 neurons per layer. For NeRF, it uses a 5-layer network with 3 layers for a shared trunk, 1 for density, and 1 for color. Weight and bias initialization follow FINER’s recommended settings, no new learnable parameters are introduced, and per-epoch training cost is described as essentially unchanged relative to FINER (Alsakabi et al., 27 Sep 2025).

Nyquist frequencies are instantiated from the domain resolution. The paper gives the following examples: $\tanh$2 for audio, so $\tanh$3; for Kodak images at $\tanh$4, Nyquist is treated as 256 cycles/image; for BSDS500, approximately 160.5 cycles/image; for voxelized $\tanh$5 3D shapes, 256 cycles/volume; and for NeRF volumes at $\tanh$6, 50 cycles/volume (Alsakabi et al., 27 Sep 2025).

4. Empirical performance across tasks

The method is evaluated on four INR tasks using the same network depth, width, and training schedule as FINER. Audio fitting uses 1,313 clips from the Spoken English Wikipedia dataset, downsampled to $\tanh$7, fitting the first 10 seconds or 40,000 samples. Image representation uses Kodak, BSDS500, and the Philips Circle Pattern. Shape representation uses Thai Statue, Armadillo, Dragon, and Asian Dragon from the Stanford 3D Scanning Repository, voxelized on a $\tanh$8 grid. NeRF synthesis uses eight Blender scenes and a $\tanh$9 volume discretization for Nyquist estimation (Alsakabi et al., 27 Sep 2025).

The headline quantitative results are summarized below.

Task	FM-FINER result	Baseline comparison
Audio MSE	$\omega_0$0	FINER: $\omega_0$1
Kodak PSNR	32.48 dB	FINER: 28.24 dB
BSDS500 PSNR	37.47 dB	FINER: 31.47 dB
3D IoU (Thai / Armadillo / Dragon / Asian Dragon)	0.991 / 0.997 / 0.996 / 0.991	FINER: 0.976 / 0.991 / 0.994 / 0.978
NeRF examples	Lego 28.11 dB, Chair 34.56 dB, Hotdog 37.41 dB	Typically wins or ranks second

In audio, FM-FINER achieves the best average MSE among the reported models, improving over FINER from $\omega_0$2 to $\omega_0$3. The paper also states that train time per epoch is very similar across methods and that FM-FINER’s cost is identical to FINER’s in this setting (Alsakabi et al., 27 Sep 2025).

In images, FM-FINER is best on Kodak and second-best on BSDS500 under the reported averages. For Kodak, MSE is 0.631 in units of $\omega_0$4, PSNR is 32.48 dB, and SSIM is 0.868. For BSDS500, the reported values are 0.201 in MSE $\omega_0$5, 37.47 dB PSNR, and 0.958 SSIM. On the Philips Circle Pattern, the paper states that FM-SIREN and FM-FINER beat the best baseline PSNR by approximately 8.5–9 dB using only 2 layers (Alsakabi et al., 27 Sep 2025).

In 3D shape representation, FM-FINER yields the best overall IoU across all four shapes: 0.991, 0.997, 0.996, and 0.991. The paper also reports identical epoch time and parameter count between FINER and FM-FINER in this experiment: 2.30 s/epoch and 132,865 parameters (Alsakabi et al., 27 Sep 2025).

In NeRF, gains are more heterogeneous but still favorable overall. FM-FINER is best on Lego at 28.11 dB, Chair at 34.56 dB, and Hotdog at 37.41 dB. The paper also notes an exception on Mic, where FM-FINER is lower at 29.81 dB. Compute efficiency improves in this setting: epoch time decreases from 106.84 s for FINER to 88.61 s for FM-FINER, a 17.06% speedup (Alsakabi et al., 27 Sep 2025).

A notable qualitative claim is that in a single-layer image reconstruction setting, FM-SIREN and FM-FINER not only outperform SIREN and FINER but also beat a classical DST baseline in PSNR using the same number of parameters, namely 2048 (Alsakabi et al., 27 Sep 2025).

5. Interpretation, strengths, and failure modes

The paper’s central interpretive claim is that structured frequency diversity reduces overlap among hidden units and thereby increases expressivity without increasing width, depth, or parameter count. The covariance heatmaps support this: SIREN and FINER exhibit widespread high covariance across neuron pairs, whereas FM-SIREN and FM-FINER show much lower covariance, especially outside the lower-frequency block. The remaining correlation in low-frequency regions is explicitly noted as expected, since low-frequency components are smoother and more overlapping (Alsakabi et al., 27 Sep 2025).

Qualitative reconstructions are reported to have sharper edges and less halo than the baselines, with lower-magnitude error maps in images and better preservation of fine geometric details in Thai Statue and Asian Dragon. In NeRF, the reported improvements are subtler, but FM-FINER is described as avoiding some blurring or ghosting seen in WIRE and showing fewer aliasing artifacts on edges. These observations are descriptive rather than theorem-level guarantees, but they are consistent with the paper’s redundancy analysis.

The method also has explicit limitations. Increasing depth beyond 2–3 layers can degrade performance for FM-SIREN and especially FM-FINER. The stated explanation is that higher layers may amplify high-frequency components introduced early, acting like noise; for FINER’s chirp activation plus Nyquist allocation, deeper networks can over-emphasize high frequencies. Width also saturates: beyond roughly 512 neurons, performance gains plateau because the Nyquist band is fixed by the signal’s sampling, so additional neurons eventually overlap frequencies and reintroduce redundancy (Alsakabi et al., 27 Sep 2025).

A further limitation is the fixed $\omega_0$6 Nyquist factor. The paper states that this constant is empirically chosen and “not theoretically optimal for all tasks.” It also assumes that signals are band-limited and that effective Nyquist rates are known or can be approximated from resolution. For highly non-stationary or irregularly sampled signals, the simple linear spacing of frequencies may not be ideal (Alsakabi et al., 27 Sep 2025).

6. Usage, practical guidance, and research directions

FM-FINER is presented as suitable when FINER or SIREN is already being used for audio or time-series INRs, image or video INRs, 3D shape or occupancy fields, and NeRF-like volume rendering. The practical recipe in the paper is straightforward: retain the same architecture and $\omega_0$7 settings as FINER, compute a per-domain Nyquist frequency from sampling or resolution, define linearly spaced neuron frequencies over $\omega_0$8, and multiply them by $\omega_0$9 for FM-FINER to avoid aliasing (Alsakabi et al., 27 Sep 2025).

The most important practical implication is that the method seeks better spectral coverage without extra model complexity. No new learnable parameters are introduced, no additional network depth is required, and parameter count is unchanged. This is why the paper characterizes FM-FINER as a “drop-in replacement” for FINER. A plausible implication is that its main attraction is not absolute novelty in activation design, but a specific reallocation of representational bandwidth inside otherwise standard periodic-activation MLPs.

The future directions stated in the paper include extending the Nyquist-informed design to other activation families beyond sinusoids and chirps, including wavelets, Gabor, and other periodic nonlinearities; exploring dynamic or learned frequency allocation rather than fixed linear spacing; and combining the approach with larger frameworks such as mixture-of-experts and Fourier reparameterization (Alsakabi et al., 27 Sep 2025).

Within the INR literature represented by the paper, FM-FINER therefore occupies a specific position: it is a frequency-allocation scheme for chirp-based periodic activations, motivated by redundancy reduction, implemented with deterministic neuron-wise multipliers, and empirically validated as a parameter-neutral improvement over standard FINER across multiple continuous signal representation tasks (Alsakabi et al., 27 Sep 2025).