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\title{Spectral Neural Network Compression via Discrete Fourier Transform: A Post Hoc and Lightweight Approach}


\author[1]{Sghaier Samia\thanks{Auteur correspondant: \texttt{samia.sghaier21@gmail.com}}}
\author[2]{Nfata Houda}
\affil[1]{Université de Tunis El Manar, Tunis, Tunisie}
\affil[2]{Université de Tunis El Manar, Tunis, Tunisie}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
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\keywords{weight compression, discrete Fourier Transform, frequency-domain filtering, complex-valued neural networks, deep learning.}

\thanks{}
\begin{abstract}
We introduce a spectral post hoc compression method for neural networks based on Discrete Fourier Transform (DFT) of complex weights. The approach filters low-magnitude frequencies to obtain sparse spectral representations while preserving accuracy. Theoretical results quantify energy preservation and output perturbation. We propose a principled thresholding rule, and demonstrate competitive performance compared to DCT and wavelets. Experiments on MNIST, CIFAR-10 and ResNet show 10–15× compression with negligible loss. Hardware metrics confirm reduced memory usage and improved inference latency. The method is lightweight, requires no retraining, and suits embedded AI.
\end{abstract}

\section{Introduction}

Deep neural networks have revolutionized numerous fields but often come with significant storage and computational costs. These challenges limit deployment on resource-constrained devices such as smartphones, embedded systems, and IoT hardware. Consequently, model compression techniques like pruning, quantization, and low-rank approximations have become essential tools to reduce model size and accelerate inference \cite{han2016deep, ullrich2017soft, denton2014exploiting}.

In this context, we ask a simple yet impactful question: \textit{What if we could compress neural networks like we compress signals—by removing unnecessary frequencies?}

In this work, we introduce a frequency-domain compression method leveraging the Discrete Fourier Transform (DFT). We empirically observe that early layers in trained models often exhibit spectral sparsity, with most energy concentrated in low-frequency components—a property that tends to diminish in deeper or highly specialized layers.

We provide a theoretical analysis of information loss related to frequency truncation and interpret the method as an implicit spectral regularizer. Empirical validation on standard benchmarks (MNIST, CIFAR-10) and more complex datasets (ImageNet) demonstrates the method’s robustness and superior compression–accuracy tradeoff compared to baseline pruning techniques.

This work seeks to revisit classical signal processing techniques, such as frequency-domain filtering, within the modern context of neural network compression, highlighting their untapped potential and practical relevance.

Our contributions include:
\begin{itemize}
    \item A simple and non-intrusive DFT-based compression method applicable post hoc to pretrained models.
    \item Theoretical insights connecting frequency-domain filtering to implicit regularization.
    \item Comprehensive experimental evaluation across datasets and architectures, highlighting practical deployment benefits.
    \item Introduction of an offline-learned spectral mask to adaptively filter frequencies based on validation feedback, improving compression–accuracy tradeoffs without retraining.
\end{itemize}


Unlike previous DFT-based methods that typically integrate spectral filtering during training or require architecture-specific modifications \cite{wang2021spectral, li2022frequency}, our approach is designed to be fully post hoc and model-agnostic. It applies spectral filtering directly to pretrained weights without retraining or altering the original network structure. Moreover, the introduction of offline-learned spectral masks allows us to balance compression and accuracy adaptively, without incurring training overhead, which distinguishes our method from earlier static filtering strategies.


While our method is algorithmically simple and efficient, it is also grounded in solid mathematical principles. In this work, we not only demonstrate the empirical effectiveness of frequency-domain weight truncation, but also provide formal guarantees. More precisely, we prove that:

\begin{itemize}
    \item the relative energy loss due to truncation can be explicitly quantified (Theorem~\ref{sec:theory}),
    \item the compression operator acts as an orthogonal projection in the frequency domain,
    \item and the output perturbation of a neuron under truncated weights is provably bounded in terms of the truncation error and the input norm.
\end{itemize}

These theoretical results explain the robustness of the proposed technique and offer a principled interpretation of its effect as a form of implicit regularization.


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Related Work}

Model compression for neural networks remains a critical research area due to the increasing demand for deploying deep learning models on resource-constrained devices. Classical strategies include pruning \cite{han2016deep}, quantization \cite{ullrich2017soft}, low-rank approximations \cite{denton2014exploiting}, and knowledge distillation \cite{hinton2015distilling}.

Pruning removes redundant weights, resulting in sparse models. However, these often require specialized hardware or libraries to benefit from sparsity during inference. Quantization reduces numerical precision to decrease model size and accelerate computation. Low-rank factorization decomposes weight matrices to exploit redundancy, effectively compressing models while maintaining accuracy.

Recently, frequency-domain approaches have gained attention as promising tools for compression. For example, \cite{yang2019transform} and \cite{shi2020compressed} explore transform coding techniques inspired by signal processing, applying discrete cosine and wavelet transforms for weight and activation compression. These methods, however, often require modifications to the training process or impose architectural constraints to be effective.

Several studies have focused specifically on the Discrete Fourier Transform (DFT) for compression. \cite{wang2021spectral} demonstrates that weight vectors in trained models exhibit spectral sparsity, enabling high-frequency truncation to act as an implicit regularizer. Similarly, \cite{li2022frequency} applies Fourier filtering to reduce parameter counts in convolutional layers with minimal loss.

Despite their promise, DFT-based methods also face limitations. First, the global nature of the Fourier basis makes it less suited to representing localized patterns, such as edge filters in CNNs, potentially leading to suboptimal reconstruction. Second, the application of DFT assumes a degree of weight smoothness that may not hold in all architectures or training regimes. Third, most prior work applies DFT compression during training or integrates it into custom layers, which can hinder plug-and-play usage and complicate deployment.

To contextualize our method, we compare it numerically to standard approaches. For instance, on the MNIST dataset, magnitude pruning at 80\% sparsity typically reduces accuracy to 95.2\% \cite{han2016deep}, while 8-bit quantization achieves around 97.8\% \cite{jacob2018quantization}. In contrast, our DFT-based method retains 97.0\% accuracy at the same compression level, without retraining or quantization overhead. On CIFAR-10 using ResNet-18, we observe less than 3\% degradation in accuracy under 70\% compression, outperforming unstructured pruning which tends to degrade more sharply beyond 60\%.

Other works explore adaptive frequency-domain filters or learnable spectral masks jointly optimized during training \cite{zhang2023learnable}, allowing compression–accuracy tradeoffs to be tuned dynamically. However, such strategies often require complex training regimes or architectural changes.

In summary, while frequency-based compression has shown substantial potential, many prior methods require architectural modifications or integration into the training pipeline. Our approach distinguishes itself by being post hoc, non-intrusive, and model-agnostic, providing a lightweight yet effective solution for compressing pretrained neural networks.

Most existing frequency-based approaches rely on localized transforms such as the Discrete Cosine Transform (DCT) or wavelets. In contrast, our work revisits the classical Fourier basis due to its analytical simplicity and wide software support. This choice enables efficient implementation but may miss localized patterns. Future work may explore hybrid strategies combining the DFT with wavelets or learned bases to capture both global and local structures.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methodology}

\subsection{Discrete Fourier Transform of Weight Vectors}

Let \( w \in \mathbb{R}^n \) be a weight vector from a fully connected or convolutional layer. Its Discrete Fourier Transform (DFT) is defined as:
\[
\hat{w}_k = \sum_{j=0}^{n-1} w_j \, e^{-i 2 \pi \frac{jk}{n}}, \quad \text{for } k = 0, \dots, n-1,
\]
where \( \hat{w} \in \mathbb{C}^n \) is the spectral representation of \( w \). The inverse transform reconstructs the original vector:
\[
w_j = \frac{1}{n} \sum_{k=0}^{n-1} \hat{w}_k \, e^{i 2 \pi \frac{jk}{n}}.
\]
Note: The use of the DFT implicitly assumes periodic boundary conditions. While this assumption may not strictly hold for weight vectors, its practical impact is mitigated by the observed smoothness in early layers.

\subsection{Frequency Filtering and Compression}

We apply a frequency filter \( F: \mathbb{C}^n \to \mathbb{C}^n \) to zero out selected frequency coefficients:
\[
\hat{w}'_k = 
\begin{cases}
\hat{w}_k, & \text{if } k \in \mathcal{S} \subset \{0, \dots, n-1\} \\
0, & \text{otherwise}
\end{cases}
\]
for some index set \( \mathcal{S} \) of retained frequencies (\( |\mathcal{S}| = k \ll n \)). The compressed vector is reconstructed via inverse DFT:
\[
w' = \text{IDFT}(\hat{w}') = \left( w_0', \dots, w_{n-1}' \right).
\]

\subsection{Filter Design Strategies: Offline Optimization of Spectral Masks}

We evaluate three filtering strategies:
\begin{itemize}
    \item \textbf{Low-pass filtering:} Retain the first \( k \) frequencies in order.
    \item \textbf{Magnitude thresholding:} Retain coefficients with largest \( |\hat{w}_k| \).
    \item \textbf{Learned spectral masks:} Define a learnable vector \( m \in [0,1]^n \) and optimize it (e.g., via gradient descent) to select informative frequencies. The compressed spectrum becomes \( \hat{w}' = m \odot \hat{w} \). In our post hoc setting, we also explore offline optimization of \( m \) using a validation set to balance compression and accuracy.
\end{itemize}

As shown in Table~\ref{tab:mask_accuracy}, learned masks provide a superior tradeoff between compression and accuracy compared to fixed filters or simple thresholding.

\begin{table}[H]
\centering
\begin{tabular}{lcc}
\toprule
\textbf{Mask Type} & \textbf{Compression Rate} & \textbf{Test Accuracy (CIFAR-10)} \\
\midrule
Low-pass (fixed) & 70\% & 86.2\% \\
Thresholding & 70\% & 87.1\% \\
\textbf{Learned Mask (offline)} & \textbf{70\%} & \textbf{88.3\%} \\
\bottomrule
\end{tabular}
\caption{Comparison of different spectral masks at 70\% compression on CIFAR-10 using ResNet-18.}
\label{tab:mask_accuracy}
\end{table}

\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
    width=0.8\linewidth,
    xlabel={Frequency Index},
    ylabel={Mask Value},
    ymin=0, ymax=1.1,
    ytick={0,0.5,1.0},
    xtick={0,10,...,90},
    grid=both,
    title={Learned Spectral Mask (ResNet-18)}
]
\addplot[blue, thick] coordinates {
    (0,1)(1,1)(2,0.95)(3,0.9)(4,0.88)(5,0.85)(6,0.6)(7,0.5)
    (8,0.3)(9,0.2)(10,0.1)(11,0.05)(12,0.02)(13,0)(14,0)(15,0)
};
\end{axis}
\end{tikzpicture}
\caption{Example of an offline-learned spectral mask showing decreasing importance of high-frequency coefficients.}
\label{fig:learned_mask}
\end{figure}

\subsection{Offline Optimization of Learnable Spectral Masks}\label{sec:mask_opt}

To introduce a degree of adaptivity without retraining the full model, we propose an offline optimization of the spectral mask \( m \in [0,1]^n \), performed independently for each layer. This vector selectively scales the frequency components \( \hat{w}_k \) of each weight vector. The filtered spectrum is computed as:
\[
\hat{w}' = m \odot \hat{w},
\]
where \( \odot \) denotes element-wise multiplication.

The goal is to optimize \( m \) such that the reconstructed weights \( w' = \text{IDFT}(\hat{w}') \) minimize task-specific loss (e.g., validation cross-entropy) while promoting sparsity in \( m \). This leads to the following optimization problem:
\[
\min_{m \in [0,1]^n} \mathcal{L}(f(w'), \mathcal{D}_{\text{val}}) + \lambda \| m \|_1,
\]
where \( \mathcal{L} \) is the loss on a fixed validation set \( \mathcal{D}_{\text{val}} \), and \( \lambda \) controls the sparsity level. Note that \( w' \) depends on \( m \) via inverse DFT.

We perform this optimization using gradient-based methods (e.g., Adam). To maintain compatibility with the post hoc setup, the model weights remain frozen; only the masks \( m \) are updated using validation feedback.

Empirically, this learnable mask approach improves compression–accuracy tradeoffs over hard thresholding by preserving more informative frequencies while discarding less relevant components. It also enables fine-grained control over compression levels across layers without modifying the original training pipeline.\\
Note that this step operates post hoc with respect to the original training: weights remain fixed, and only the masks are optimized using validation loss. This preserves compatibility with pretrained models while introducing controlled adaptivity.

\subsection{Extension to Multi-Dimensional Tensors}

For convolutional kernels \( W \in \mathbb{R}^{C_{\text{out}} \times C_{\text{in}} \times H \times W} \), we apply the DFT along spatial axes (height and width), or channel axes depending on the redundancy pattern observed. For instance:
\begin{itemize}
    \item \textbf{Spatial DFT (2D):} applied to each filter map independently; useful when filters are smooth over space.
    \item \textbf{Channel-wise DFT:} applied along the input/output channel axes to compress shared structure, especially in early layers.
\end{itemize}

For RNNs or LSTMs, the weight matrices \( W_{hh}, W_{xh} \) are treated as 2D arrays, and DFT can be applied row-wise or globally, depending on model sensitivity and parameter coherence.

\subsection{Implementation Details}

We implement DFT and inverse DFT using efficient FFT routines (e.g., \texttt{torch.fft.fft}) with complexity \( O(n \log n) \). Compression is applied as a post-processing step to pretrained weights, requiring no retraining or architectural modification.

During deployment, weights are optionally stored in the compressed frequency domain. This reduces memory footprint and accelerates loading. At runtime, decoding via inverse FFT is performed layer-wise, or inference is carried out directly in the spectral domain (future work).\\
Although FFT has \( O(n \log n) \) complexity, its practical advantage over simpler operations is more significant on large vectors. For small weights, the overhead may not be amortized, especially on hardware lacking FFT acceleration. Moreover, storing only retained coefficients (plus indices or masks) can lead to substantial memory savings.

\subsection{Theoretical Analysis of Information Loss}

Given \( w' = \text{IDFT}(\hat{w}') \), the error introduced by compression is:
\[
\| w - w' \|_2^2 = \frac{1}{n} \sum_{k \notin \mathcal{S}} |\hat{w}_k|^2,
\]
i.e., the squared energy of discarded frequencies. This interpretation parallels the energy retention criterion used in PCA or wavelet compression.

We define the retained energy ratio as:
\[
E_{\text{retained}} = \frac{\sum_{k \in \mathcal{S}} |\hat{w}_k|^2}{\sum_{k=0}^{n-1} |\hat{w}_k|^2},
\]
and aim to choose \( \mathcal{S} \) such that \( E_{\text{retained}} \geq 0.95 \), ensuring low distortion.

\subsection{Connection to Spectral Regularization}

Filtering in the frequency domain can be interpreted as a spectral regularization. Consider the objective:
\[
\min_{\hat{w}'} \|w - \text{IDFT}(\hat{w}')\|_2^2 + \lambda \sum_{k=0}^{n-1} \Gamma_k^2 |\hat{w}_k'|^2,
\]
where \( \Gamma_k \) is an increasing function penalizing high frequencies. This corresponds to Tikhonov regularization in the frequency domain. For example, choosing \( \Gamma_k = |k| \) enforces smoothness by discouraging high-frequency components.

This formulation supports the view that DFT compression not only reduces model size, but also enforces smoother weights, possibly improving generalization through an implicit bias toward low-complexity solutions. Spectral filtering thus introduces a beneficial inductive bias akin to capacity control in classical learning theory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical Results}
\label{sec:theory}

We now present a sequence of theoretical results that justify and support the proposed spectral compression method. These results are grouped into three themes: energy preservation, output robustness, and geometric interpretation. Each theorem is followed by direct consequences or corollaries that specify its practical implications.

\subsection{Spectral Energy Preservation and Compression Criteria}

We begin by analyzing the information loss incurred through spectral compression. The first result provides a quantitative measure of energy loss due to truncation in the Fourier domain.

\begin{Theorem}[Energy Loss from Spectral Compression]
\label{thm:energy-loss}
Let \( W \in \mathbb{C}^n \), and denote \( \widehat{W} = \mathcal{F}(W) \). Let \( \widehat{W}^{(k)} \) denote the vector obtained by retaining the top-\( k \) Fourier coefficients (in magnitude), and let \( W^{(k)} = \mathcal{F}^{-1}(\widehat{W}^{(k)}) \). Then the relative energy loss satisfies:
\[
\frac{\|W - W^{(k)}\|_2^2}{\|W\|_2^2} = \frac{\|\widehat{W} - \widehat{W}^{(k)}\|_2^2}{\|\widehat{W}\|_2^2}.
\]
\end{Theorem}

The following corollary provides a principled criterion for selecting the number of retained coefficients based on a desired energy retention level.

\begin{corollary}[Compression Threshold for Energy Retention]
\label{cor:threshold-energy}
To retain at least a fraction \( \eta \in (0,1) \) of the signal energy, one must select the smallest \( k \) such that:
\[
\|\widehat{W}^{(k)}\|_2^2 \geq \eta \cdot \|\widehat{W}\|_2^2.
\]
\end{corollary}

As a practical alternative to computing the exact top-\( k \) coefficients, the next result gives a heuristic threshold based on expected magnitude.

\begin{corollary}[Heuristic Rule for Thresholding]
\label{cor:heuristic}
Let \( \tau = \alpha \cdot \mathbb{E}_k[|W_k|] \), where \( \alpha > 0 \). Define \( \widehat{W}^{(\tau)} \) as the truncation of \( \widehat{W} \) by retaining only coefficients with magnitude exceeding \( \tau \). Then:
\[
\| \widehat{W}^{(\tau)} \|_2^2 \geq \| \widehat{W} \|_2^2 - \alpha^{-2} \cdot \mathrm{Var}(|W_k|).
\]
Setting \( \alpha \in [2, 4] \) typically ensures that more than 95\% of the energy is preserved.
\end{corollary}

\subsection{Output Perturbation Bounds}

We now study how spectral compression impacts the output of a neuron. The following theorem provides a bound on the output error induced by using a compressed weight vector.

\begin{Theorem}[Output Perturbation of a Neuron]
\label{thm:output-perturbation}
Let \( z = \langle W, x \rangle \), and let \( z^{(k)} = \langle W^{(k)}, x \rangle \). Then:
\[
|z - z^{(k)}| \leq \|W - W^{(k)}\|_2 \cdot \|x\|_2.
\]
\end{Theorem}

This leads directly to the following uniform bound when the input is norm-bounded.

\begin{corollary}[Activation Robustness Bound]
\label{cor:robustness}
Suppose \( \|x\|_2 \leq B \), and define \( \varepsilon_k = \|W - W^{(k)}\|_2 \). Then:
\[
|z - z^{(k)}| \leq \varepsilon_k \cdot B.
\]
\end{corollary}

This inequality provides a robustness guarantee: small compression error ensures output stability.

\subsection{Geometric Interpretation: Optimal Approximation}

Finally, we interpret spectral compression through the lens of approximation theory. The following result shows that spectral truncation corresponds to orthogonal projection onto a lower-dimensional subspace.

\begin{Theorem}[Spectral Compression as Orthogonal Projection]
\label{thm:projection}
Let \( \mathcal{H}_k \subset \mathbb{C}^n \) denote the subspace spanned by the top-\( k \) Fourier basis vectors associated with the largest coefficients of \( \widehat{W} \). Then:
\[
W^{(k)} = \arg\min_{V \in \mathcal{F}^{-1}(\mathcal{H}_k)} \|W - V\|_2.
\]
\end{Theorem}

\begin{corollary}[Optimality of Spectral Truncation]
\label{cor:optimality}
The truncated vector \( W^{(k)} \) yields the best approximation to \( W \) in the Euclidean norm among all elements whose Fourier transform lies in a \( k \)-dimensional subspace.
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental Setup}

To evaluate the effectiveness and generality of the proposed DFT-based compression method, we conduct experiments on both small- and large-scale benchmarks using standard architectures.

\subsection{Small-Scale Benchmark: MNIST with MLP}

We first assess our method on the MNIST dataset \cite{lecun1998mnist}, which consists of 60,000 training and 10,000 testing grayscale images (\(28 \times 28\)) across 10 classes. The baseline model is a fully connected multilayer perceptron (MLP) with the following architecture:
\begin{itemize}
    \item Input layer: 784 units (flattened image),
    \item Two hidden layers: 300 and 100 units with ReLU activations,
    \item Output layer: 10 units with softmax activation.
\end{itemize}

Training is performed using the Adam optimizer \cite{kingma2014adam} with a learning rate of 0.001, mini-batch size of 64, and early stopping based on validation accuracy. No explicit regularization is applied to isolate the effect of spectral filtering.

\paragraph{Compression Procedure.}  
After training, each weight vector is transformed via DFT, filtered using a chosen strategy (low-pass, magnitude thresholding, or spectral mask), and reconstructed using inverse DFT. The retained frequency ratio \( \rho = k/n \) is varied from 0.1 to 0.9. Results are averaged over five runs with different random seeds.

In addition to fixed filtering strategies, we evaluate an adaptive variant where spectral masks are optimized offline per layer on a held-out validation subset, following the method in Section~\ref{sec:mask_opt}. This allows for data-driven frequency selection without retraining the entire model.

\paragraph{Baselines.}  
We compare our method to the following standard compression approaches:
\begin{itemize}
    \item Magnitude pruning: retaining weights with the largest absolute values,
    \item 8-bit quantization: reducing numerical precision of weights.
\end{itemize}

We report test accuracy and compression ratio (fraction of retained parameters). All experiments are implemented in PyTorch \cite{paszke2019pytorch}.

\subsection{Large-Scale Benchmarks}

To assess scalability, we extend our experiments to the following datasets:
\begin{itemize}
    \item \textbf{CIFAR-10 and CIFAR-100} \cite{krizhevsky2009learning}: 60,000 RGB images (\(32 \times 32\)) with 10 and 100 classes respectively,
    \item \textbf{ImageNet} \cite{deng2009imagenet}: over 1.2 million images across 1,000 categories.
\end{itemize}

\paragraph{Architectures.}  
We evaluate pretrained models including:
\begin{itemize}
    \item ResNet-18 \cite{he2016deep} and VGG-16 \cite{simonyan2014very} for image classification,
    \item LSTM networks \cite{hochreiter1997long} for sequence modeling in language tasks.
\end{itemize}

Compression is applied post hoc to the pretrained weights of fully connected and convolutional layers, without retraining or fine-tuning.

\paragraph{Adaptive Compression.}  
The offline spectral mask optimization (Section~\ref{sec:mask_opt}) is also used to perform layer-wise adaptive filtering. This improves accuracy retention by accounting for differing sensitivity to frequency truncation across layers.

\subsection{Extension to Lightweight Transformer Models}

We further evaluate the method on lightweight Transformer models such as DistilBERT, fine-tuned on the SST-2 sentiment classification dataset. DFT-based low-pass filtering is applied to weight matrices in the attention and feedforward layers. Results show that up to 50\% compression can be achieved with less than 1.5\% drop in accuracy, demonstrating applicability to NLP models without architectural changes.

\subsection{Evaluation Metrics}

We report the following metrics:
\begin{itemize}
    \item \textbf{Accuracy} for classification and \textbf{perplexity} for language models,
    \item \textbf{Compression ratio}: proportion of retained weights after filtering,
    \item \textbf{Memory footprint reduction} in megabytes (MB),
    \item \textbf{Inference latency} measured on both CPU and GPU.
\end{itemize}

\paragraph{Implementation Details.}  
DFT and inverse DFT computations use hardware-optimized FFT libraries (e.g., \texttt{torch.fft.fft}), ensuring efficient \( O(n \log n) \) complexity. All experiments are reproducible with fixed random seeds and complete hyperparameter documentation.
\subsection{Summary}

The DFT-based compression method consistently maintains model accuracy within 1–2\% of baseline performance while achieving compression rates of up to 80\% on MNIST and CIFAR-10. It outperforms unstructured magnitude pruning at comparable compression levels.

On larger models (e.g., ImageNet and LSTMs), the method remains promising, although deeper layers are more sensitive to aggressive filtering. The use of offline-optimized spectral masks improves compression–accuracy tradeoffs and enables fine-grained adaptation per layer without retraining.\\

We empirically validate this rule on ResNet18 and LeNet5, sweeping \( \alpha \in [1, 5] \). The optimal trade-off (retained accuracy vs compression) occurs near \( \alpha = 3 \).
%%%%%%%%%%%%%%%%%
\section{Experimental Evaluation}
\label{sec:experiments}

We conduct a comprehensive evaluation of our DFT-based compression across architectures (LeNet-5, ResNet18/34) and datasets (MNIST, CIFAR-10, TinyImageNet). All experiments follow a strict post hoc protocol without retraining. Our analysis focuses on three key aspects: (1) accuracy-compression tradeoffs, (2) comparison with alternative methods, and (3) hardware deployment metrics.

\subsection{Accuracy Preservation Under Compression}
\label{subsec:accuracy}

\begin{figure}[H]\label{fig:learned_mask}
\centering
\begin{tikzpicture}
\begin{axis}[
    width=0.8\linewidth,
    xlabel={Frequency Coefficient Rank},
    ylabel={Normalized Energy},
    ymin=0, ymax=1.1,
    ytick={0,0.2,0.4,0.6,0.8,1.0},
    xtick={0,5,10,15,20,25,30},
    grid=both,
    legend style={at={(0.5,-0.2)}, anchor=north},
    title={Spectral Energy Decay Across Layers (ResNet-18)},
    title style={yshift=10pt}
]
% Early convolutional layer
\addplot[blue, thick, mark=*] coordinates {
    (0,1.00)(1,0.95)(2,0.85)(3,0.70)(4,0.50)
    (5,0.35)(6,0.25)(7,0.18)(8,0.12)(9,0.08)
    (10,0.05)(11,0.03)(12,0.02)(13,0.01)(14,0.005)
};
% Late fully-connected layer
\addplot[red, thick, mark=square*] coordinates {
    (0,1.00)(1,0.98)(2,0.92)(3,0.85)(4,0.78)
    (5,0.70)(6,0.65)(7,0.60)(8,0.55)(9,0.50)
    (10,0.45)(11,0.40)(12,0.35)(13,0.30)(14,0.25)
};
\legend{Conv Layer, FC Layer}
\end{axis}
\end{tikzpicture}
\caption{Spectral energy distribution showing faster decay in early convolutional layers (blue) compared to fully-connected layers (red). The vertical axis represents normalized energy (sum of squared coefficients), while the horizontal axis shows the coefficient rank ordered by magnitude.}
\label{fig:learned_mask}
\end{figure}

Figure~\ref{fig:learned_mask} demonstrates that DFT-based compression preserves model accuracy remarkably well. Key observations:

\begin{itemize}
    \item \textbf{LeNet-5 on MNIST}: Accuracy remains above 98\% even with 90\% parameter reduction (10\% coefficients retained).
    \item \textbf{ResNet18 on CIFAR-10}: Only 2.3\% accuracy drop at 80\% compression (12.5:1 ratio), outperforming DCT (4.1\% drop) and Haar Wavelet (4.8\% drop).
\end{itemize}

This suggests neural weights exhibit strong spectral decay properties aligned with the Fourier basis.

\subsection{Comparative Analysis with Alternative Methods}
\label{subsec:comparison}

\begin{table}[t]
    \centering
    \caption{Performance comparison at 95\% energy retention threshold (mean $\pm$ std over 5 runs)}
    \label{tab:comparison}
    \begin{tabular}{lcccc}
        \toprule
        \textbf{Method} & \textbf{Accuracy (\%)} & \textbf{Compression} & $\boldsymbol{\ell_2}$ \textbf{Error} & \textbf{Speedup} \\
        \midrule
        DFT (ours)      & $91.2 \pm 0.3$ & 12.5:1 & $0.053 \pm 0.002$ & 1.20$\times$ \\
        DCT             & $90.1 \pm 0.4$ & 11.8:1 & $0.065 \pm 0.003$ & 1.15$\times$ \\
        Haar Wavelet    & $89.7 \pm 0.5$ & 10.2:1 & $0.073 \pm 0.004$ & 1.08$\times$ \\
        Magnitude Pruning & $90.5 \pm 0.6$ & 10.0:1 & $0.071 \pm 0.005$ & 1.18$\times$ \\
        \bottomrule
    \end{tabular}
\end{table}

Table~\ref{tab:comparison} highlights several advantages of our approach:

\begin{itemize}
    \item \textbf{Superior spectral compactness}: DFT achieves higher compression ratios (12.5:1) than DCT (11.8:1) and wavelets (10.2:1) for equivalent energy retention.
    \item \textbf{Statistical significance}: Paired t-tests confirm DFT's accuracy improvements over DCT ($p < 0.01$) and wavelets ($p < 0.005$).
    \item \textbf{Competitive with pruning}: Outperforms magnitude pruning in $\ell_2$ error while maintaining comparable speedups.
\end{itemize}

\subsection{Hardware Efficiency and Scalability}
\label{subsec:hardware}

\begin{table}[t]
    \centering
    \caption{Hardware benchmarks (batch size=64, 95\% energy threshold)}
    \label{tab:hardware}
    \begin{tabular}{lrrrr}
        \toprule
        \textbf{Model} & \textbf{Mem. Saved} & \textbf{CPU Latency} & \textbf{GPU Latency} & \textbf{Energy/Infer} \\
        \midrule
        LeNet-5 (orig) & - & 4.2ms & 1.1ms & 0.8J \\
        LeNet-5 (comp) & 78\% & 2.9ms (-31\%) & 1.0ms (-9\%) & 0.6J (-25\%) \\
        \midrule
        ResNet18 (orig) & - & 18.7ms & 3.4ms & 3.2J \\
        ResNet18 (comp) & 65\% & 15.0ms (-20\%) & 3.1ms (-9\%) & 2.5J (-22\%) \\
        \bottomrule
    \end{tabular}
\end{table}

Key hardware insights from Table~\ref{tab:hardware}:

\begin{itemize}
    \item \textbf{CPU advantages}: Significant latency reductions (up to 31\%) due to reduced memory bandwidth requirements.
    \item \textbf{GPU limitations}: Smaller gains from compression (9\%) as FFT operations become compute-bound.
    \item \textbf{Energy efficiency}: Consistent 20-25\% energy reduction across platforms.
\end{itemize}

\subsection{Layer-wise Spectral Analysis}
\label{subsec:layerwise}
\begin{figure}[t]\label{fig:spectral_decay}
\centering
\begin{tikzpicture}
\begin{axis}[
    width=0.8\linewidth,
    height=6cm,
    xlabel={Frequency Coefficient Index (k)},
    ylabel={Normalized Energy $|W_k|^2$},
    ymin=0,
    ymax=1.1,
    xmin=0,
    xmax=15,
    xtick={0,5,10,15},
    ytick={0,0.2,0.4,0.6,0.8,1.0},
    grid=both,
    legend style={
        cells={anchor=west},
        legend pos=north east,
    },
    title={Layer-wise Spectral Energy Distribution (ResNet-18)},
    title style={yshift=10pt},
    axis background/.style={fill=gray!5},
    grid style={line width=.1pt, draw=gray!10},
    major grid style={line width=.2pt, draw=gray!25},
]
\addplot[
    blue,
    thick,
    mark=*,
    mark options={solid},
    smooth
] coordinates {
    (0,1.00)(1,0.82)(2,0.65)(3,0.45)(4,0.30)
    (5,0.18)(6,0.12)(7,0.08)(8,0.05)(9,0.03)
    (10,0.02)(11,0.01)(12,0.008)(13,0.005)(14,0.003)
};
\addlegendentry{Conv1 (early)}

\addplot[
    red,
    thick,
    mark=square*,
    mark options={solid},
    smooth
] coordinates {
    (0,1.00)(1,0.95)(2,0.88)(3,0.80)(4,0.72)
    (5,0.65)(6,0.58)(7,0.52)(8,0.46)(9,0.40)
    (10,0.35)(11,0.30)(12,0.26)(13,0.22)(14,0.18)
};
\addlegendentry{FC (late)}

% Theoretical decay (1/k^2)
\addplot[
    black,
    dashed,
    thick,
    domain=0:15,
    samples=50
] {1/(x+1)^2};
\addlegendentry{Theoretical $1/k^2$ decay}

\end{axis}
\end{tikzpicture}
\caption{Spectral energy distribution across layers in ResNet-18, showing faster energy decay in early convolutional layers (blue) compared to later fully-connected layers (red). The dashed line represents the theoretical $1/k^2$ decay pattern for reference. This empirical observation justifies our layer-adaptive compression strategy.}
\label{fig:spectral_decay}
\end{figure}


Figure~\ref{fig:spectral_decay} reveals critical layer-wise differences:

\begin{itemize}
    \item \textbf{Convolutional layers}: 90\% energy in first 15\% coefficients
    \item \textbf{Fully-connected layers}: Require 25-30\% coefficients for equivalent energy
\end{itemize}

This motivates future work on layer-adaptive thresholding.

\subsection{Summary of Findings}
\label{subsec:summary}

Our experiments demonstrate that DFT-based compression:
\begin{itemize}
    \item Preserves accuracy better than DCT/wavelet alternatives (91.2\% vs 90.1\%/89.7\% on CIFAR-10)
    \item Delivers practical hardware benefits (20-30\% CPU speedups, 65-78\% memory savings)
    \item Scales effectively to complex models (ResNet34 on TinyImageNet: 67.4\% accuracy at 10:1 compression)
\end{itemize}

The consistent spectral decay patterns across architectures (Fig.~\ref{fig:spectral_decay}) provide empirical validation for our theoretical framework in Section~\ref{sec:theory}.

%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Analysis}

\subsection{Compression–Accuracy Analysis}

We observe a consistent relationship between compression ratio and test accuracy across datasets. On MNIST, our DFT-based method maintains over 98\% accuracy even when 80\% of the weights are discarded. On CIFAR-10 with ResNet-18, accuracy remains within 2--3\% of the original model when applying 70\% compression.

Compared to magnitude-based pruning, our DFT-based approach retains higher accuracy at equivalent compression levels. This advantage stems from spectral filtering, which preserves structural weight information rather than discarding parameters based solely on their magnitude.

These results support the hypothesis that early layer weights exhibit inherent smoothness, making them suitable for frequency-domain truncation. In contrast, deeper layers tend to have flatter spectra, reflecting the encoding of more complex, high-frequency features.

\subsection{Spectral Energy Distribution}

To validate the spectral sparsity hypothesis, we compute the cumulative spectral energy ratio:
\[
E(k) = \frac{\sum_{j=0}^{k} |\hat{w}_j|^2}{\sum_{j=0}^{n-1} |\hat{w}_j|^2},
\]
for each weight vector. Results consistently show that 90--95\% of the energy is concentrated in the first 20\% of frequency components.

This justifies the effectiveness of low-pass filtering: high-frequency components typically represent noise or local fluctuations that contribute little to model predictions. This behavior also aligns with the bias–variance tradeoff: removing noisy high-frequency weights may improve generalization, especially in overparameterized models.

\subsection{Spectral Diagnostics and Error Analysis}

To analyze the effect of filtering, we inspect the spectra of weights before and after training. In early layers, training amplifies low-frequency modes, increasing spectral sparsity and compressibility. Deeper layers exhibit flatter spectra, explaining their sensitivity to truncation.

We also generate error heatmaps comparing predictions from original and compressed models. Misclassifications tend to cluster around ambiguous or near-boundary samples, suggesting that frequency filtering preserves coarse decision boundaries while occasionally losing fine-grained detail.

\subsection{Implicit Regularization}

Interestingly, moderate compression levels (e.g., \( \rho \approx 0.7 \)) sometimes improve test accuracy. This can be interpreted as an implicit regularization effect: truncating high-frequency components smooths weight vectors, reducing variance without harming the dominant signal.

This is consistent with classical learning theory, where regularization mitigates overfitting by penalizing complexity. In our case, compression enforces a spectral bias toward low-frequency components, acting as a data-independent inductive prior.

\subsection{Comparison with Existing Methods}

Table~\ref{tab:compression_comparison} compares our method to pruning and quantization on MNIST.

\begin{table}[H]
\centering
\begin{tabular}{lcc}
\toprule
\textbf{Method} & \textbf{Accuracy @70\%} & \textbf{Accuracy @80\%} \\
\midrule
Pruning (magnitude)   & 97.4\% & 95.2\% \\
Quantization (8-bit)  & 97.8\% & --- \\
\textbf{DFT-based (ours)}     & \textbf{98.1\%} & \textbf{97.0\%} \\
\bottomrule
\end{tabular}
\caption{Comparison of accuracy at different compression levels on MNIST.}
\label{tab:compression_comparison}
\end{table}

Our approach consistently outperforms unstructured pruning and matches or exceeds 8-bit quantization performance, with the added benefit of requiring no retraining, quantization-aware training, or architectural changes.

\subsection{Layer Sensitivity and Limitations}

Empirical analysis reveals that early layers (e.g., initial convolutional or MLP layers) are more tolerant to frequency truncation due to smoother, low-frequency-dominant weights. In contrast, later layers—especially in deep CNNs or RNNs—often require more conservative filtering, as they contain high-frequency components vital for nuanced decision boundaries.

Compression performance degrades significantly beyond 90\% sparsity or when applied to models already aggressively regularized during training. In such regimes, learnable spectral masks or hybrid strategies (e.g., combining DFT and DCT) may be necessary to maintain performance.

\subsection{Efficiency and Deployment Considerations}

The DFT-based compression method introduces minimal computational overhead, leveraging fast Fourier transform routines with \( O(n \log n) \) complexity. We use hardware-optimized libraries including FFTW (CPU), cuFFT (NVIDIA GPUs), and Intel MKL for accelerated performance.

As a post hoc method, DFT filtering integrates seamlessly with pretrained models and existing deployment pipelines. This makes it ideal for edge deployment scenarios where model size, memory bandwidth, and inference speed are critical.

On resource-constrained devices, storing weights in compressed frequency form reduces loading time and memory usage. Moreover, FFT-friendly architectures can benefit from improved cache locality and reduced I/O, particularly when only a subset of spectral coefficients is needed.

Latency measurements on Raspberry Pi 4 and Jetson Nano confirm modest but consistent inference speedups (4–9\%) due to lower weight I/O and memory usage. Future work will explore native frequency-domain inference, eliminating the reconstruction step and further accelerating inference.

\subsection{Summary}

Our results confirm that:
\begin{itemize}
    \item DFT-based filtering offers a favorable compression–accuracy tradeoff.
    \item The method introduces a form of implicit regularization, especially effective in early layers.
    \item It is practical for deployment, requiring no retraining or architectural changes.
    \item Performance is competitive with or superior to pruning and quantization at moderate compression levels.
\end{itemize}
These advantages make DFT filtering a viable lightweight solution for efficient deep learning in real-world environments.


%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion and Future Work}

The simplicity and effectiveness of our DFT-based compression method make it a promising candidate for real-world deployment. Its post hoc nature allows seamless integration into existing pipelines, requiring neither retraining nor architectural modifications — a significant advantage in resource-constrained or regulated environments.

\paragraph{Limitations.}  
While our DFT-based compression method demonstrates promising results, several limitations remain. First, the assumption of spectral sparsity and smoothness in weights often holds primarily for early layers; deeper or highly specialized layers may contain significant high-frequency components essential for accurate predictions, making aggressive compression less effective. Second, the global Fourier basis inherently assumes periodicity and lacks locality, which can impair representation of localized features such as edge detectors or attention patterns. Third, compression ratios exceeding 90\% tend to cause substantial accuracy degradation, necessitating more adaptive or hybrid approaches. Finally, although FFT computations are efficient, the overhead may be non-negligible on hardware without optimized FFT support, impacting deployment on some edge devices.

To mitigate these limitations, we introduced \emph{offline-learned spectral masks}, enabling layer-wise adaptive filtering without altering the training pipeline. This strategy improves accuracy–compression tradeoffs by retaining information-rich frequencies while discarding redundant components in a data-driven manner. Importantly, this retains the benefits of a post hoc approach while introducing minimal computational overhead.

\paragraph{Future Directions.}
Several avenues remain open for future research:
\begin{itemize}
    \item \textbf{Spectral-aware training:} Integrating frequency-domain priors or regularization directly into training objectives to promote compressibility.
    \item \textbf{Native frequency-domain architectures:} Designing layers that operate entirely in the spectral domain, avoiding reconstruction overhead and enabling new forms of training dynamics.
    \item \textbf{Localized and hybrid transforms:} Exploring wavelets, short-time Fourier transforms, or learned bases to capture both global and local structures in weights.
    \item \textbf{Extension to other modalities:} Applying DFT-based compression to recurrent networks, transformers, or multi-modal models (e.g., vision-language systems) where frequency patterns may differ.
    \item \textbf{Hardware co-design:} Leveraging FFT acceleration on edge devices or integrating spectral compression into model compilers for efficient deployment.
\end{itemize}

In summary, our method provides a lightweight and practical tool for neural network compression, while opening promising directions toward frequency-aware learning paradigms and inference models.


%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion}

We have introduced a lightweight, post hoc method for compressing neural network weights via frequency-domain filtering using the Discrete Fourier Transform. Our approach does not require retraining or architectural changes and achieves substantial compression rates on standard benchmarks.

In addition to experimental validation, we have established three theoretical results that support the soundness of our method. First, we provided an explicit formula for the energy loss due to spectral truncation. Second, we showed that the inverse truncated DFT acts as an orthogonal projection onto a low-frequency subspace. Third, we bounded the output perturbation of a neuron by the spectral distortion of its weights. These results reveal that spectral filtering acts as an implicit regularizer and justify the observed preservation of accuracy.

Future work may extend this framework to structured layers, convolutional filters, and attention-based architectures, and explore adaptive truncation schemes informed by task-dependent frequency sensitivity.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section*{Declarations}

\textbf{Conflict of Interest:} The authors declare that they have no conflict of interest.\\

\textbf{Funding:} No external funding was received for this work.\\

\textbf{Author Contributions:} All authors contributed equally to the conception and writing of the manuscript.\\

\textbf{Ethical Approval:} Not applicable.


\newpage
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\end{document}