Temporal & Depthwise Separable Convolutions

• Temporal and depthwise separable convolutions are CNN factorization techniques that decompose standard convolutions to reduce parameters while preserving efficiency.
• They enable efficient sequential and spatiotemporal modeling across modalities such as audio, video, sEMG, and language with notable computational savings.
• Empirical studies demonstrate these methods achieve competitive accuracy in applications like speech recognition, gesture classification, and video analysis with substantial parameter reduction.

Temporal and depthwise separable convolutions are convolutional neural network (CNN) factorization strategies that substantially reduce parameter count and computation. These operations decompose standard convolutions along temporal, spatial, and channel axes, enabling highly efficient, expressive, and parallelizable architectures for sequential and spatiotemporal modeling across modalities including audio, video, electromyography (sEMG), and natural language. Modern neural models often integrate these techniques to meet stringent deployment and performance requirements without sacrificing accuracy.

1. Mathematical Formulation of Temporal and Depthwise Separable Convolutions

A standard convolution over a D-dimensional tensor feature map (e.g., 1D for time, 2D for images, 3D for video) with $C_\text{in}$ input channels and $C_\text{out}$ output channels, kernel size $K$, has parameter count $C_\text{in} \cdot C_\text{out} \cdot K^D$ and high computational complexity.

Depthwise separable convolution factorizes any $D$D convolution into sequential application of:

• A depthwise convolution: applies a single $K^D$ filter per input channel (no cross-channel mixing).
• A pointwise convolution: $1 \times 1$ kernel (or $1 \times 1 \times 1$ in 3D) mixes channels linearly.

For 1D temporal convolutions (common in audio or time-series):

• Depthwise step: $C_\text{in} \cdot K$ parameters.
• Pointwise step: $C_\text{in} \cdot C_\text{out}$ parameters.
• Total: $C_\text{in} \cdot (K + C_\text{out})$ ($\ll C_\text{in} \cdot C_\text{out} \cdot K$), often achieving a $>10 \times$ reduction for modest $K$ and large $C_\text{out}$ (Rahimian et al., 2019, Kriman et al., 2019).

Dilated temporal convolution further generalizes the 1D convolution: spacing kernel taps by a dilation $d$,

$y(t) = \sum_{m=0}^{K-1} x(t - d \cdot m) w(m)$

extending the receptive field to $(K-1) d + 1$ with no increase in parameter count, and can be stacked for even broader context (Drossos et al., 2020).

Standard multi-dimensional generalizations exist for 2D and 3D (spatial or spatiotemporal) separable convolution (Nguy et al., 2023), decomposing $$K_T \times K_H \times K_W \times C_\text{in} \to C_\text{out}$$ into depthwise and pointwise as well.

2. Architectural Use and Integration

These factorized convolutions are now core primitives in several high-performance deep learning models:

• QuartzNet (Kriman et al., 2019): Each block comprises 1D time-channel (temporal depthwise with pointwise mixing) separable convolutions, batch normalization, and ReLU, enabling a deep ASR model ($\sim$19M parameters) competitive with models >10$\times$ larger.
• XceptionTime (Rahimian et al., 2019): Stacks parallel temporal (1D) depthwise separable convolutions of varied kernel lengths within each module, concatenated with a pooled skip pathway and adaptive pooling to ensure variable window handling in sEMG gesture classification.
• 3D Spatiotemporal CNNs (Nguy et al., 2023): Replaces each 3D spatiotemporal convolution with depthwise (per-channel 3D spatial-temporal) followed by pointwise, preserving spatial-temporal context at $94\%$ parameter reduction in eye blink detection without loss of F1.
• SliceNet (Kaiser et al., 2017): Applies temporal (1D) depthwise-separable convolutions throughout both encoder and decoder stacks for neural machine translation, replacing both attention and recurrence for long-context modeling.

A canonical integration pattern involves stacking several separable convolutional layers/blocks, with non-linearities and normalization, sometimes with skip or residual connections. In hybrid models, depthwise separable convolutions often replace standard convolutions and RNNs (such as GRUs, LSTMs), especially where long-term context is key and sequence parallelism is critical (Drossos et al., 2020, Pfeuffer et al., 2019).

3. Comparative Complexity and Receptive Field Analysis

The primary advantage is the marked reduction in both parameter count and throughput cost:

• Parameter ratio (1D temporal, $K \ll C$): $K C^2 \to (K + C)C$ ($\sim 1/K$ overhead for large $C$) (Rahimian et al., 2019, Kriman et al., 2019, Kaiser et al., 2017).
• FLOP count: Drops by a similar factor, as most computation in standard convs is quadratic in channel count.

Empirical tabulations (from (Rahimian et al., 2019)), for equal input/output channel count $f$:

Block	DSC params	Standard params	Reduction
Block 4 ($f=128$)	$128 \cdot 73 + 3 \cdot 128^2 = 58,\!496$	$128 \cdot 128 \cdot 73 = 1,\!196,\!032$	$20.4\times$

Similarly, for 3D CNNs (Nguy et al., 2023):

Model	#Params	% vs. Baseline
3D-P3B3	7.6 M	100%
DWS-P3B3	0.46 M	6%

Receptive field: Dilated or wide separable convolutions (large $K$, large $d$) allow growing the model's context window while keeping parameters constant. E.g., a single dilated conv layer with $K=7$, $d=10$ offers a $61$-frame window (Drossos et al., 2020).

A plausible implication is that the massive parameter savings can be directly converted to greater network depth, larger kernel sizes, or broader receptive fields, substantially increasing representational power under fixed computational budgets.

4. Empirical Performance and Application Domains

Depthwise separable and temporal convolutions have been empirically validated across diverse domains:

• Sound event detection: Using both depthwise-separable and 1D dilated convolutions, achieves $+4.6\%$ absolute framewise F1 and $-3.8\%$ error rate, with $85\%$ fewer parameters and $78\%$ faster per-epoch training compared to standard CRNNs (Drossos et al., 2020).
• Automatic Speech Recognition: QuartzNet attains $2.96\%$/ $8.07\%$ (test-clean/other) WER on LibriSpeech with $15\times$ parameter reduction compared to Jasper (Kriman et al., 2019).
• Hand gesture recognition: XceptionTime yields a $5.71\%$ absolute accuracy gain (window $=200$ms) versus prior sEMG CNNs at $\sim4-20\times$ smaller size (Rahimian et al., 2019).
• Video analysis: Spatiotemporal CNNs for blink detection retain F1 parity (within $0.3\%$) after a $94\%$ parameter reduction (Nguy et al., 2023).
• Machine Translation: SliceNet surpasses ByteNet, raising BLEU from $23.8$ to $25.5$ (En→De, newstest2014) and reducing non-embedding parameter count by $38\%$ (Kaiser et al., 2017).
• Video Segmentation with convLSTM: Separable convLSTM yields $88\%$ parameter/FLOP reductions, $10–15\%$ faster inference, and negligible accuracy drop (≤$1\%$ mIoU) (Pfeuffer et al., 2019).

A consistent observation is that parameter savings typically do not incur performance penalties, and in many cases, yield regularization benefits and accuracy improvements due to reduced overfitting capacity.

5. Trade-Offs, Limitations, and Design Principles

While depthwise separable and temporal convolutions offer efficiency and scalability, several domain-specific trade-offs have been observed:

• Loss of expressiveness: Since depthwise steps do not mix channels, representational richness may be reduced if over-factored, especially with low output channel counts (Kriman et al., 2019, Pfeuffer et al., 2019).
• Channel or kernel size dependence: For very small $C$ or $K$, parameter and FLOP savings diminish—DWS may offer limited advantage when $C_o \approx 1$ or $K=1$ (Drossos et al., 2020).
• Receptive field sparsity: Excessive dilation ($d_t > 100$) can cause "gridding" (missing local detail), so empirically, $d_t$ in $[10,50]$ yields the best accuracy for long-range dependencies (Drossos et al., 2020).

Guidelines reported include:

• Use depthwise separable convolutions wherever model or compute limits are stringent (mobile, embedded).
• Match dilation × kernel size to the expected temporal or spatial event duration.
• Prefer larger, non-dilated separable kernels (where feasible) over aggressive dilation, as the increased context is less sparse and better at local detail (Kaiser et al., 2017).

6. Extensions and Hybridizations

Advanced architectural extensions include:

• Super-separable convolution: Groups channels and applies separable convs per group, reducing the $1\times1$ mixing parameters by group factor while maintaining cross-group communication in deeper stacks (Kaiser et al., 2017). This further reduces the $O(D^2)$ cost to $O(D^2/g)$.
• Hybrid models: Integration of separable conv blocks with attention, RNNs, or Transformer modules can combine the strengths of efficient local context aggregation with global sequence modeling (Kriman et al., 2019).
• Separable convLSTM: Embeds separable convolutional operations for all gates inside LSTM cells for video and spatiotemporal sequence modeling (Pfeuffer et al., 2019).

Applications have rapidly proliferated: speech and audio recognition, video segmentation and object tracking, sEMG-based biomedical sensing, neural machine translation, and low-latency real-time inference scenarios.

7. Summary Table: Parameter Reduction and Performance

Model/Application	Parameter Reduction	Speedup	Performance Impact	Reference
Sound Event Detection (SED)	85%	78% faster	+4.6% abs. F1, -3.8% error rate	(Drossos et al., 2020)
QuartzNet ASR	$15\times$	-	Within 1% WER of Jasper baseline	(Kriman et al., 2019)
XceptionTime (gesture)	$4-20\times$	-	+5.71% accuracy	(Rahimian et al., 2019)
3D Eye Blink Detection	94%	Much faster	$\leq 0.3\%$ F1 drop or gain	(Nguy et al., 2023)
Video Segmentation (convLSTM)	$88\%$	$10–15\%$	≤1% mIoU drop	(Pfeuffer et al., 2019)
Translation (SliceNet)	38%	-	+1.7 BLEU vs. ByteNet	(Kaiser et al., 2017)

Implementations adopting temporal and depthwise separable convolution exhibit robust empirical gains, scalable architectural flexibility, and operational efficiency, with modest expressiveness trade-offs that can be compensated through architectural choices or hybridization. The strategy enables modern deep models to meet high-performance criteria across multiple sequence and spatiotemporal recognition tasks.