SSDPT: Transformer & DSTBC Systems

• SSDPT is a dual-topic framework that encompasses a self-supervised dual-path Transformer for anomalous sound detection and single-symbol ML decodable DSTBCs for cooperative networks.
• The Transformer component leverages alternating time and frequency encoders with random masking and reconstruction tasks, achieving notable performance improvements in anomaly detection metrics.
• For DSTBC applications, SSDPT ensures independent decodability of real channel symbols by enforcing algebraic decoupling, thus reducing ML complexity and optimizing coding gains in cooperative networks.

SSDPT refers to two distinct topics in the scientific literature: (1) the Self-Supervised Dual-Path Transformer for anomalous sound detection in machine condition monitoring (Bai et al., 2022), and (2) Single-Symbol ML Decodable Distributed Space-Time Block Codes for partially-coherent cooperative networks (“SSD DSTBCs” in (0708.3019)). This entry addresses both, given their prominent association with the acronym “SSDPT.”

1. Self-Supervised Dual-Path Transformer (SSDPT) for Anomalous Sound Detection

SSDPT is a machine learning architecture for unsupervised anomaly detection in industrial machine audio, focusing on scenarios where only normal-operation sound is available for training. The framework combines dual-path Transformer modeling of time and frequency dependencies with self-supervised learning objectives, achieving state-of-the-art performance under domain shift conditions (Bai et al., 2022).

Key Principles

SSDPT is predicated on two innovations:

• Dual-Path Transformer (DPT): Stacks Transformer encoders that alternately aggregate temporal and spectral (frequency) relations within short spectrogram segments.
• Self-supervised learning: Uses only normal recordings, with random masking and reconstruction as an auxiliary task, and machine-ID classification as the primary task.

This architecture supports training in environments where anomaly instances are rare or unavailable.

2. Acoustic Feature Extraction and Segmentation

• Input: Audio at 16 kHz.
• Transformation: Short-Time Fourier Transform (STFT, 1024 window, 512 hop), then mapped to 128 Mel bins, followed by $\log$ scaling to produce $W \in \mathbb{R}^{T \times F}$, $F=128$.
• Segmentation: $W$ is divided into overlapping $P \times F$ patches $X_t$, segment length $P \in \{64, 128, 256\}$, training hop $H=8$, testing hop $H=1$.
• Assignment: Each segment inherits machine-ID label $\ell \in \{1, \dotsc, I\}$.

Segmentation enables the DPT to model both local and contextual structure in the acoustic data.

3. Dual-Path Transformer Block Architecture

• Composition: Each DPT block processes segment $W \in \mathbb{R}^{T \times F}$0 via two sequential Transformer encoders:
  • Time Encoder $W \in \mathbb{R}^{T \times F}$1: Processes $W \in \mathbb{R}^{T \times F}$2-length sequence of $W \in \mathbb{R}^{T \times F}$3-dimensional vectors.
  • Frequency Encoder $W \in \mathbb{R}^{T \times F}$4: After transposing, processes $W \in \mathbb{R}^{T \times F}$5-length sequences of $W \in \mathbb{R}^{T \times F}$6-dimensional vectors; result is transposed back.
• Transformer mechanics (per encoder, input $W \in \mathbb{R}^{T \times F}$7):

1. Multi-Head Scaled Dot-Product Attention (MHSA): For head $W \in \mathbb{R}^{T \times F}$8, queries, keys, and values are projected and computed as:

$W \in \mathbb{R}^{T \times F}$9

2. Residual and LayerNorm: $F=128$0. 3. Feed-Forward Network (FFN): $F=128$1. 4. Second Residual and LayerNorm: $F=128$2.

• Stacking: $F=128$3 DPT blocks are stacked, alternating between time and frequency encoding.

This cascading interaction allows for interleaved modeling of temporal and spectral dependencies, both crucial for anomalous sound discrimination.

4. Self-Supervised Training Objectives

SSDPT employs two concurrent self-supervised tasks on normal sound segments $F=128$4:

• 4.1. Random Masking & Reconstruction: Applies $F=128$5 square patch masks (e.g., $F=128$6, $F=128$7) to $F=128$8 yielding $F=128$9, with DPT reconstructing $W$0. The mean squared error (MSE) reconstruction loss:

  $W$1

• 4.2. Machine-ID Classification: After final DPT block, max-pool over $W$2 to obtain $W$3; softmax classifier outputs machine-ID predictions $W$4. Cross-entropy loss:

  $W$5

• 4.3. Joint Loss: For $W$6 segments:

  $W$7

  with $W$8.

The auxiliary reconstruction task complements the classification, improving generalization under domain shift (Bai et al., 2022).

5. Evaluation Protocol and Performance

• Dataset: DCASE2021 Task 2 (seven machine types, two domains, three sections). Training: 1000 source-domain normal, 3 target normal; Testing: 100 normals + 100 anomalous per domain/section.
• Anomaly Scoring: Each test clip is segmented. For each segment, compute:

  • Classification score:

    $W$9

    where $P \times F$0 is softmax probability for correct machine-ID.

  • Reconstruction score:

    $P \times F$1

  • Final anomaly score:

    $P \times F$2

    with $P \times F$3.

• Metrics:
  • Area under ROC curve (AUC)
  • Partial AUC at FPR $P \times F$4 (pAUC)
  • Harmonic mean AUC/pAUC (h-AUC, h-pAUC) per machine type.

Method	h-AUC	h-pAUC
Baseline AE	0.619	0.533
Baseline MobileNetV2	0.597	0.564
MNv2 + LOF	0.694	0.616
WaveNet-ensemble	0.705	0.625
SSDPT (no auxiliary)	0.713	0.607
SSDPT (full)	0.722	0.611
SSDPT-ensemble	0.739	0.626

The DPT model alone outperforms prior methods. The auxiliary task brings a further $P \times F$5 h-AUC, and ensembling pushes h-AUC to $P \times F$6, an increase of $P \times F$7 points over the previous state-of-the-art (Bai et al., 2022).

6. Connections: Distributed STBCs and SSD Property

The acronym SSDPT can also refer to Single-Symbol ML Decodable Distributed Space-Time Block Codes (DSTBCs) for partially-coherent relay networks (0708.3019). Here, the “SSD” property ensures each real channel symbol is decodable independently with maximum-likelihood complexity $P \times F$8 rather than exponential in $P \times F$9.

• Classical SSD conditions (co-located): Decoupling (block diagonal) conditions on code matrices $X_t$0 over $X_t$1 transmit antennas, ensuring the equivalent Gramian is block-diagonal.
• Distributed extension (PCRC): In cooperative amplify-and-forward with partially-coherent relays (backward phase CSI only), additional “three-relay” block-diagonal constraints ensure that both signal and noise covariances decouple, preserving single-symbol ML decodability.
• Code constructions: Alamouti $X_t$2, coordinate-interleaved ODs $X_t$3, and stacked ROD-based complex ODs (rate $X_t$4, any $X_t$5) meet the necessary and sufficient conditions for SSD in the distributed partially-coherent setting.

This alternative use of SSDPT highlights its relevance for signal processing in communication systems, independent of anomaly detection.

7. Practical Considerations and Design Trade-offs

For anomalous sound detection (Bai et al., 2022):

• Hyperparameters: $X_t$6 DPT blocks, $X_t$7, $X_t$8 for train (test), embedding $X_t$9, $P \in \{64, 128, 256\}$0 MHSA heads, FFN $P \in \{64, 128, 256\}$1.
• Optimization: AdamW with $P \in \{64, 128, 256\}$2 initial learning rate.
• Masking: $P \in \{64, 128, 256\}$3 patches of $P \in \{64, 128, 256\}$4 for the self-supervised task.

For distributed STBCs in PCRC (0708.3019):

• Optimal SSD-ML: Satisfy both co-located and three-relay constraints; phase compensation at relays raises rate from $P \in \{64, 128, 256\}$5 to $P \in \{64, 128, 256\}$6.
• Coding gain: Sub-optimal SSD (neglecting noise covariance) retains full diversity but loses $P \in \{64, 128, 256\}$7–$P \in \{64, 128, 256\}$8 dB in coding gain.
• Design guidelines: Choose codes satisfying all algebraic decoupling constraints; verify additional PCRC constraints for best performance.

In both contexts, the unifying theme is the exploitation of algebraic structures (Transformer or code orthogonality) and self-supervision (masking or phase compensation) to provide efficient, robust modeling and inference.