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Õ

= − (26)

Where is the learning rate, and the learning rate and is the sample size.

3.2. TSA detection model
The overall structure of the VNN-DM model consists of two components: an encoder and a decoder. The
encoderisprimarilyresponsiblefordataextractionandrepresentation,whilethedecoderenhancesthemodel’s
representational capacity. This section provides a detailed explanation of the detection model’s structure.

3.2.1 Encoder
The encoder consists of Conv, PrimCap, ConvCap, and FCCap layers. Multiple convolutional layers are em-
ployed to extract features from the input data. When the measured data is fed into the convolutional layers,
the local feature detectors are activated by the rectified linear unit (ReLU) to capture the underlying features
of the data:

!
Õ
= ReLU · + (27)

Here, represents the weight matrix of the feature map in the -th capsule. For the fully connected capsule,
the input is two-dimensional, and the positional relationship between the data is retained, making it crucial in
ensuring the effective use of capsule networks for feature extraction.

The initial capsule feature vector is a high-dimensional compressed data vector combined with multiple pa-
rameters based on actual input data, maintaining the correlation between the amplitude and positional phase.
The relevant equation is:

!
Õ

= · · + (28)

Where is the coefficient calculated by the machine during dynamic routing. A fully connected capsule
network will predict the output of each capsule layer differently, requiring different weight matrices , to
normalize the prediction result of each capsule using the matmul function:

Due to the dimensional differences between the fully connected capsule layer and the convolutional capsule
layer, a pose matrix is required. This matrix is composed of the direction cosines of two different sets of
orthonormal basis vectors:

(29)
= ·

Here, (·) represents the matrix product between the prediction matrix and the corresponding matrix

in the capsule layers. Each = represents a class, while = denotes the probability of belonging

2
to that class. This allows for determining whether the measured values are normal or influenced by a TSA
attack.