## 3 Proposed Model
### Similarity Updating  
The entire network was smoothed through the aggregation of a convolution operation. Ultimately, as illustrated in Figure 2, it was produced via the convolution layer as well as the polynomial kernel.  
Every trainable characteristic is changed in the deleterious direction of the gradient with an adjustable step size given by a hyperparameter collectively called rate, and the gradients of the weight vector give the direction in which the function does have the sharpest rate of rise. The gradients are either of both the losses with regard to every learnable variable and a parametric iteration is written as follows in Equation (21) as well as Equation (22):  
\[x(l)=x(l)-\beta\frac{\partial J(x,a)}{\partial x(l)}=x(l)-\beta\sum_{i=1}^{n} \frac{\partial J(x,a;r(i),s(i)}{\partial x(l)}-\Omega x \tag{21}\]  
\[y(l)=y(l)-\beta\frac{\partial J(x,a)}{\partial x(l)}=y(l)-\beta\sum_{i=1}^{n} \frac{\partial J(y,a;r(i),s(i)}{\partial y(l)} \tag{22}\]  
Figure 3 demonstrates the working flow of the proposed AUTO-NN for forecasting.  
Where, \(\beta\) is the parametric update rate, \(x\) and \(y\) are the weighted sum and offset vector for each tier, correspondingly, and \((a(i),b(i)),1\leq i\leq N\) is a collection of samples Assuming the softmax has K neurons, with both the jth neurons assessing the prediction probability of class \(j\), provided the participation of \(xH\), which is the past layer's output, and related to loads \(W(j)\) s as well as predilection \(b(j)\), as shown in Equation (23):  
\[P\bigg{(}y=\frac{j}{x}\bigg{)}=\frac{exp(bb_{s}^{j}+X_{HH}^{T}W_{s}^{(j)})}{ \sum_{k=l}^{k}exp(bb_{s}^{(k)}+X_{H}^{T}~{}W_{S}^{(k)})} \tag{23}\]  
wherein, \(xH\) is the previous layer's payout. The created providing individualized expectations as shown in Equation (24) based on the probability assessment:  
\[f(x)=argmax_{1\leq j\leq k}p(y=\frac{j}{x}) \tag{24}\]  
Over the large, statistical gradient decline greatly improved the variables 1, 2,..., \(H\), \(s\) of the log-likelihood tragedy over the testing phase St. Parallel to ERReLU, our focus in this working framework, the irregularity factors (\(V\), \(H\)) take esteem issues (\(v\), \(h\)) \(\in\) {0, 1}\(m\)+, and the combined probability distribution underneath the model. It is offered by the Gibbs circulatory with the vitality task presented in circumstance as shown in Equations (25) and (26)  
\[p(v,h)=\frac{1}{z}e^{-Ec(v,h)} \tag{25}\]  
Figure 2: Architecture of AUTO-encoder-based Neural Network (AUTO-NN).  
\[E(v,h)=-\sum_{i=1}^{n}\sum_{j=1}^{m}w_{ij}h_{ii}V_{jj}-\sum_{j=1}^{m}b_{ii}V_{jj}- \sum_{j=1}^{nm}c_{ii}hi \tag{26}\]  
\(W_{jj}\) is a legitimate valued weight connected to the margin among components \(V_{jj}\) and \(H_{ii}\) for all \(i\) 1,..., \(n\) and \(jj\) 1,..., \(m\), and \(b_{ji}\) and \(c_{ii}\) are genuine esteemed inclination terms linked to the \(j\)th. The predictor built from an ERReLU network has only correlations seen between the layer of hidden and noticeable components but among two factors from the same layer. As shown in Equations (27)-(29), the restricted probability of a single parameter becoming one could be calculated as the termination frequency of a (probability) neurons having parabolic initiating employment:  
\[\sigma(x_{i})=\frac{1}{\left(1+e^{-xi}\right)} \tag{27}\]  
\[p(H_{i}=\frac{1}{v})=\sigma(\sum_{j=1}^{m}w_{ij}v_{j}+c_{i}) \tag{28}\]  
\[p(V=\frac{1}{h})=\sigma(\sum_{i=1}^{n}w_{ij}h_{j}+b_{i}) \tag{29}\]  
Gibbs sampling is particularly straightforward due to the independence of the components in a single layer: Rather than examining new qualities for all elements individually, the circumstances of all components in a thin layer can be simultaneously evaluated. As a result, Gibbs inspection can be done in just two phases: evaluating a state v for the obvious layer depending on \(p(v\,|\,h)\) as well as analyzing another state h for the input layers depending on \(p(h\,|\,v)\). Square Gibbs sample is another term for all of this.