Enhancing Unmanned Aerial Vehicle Object Detection via Tensor Decompositions and Positive–Negative Momentum Optimizers

Abstract

The current development of machine learning has advanced many fields in applied sciences and industry, including remote sensing. In this area, deep neural networks are used to solve routine object detection problems, satisfying the required rules and conditions. However, the growing number and difficulty of such problems cause the developers to construct machine learning models with higher computational complexities, such as an increased number of hidden layers, epochs, learning rate, and rate decay. In this paper, we propose the Yolov8 architecture with decomposed layers via canonical polyadic and Tucker methods for accelerating the solving of the object detection problem in satellite images. Our positive–negative momentum approaches enabled a reduction in the loss in precision and recall assessments for the proposed neural network. The convolutional layer factorization reduces the shapes and accelerates the computations at kernel nodes in the proposed deep learning models. The advanced optimization algorithms achieve the global minimum of loss functions, which makes the precision and recall metrics superior to the ones for their known counterparts. We examined the proposed Yolov8 with decomposed layers, comparing it with the conventional Yolov8 on the DIOR and VisDrone 2020 datasets containing the UAV images. We verified the performance of the proposed and known neural networks on different optimizers. It is shown that the proposed neural network accelerates the solving object detection problem by 44–52%. The proposed Yolov8 with Tucker and canonical polyadic decompositions has greater precision and recall metrics than the usual Yolov8 with known analogs by 0.84–0.94 and 0.228–1.107 percentage points, respectively.

1. Introduction

One of the key problems in remote sensing, computer vision, and artificial intelligence is object detection in images. This problem arises in various scientific, health, and industrial areas, affecting the quality of modern intellectual systems. Since the 1990s, many researchers have been developing methods for solving the object detection problem. Avionics, radar, space radio, and telescope systems utilize expert system models to increase the accuracy of feature detection by researchers. Nowadays, the rapid development of UAV systems equipped with cameras requires more advanced technologies for object detection on Earth. Instead of expert systems, researchers and engineers prefer to utilize deep learning models. This technique can be found in various areas of human activity, such as remote sensing [1], geolocation [2], astronomy [3], and others.

Due to the development of deep neural networks, valuable gains can be made when using the features of convolutional neural networks (R-CNN) [4]. Such machine learning models contain multiple convolution, pooling, and full-dense layers. The convolution operation analyzes the patterns in the image by compressing it to digital data, which the multilayer perceptron block processes and transfers to backpropagation. Next, the loss function determines the distinction between the true and obtained results. Finally, the backpropagation algorithm updates the weight values in the convolutional and linear layers via gradient-based optimization algorithms. The expressivity and robustness of self-learning algorithms allow us to learn informative object representations without the need to design features manually. The R-CNN improved the quality of solving the object detection problem in images of arbitrary datasets. Since 2014, there have been many promising neural network architectures that detect objects in images relatively fast and with the required precision. Scholars often utilize the fast R-CNN for small object detection in optical remote sensing images. For example, the authors of [5] used SPP-Net to detect the object in speckle noise conditions. Multi-scale object detection in remote sensing imagery was solved using the FRCN model. However, one of the best models that solve the object detection problem is You Only Look Once (Yolov8) neural network [6]. The Yolov8 architecture proved its stability and performance in object detection in images and videos. Moreover, this model is suitable for working with data that have been noised.

1.1. Motivation

Modern neural network architectures that solve the object detection problem [7] with the required precision consume too much computational time and hardware resources. For the acceleration of neural network training, many researchers perform dataset configuration [8], such as compressing [9] and reducing [10]. To reduce the training time of deep neural networks, the authors of paper [11] made the computations using the Strassen and Strassen–Winograd matrix multiplication algorithm, thus obtaining higher performance and preserving precision. These methods made computation in convolutional layers faster. However, such an approach consumes even more hardware resources. In one study [12], the authors use the Ozaki matrix multiplication method in convolutional layers. This approach reduces the computational and time costs. However, it cannot maintain the required precision. Therefore, one needs an approach that may accelerate deep neural network training while maintaining accuracy. One of the most suitable techniques is convolutional layer conversion by tensor decomposition. The idea of tensor-based neural networks [13] came from quantum computing, where the operation over wave functions is similar to analogs, which can be found in tensor analysis. In [14], the authors achieved a higher training speed of neural networks with a minimal loss of accuracy using tensor decompositions. The most utilized tensor decompositions are canonical polyadic (CP) and Tucker. Such approaches consider the convolutional layer as a three-order tensor and decompose it into a sublayer with smaller shapes. It reduces the computational costs and time consumption. However, it remains difficult to maintain the required precision and recall metrics. This problem can be solved by using advanced loss function optimization algorithms. The state-of-the-art (SOTA) stochastic gradient descent (SGD) and adaptive moment estimation (Adam) can achieve the required precision in the case of CP and Tucker decompositions. SGD finds the minimum, considering the gradient directions. Adam uses the exponential moving averages of the gradient and its square. These techniques do not necessarily attain the global minimum of loss function because they do not distinguish the local and global extremes. For example, such flaws can be observed in the minimization of Rastrigin and Rosenbrock test functions. Therefore, one needs to use the optimizer, which can converge to a global minimum for a lesser number of epochs. In papers [15,16], we developed positive–negative momentum approaches such as DiffPNM, YogiPNM, and PNMBeleif. DIffPNM let the ensemble learning model solve the pattern recognition problem in satellite images from the UC-Merced dataset. YogiPNM and PNMBelief allowed the multi-modal neural network to solve the skin lesion recognition problem better than existing analogs. In this work, we propose the Yolov8 architectures with the convolutional decomposition by CP-Astrid and Tucker-2 models and train them using DiffPNM, YogiPNM, and PNMBelief. We clearly demonstrate that the usual Yolov8 with SGD and Adam is inferior in precision and recall properties to accelerated Yolov8 with positive–negative momentum approaches.

1.2. Our Contribution

The main contributions of our study can be summarized as follows:

(i) New Yolov8-CP-Astrid and Yolov8-Tucker-2 neural networks are proposed. Such models can train faster because of the decomposed convolutional layers.

(ii) We integrated our optimization methods DiffPNM, YogiPNM, and PNMBelief into the proposed Yolov8-CP-Astrid and Yolov8-Tucker-2 neural networks. This technique allowed us to minimize the precision and recall losses.

(iii) In the experimental part of the study, we applied the proposed Yolov8-CP-Astrid and Yolov8-Tucker-2 to solve the object detection problem on the DIOR dataset. Our models achieved greater precision and recall metrics than conventional Yolov8 models with SGD and Adam by 0.84–0.94 and 0.228–10.107 percentage points, correspondingly, and reduce the training time by 44–52%.

Thus, the proposed Yolov8-CP-Astrid and Yolov8-Tucker-2 neural networks allowed to improve the accuracy and rate of solving the object detection problem on the DIOR dataset. It should be noted that any deep neural network can be decomposed in several tensor decomposition models. The application of advanced optimization algorithms can minimize precision and recall losses.

The rest of the paper is organized as follows. In Section 2, Yolov8 architecture, tensor decompositions, and optimization methods are described. In Section 3, we propose the Yolov8-CP-Astrid and Yolov8-Tucker-2 neural networks. In Section 4, we demonstrate the results of solving the object detection problem in images from the DIOR dataset using known and proposed neural network architectures. Section 5 discusses the further improvements and applications of the proposed Yolov8-CP-Astrid and Yolov8-Tucker-2 neural networks. Finally, in Section 6, some conclusions on the obtained results of solving the object detection problem via the proposed Yolov8-CP-Astrid and Yolov8-Tucker-2 neural networks are given.

2. State of the Art

2.1. Yolov8 Architecture

The main feature of Yolo-based neural network architectures is the sequential utilization of special blocks: backbone, neck, and head. The backbone block is the core part of the YOLOv8. It extracts features from the input RGB color images. The neck block aggregates and processes the features of different scales extracted by the backbone network. This block embraces a feature pyramid network structure [6], which effectively fuses features of various scales to construct a more comprehensive representation. The head block is the main part of the YOLOv8 because it locates and identifies objects in the images. The output head contains several detectors, which predict the position and category of objects. The backbone, neck, and head blocks consist of the following modules: Conv, SPPF, BottleNeck, C2f, and Detect. The Yolov8 uses a similar backbone to Yolov5 but with a modified C2f module. The C2f module consists of the two-convolution cross-stage partial bottleneck and combines high-level features with contextual information to enhance detection accuracy. Also, Yolov8 engages the decoupled heads anchor-free model, which helps to handle object detection, classification, and regression tasks independently. Such an architecture permits each branch to focus on its specific task, which improves the overall model accuracy. The output layer of Yolov8 contains the sigmoid function SiLU for object scores, indicating the probability of an object being present within the bounding box. For the class probabilities representation, Yolov8 applies the softmax function, signifying the probability of an object belonging to each possible class.

One should note that Yolov8 considers three loss functions: bounded box, class, and distribution focal losses. Distributed focal is designed to raise object detection accuracy. Unlike standard loss functions, this type of loss focuses on hard-to-detect examples, assigning more weight to challenging instances and making it easier to learn from them. The bounded box loss analyzes the correctness of the model’s predicted boxes aligned with the objects in images. The class loss measures each predicted bounding box classification correctness. The network architecture of YOLOv8 is illustrated in Figure 1.

The Yolov8 model contains multiple convolutional layers: Conv2d in blocks Conv, SPPF, Bottleneck, C2f, and Detect. Therefore, to accelerate the training of the Yolov8, one needs to simplify the structure of layers. In this paper, we modify the Conv2d layers via tensor decompositions.

2.2. Tensor Decompositions

The application of tensor analysis to machine learning is a reasonable continuation of the artificial intelligence model development. The structure of such models satisfies the multilinear algebra properties, which allows the processing of more complex data. The conventional convolutional layers in neural networks can be identified via a low-rank tensor with the convolution operator. However, such a layer structure consumes a lot of computational time and resources. Therefore, one can use tensor decomposition methods to accelerate the corresponding convolutional neural networks.

Let us define the necessary notations and operations of tensor analysis [17]. Let $\underline{\mathbf{X}}=\left[x_{i_1,\cdots ,i_N}\right]\in {\mathbb{R}}^{I_1\times \cdots \times I_N}$ be an N-order tensor. The mode-n unfolding of $\underline{\mathbf{X}}$ is the rearrangement by mode-n fibers by constructing the columns of matrix ${\mathbf{X}}_n=\left[x_{i_n,j}\right]\in {\mathbb{R}}^{I_n\times {\prod }_{p\ne n}{I}_p}$ for $n\in \{1,\dots ,N\}$, where $j=1+{\sum }_{k=1,k\ne n}^N(i_k-1)j_k$ with $j_k={\prod }_{m=1,m\ne n}^{k-1}{I}_m$, and $i_n=1,\dots ,I_n$. In a general case, one reshapes a tensor $\underline{\mathbf{X}}$ into ${\mathbf{X}}_{\langle n\rangle }\in {\mathbb{R}}^{{\prod }_{p=1}^n{I}_p\times {\prod }_{r=n+1}^N{I}_r}$ ($x_{i_1,\cdots ,i_N}\to x_{i,j}$), where $i=1+{\sum }_{p=1}^n(i_p-1){\prod }_{m=1}^{p-1}{I}_m$ and $j=1+{\sum }_{r>n}^N(i_r-1){\prod }_{m=n+1}^{r-1}{I}_m$. The next useful tensor operation is the mode-$\left\{n\right\}$ product. Let $\mathbf{U}\in {\mathbb{R}}^{J\times I_n}$, then $\underline{\mathbf{Z}}=\underline{\mathbf{X}}{\times }_n\mathbf{U}$, where $\underline{\mathbf{Z}}=\left[z_{i_1,\cdots ,i_{n-1},j,i_{n+1},\cdots ,i_N}\right]\in {\mathbb{R}}^{I_1,\cdots ,I_{n-1},J,I_{n+1},\cdots ,I_N}$ and $z_{i_1,\cdots ,i_{n-1},j,i_{n+1},\cdots ,i_N}={\sum }_{i_n=1}^{I_n}{x}_{i_1,\cdots ,i_N}{u}_{j,i_n}.$ The canonical polyadic decomposition transforms an N-order tensor $\underline{\mathbf{X}}\in {\mathbb{R}}^{I_1\times \cdots \times I_N}$ into a linear combination of terms ${\mathbf{b}}_r^{\left(1\right)}\circ \cdots \circ {\mathbf{b}}_r^{\left(N\right)}$, which are first-order tensors such as

$$\underline{\mathbf{X}}\cong \sum _{r=1}^R{\lambda }_r{\mathbf{b}}_r^{\left(1\right)}\circ \cdots \circ {\mathbf{b}}_r^{\left(N\right)}$$

$$=\underline{\Lambda }{\times }_1{\mathbf{B}}^{\left(1\right)}{\times }_2\cdots {\times }_N{\mathbf{B}}^{\left(N\right)}.$$

(1)

where ${\lambda }_r$ are non-zero entries of the diagonal core tensor $\underline{\Lambda }\in {\mathbb{R}}^{R\times \cdots \times R}$ and ${\mathbf{B}}^{\left(n\right)}=[{\mathbf{b}}_1^{\left(n\right)},\dots ,{\mathbf{b}}_R^{\left(n\right)}]\in {\mathbb{R}}^{I_n\times R}$, and ∘ is the outer product. To clarify the concept, the CP decomposition of the three-order tensor is given in Figure 2.

As one can see from Figure 2, the three-order tensor $\underline{X}$ can be illustrated as the product of its three-order diagonalized core tensor $\underline{\Lambda }$ and $B^{\left(1\right)}$, $B^{\left(2\right)}$, and transposed $B^{\left(3\right)}$ matrices. Computing matrices and diagonalized core tensors separately is a less complex procedure than processing the full tensor format. The CP decomposition with the corresponding structure has the least computational complexity among other approaches.

Unlike the CP decomposition, the Tucker decomposition is a more general factorization of an Nth-order tensor into a small size, not-diagonalized core tensor and factors:

According to Figure 3, three-order tensor $\underline{X}$ is isomorphic to the product of its three-order core tensor $\underline{G}$ and $B^{\left(1\right)}$, $B^{\left(2\right)}$, and transposed $B^{\left(3\right)}$ matrices. Unlike the CP, Tucker decomposition contains full-format core tensor $\underline{G}$, which raises the computational complexity of the solution.

$$\underline{\mathbf{X}}\cong \sum _{r_1=1}^{R_1}\cdots \sum _{r_N=1}^{R_N}{g}_{r_1,\dots ,r_N}\left({\mathbf{b}}_{r_1}^{\left(1\right)}\circ \cdots \circ {\mathbf{b}}_{r_N}^{\left(N\right)}\right)$$

$$=\underline{\mathbf{G}}{\times }_1{\mathbf{B}}^{\left(1\right)}{\times }_2\cdots {\times }_N{\mathbf{B}}^{\left(N\right)},$$

(2)

$\underline{\mathbf{G}}\in {\mathbb{R}}^{R_1\times \cdots \times R_N}$ is the core tensor, and ${\mathbf{B}}^{\left(n\right)}=[{\mathbf{b}}_1^{\left(n\right)},\dots ,{\mathbf{b}}_{R_n}^{\left(n\right)}]\in {\mathbb{R}}^{I_n\times R_n}$ are the mode-n factor matrices, $n=1,\dots ,N$. In Figure 3, the Tucker decomposition of the three-order tensor is demonstrated.

Let $\underline{\mathbf{X}}\in {\mathbb{R}}^{I\times \cdots \times I}$ be a N-th order tensor, and R denote the rank of the decomposition. To better illustrate the difference between the full tensor format, CP, and Tucker-decomposed layer, the evaluation of computational complexity is given in

Table 1. Computational complexities of tensor decompositions.

Decomposition	Computational Complexity	References
Full Tensor Format	$O\left(I^N\right)$	-
Canonical Polyadic	$O\left(NIR\right)$	[18]
Tucker	$O(NIR+R^N)$	[19]
Hierarchical Tucker	$O(NIR+N{R}^3)$	[20]
Tensor Train	$O\left(NI{R}^2\right)$	[21]

.

According to the calculations given in Table 1, the CP decomposition possesses lower computational complexity than Tucker. This can be explained by the diagonalization $\underline{\Lambda }$ of the core tensor $\underline{\mathbf{G}}$ in Figure 2 and Figure 3. One should note that along with (1) and (2) decompositions, there exists a hierarchical Tucker (HT) [20] and tensor train (TT) [21]. These approaches require fewer hardware resources than Tucker decomposition and significantly accelerate the work of neural networks. However, the graph structure of HT and TT decompositions does not preserve the precision (accuracy) as the CP and Tucker approaches do. Also, if tensor decompositions reduce the number of computations, it will cause the precision loss. Therefore, it is necessary to use the positive–negative momentum optimization algorithms.

2.3. Optimization Algorithms

The quality of training in machine learning depends on the backward error propagation. In other words, precision, recall, and other metrics have high values if the loss function gives minimal values. The closer optimization algorithms descend to the global minimum, the higher the final precision is that can be achieved.

In paper [22], the authors introduced a regret-bound metric $R\left(T\right)$, which estimates the convergence rate of the optimization algorithm to the global minimum. The convergence rate metrics are defined as follows:

$$R\left(T\right):=\sum _{t=1}^T[f_t\left({\theta }_t\right)-f_t\left({\theta }^{*}\right)],$$

(3)

where ${\theta }^{*}=argmi{n}_{\theta }{\sum }_{t=1}^T{f}_t\left({\theta }_t\right)$, $f_t\left({\theta }_t\right)$ is a continuous function and $g_{1:T,i}=(g_{1,i},...,g_{T,i})\in {\mathbb{R}}^T$ is the gradient in $1,\dots ,T$ iterations and i-th dimension. The conventional optimization algorithms, namely, SGD and Adam, have the $R\left(T\right)$ as $O\left(\sqrt{T}\right)$. Such a metric means that the optimization algorithm is proper for machine learning models. Consider the Lebesgue norms $L^2\left(\mathbb{R}\right)$ and $L^{\infty }\left(\mathbb{R}\right)$, which have the corresponding norms:

$$\parallel g_{1:T,i}{\parallel }_2:={\left({\int }_{\mathbb{R}}{g}_{1:T,i}^{2}d{\theta }_{t,i}\right)}^{1/2}\le G,$$

(4)

$${\parallel {\theta }_{n,i}-{\theta }_{m,i}\parallel }_2\le D,$$

(5)

$$\parallel g_{1:T,i}{\parallel }_{\infty }:=\underset{{\theta }_{t,i}\in \mathbb{R}}{sup}\left(g_{1:T,i}\right)\le G_{\infty },$$

(6)

$${\parallel {\theta }_{n,i}-{\theta }_{m,i}\parallel }_{\infty }\le D_{\infty },$$

(7)

where $G,D,R_{\infty },D_{\infty }$ are positive real numbers and the indices are $m,n\in \{1,\dots ,T\}$. Next, we provide the positive–negative momentum optimization algorithm.

Let $f:\Omega \to \mathbb{R}$ be a smooth function over a closed convex set $\Omega \subset {\mathbb{R}}^n$ with $n\ge 2$ that contains one or more extremes. From [23], we are reminded that the most applicable optimization algorithm is stochastic gradient descent

$${\theta }_{t+1}={\theta }_t-{\alpha }_t\nabla f\left({\theta }_t\right),$$

(8)

where ${\theta }_t$ is a weight, ${\alpha }_t$ is a learning rate, $f\left({\theta }_t\right)$ is a loss function, and $\nabla f\left({\theta }_t\right)$ is a gradient. This algorithm still finds application in modern neural networks. However, such an optimizer does not achieve the global minimum neighborhood of the objective function. This fact has been checked using test Rastrigin and Rosenbrock functions. Moreover, considering only the learning rate slows the minimization process because of non-dynamical updates. For that reason, the authors proposed Adam with exponential moving averages.

The main feature of the Adam optimizer is the utilization of exponential moving averages of the gradient and its square. Such an approach considers not only gradient directions but also means of moments. The Adam algorithm can be described by the following iterative formula

$${\theta }_t={\theta }_{t-1}-{\displaystyle \frac{{\alpha }_t{\hat{m}}_t}{\sqrt{{\hat{v}}_t}+ϵ}},$$

(9)

where

$${\hat{m}}_t={\displaystyle \frac{m_t}{1-{\beta }_1^t}},{\hat{v}}_t={\displaystyle \frac{v_t}{1-{\beta }_2^t}},$$

$$m_t={\beta }_1{m}_{t-1}+(1-{\beta }_1)g_t,v_t={\beta }_2{v}_{t-1}+(1-{\beta }_2)g_t^2.$$

The parameters ${\beta }_1,{\beta }_2$ are called moments, designed for $m_t$ and $v_t$, respectively. Adam has the following regrets bound.

Theorem 1.

([22]). Let $\gamma >0$, ${\beta }_1,{\beta }_2$ are moments, ϵ is a vanishing parameter, $\eta ={\displaystyle \frac{{\beta }_1^2}{\sqrt{{\beta }_2}}}$, $\lambda \in (0,1)$, and $f_t$ is a continuous convex function, whose $g_t$ and ${\theta }_t$ with ${\theta }^{*}$ satisfy the conditions (4)–(7). Then, Adam has the following regret-bound assessment:

$$R\left(T\right)\le {\displaystyle \frac{D^2\sqrt{T}}{2\gamma (1-{\beta }_1)}}\sum _{i=1}^n\sqrt{v_{T,i}}$$

$$+{\displaystyle \frac{\gamma (1+{\beta }_1)G_{\infty }^3{G}^{-2}}{(1-{\beta }_1)\sqrt{1-{\beta }_2}{(1-\eta )}^2}}\sum _{i=1}^T{\parallel g_{1:T,i}\parallel }_2$$

$$+{\displaystyle \frac{D_{\infty }^2{G}_{\infty }\sqrt{1-{\beta }_2}}{2\gamma (1-{\beta }_1){(1-\lambda )}^2}}.$$

(10)

The exponential moment estimation accelerates the descent towards the extreme point. However, Adam cannot achieve the global minimum of objective function. Besides the local minimum avoidance problem, this optimizer does not handle the vanishing and exploding gradient issues, which impair the minimization process. This problem inspired us to propose optimization algorithms with positive–negative moment estimations. The DiffPNM was proposed in [15] as a majority voting ensemble learning model for increasing the accuracy of solving the pattern recognition problem in images from the UC-merced dataset. Such an approach is based on the positive–negative moment estimation, which is a more general technique than usual exponential moving averages. For considering the previous gradient values, DiffPNM utilizes the friction parameter

$${\xi }_t={\displaystyle \frac{1}{1+exp(-|g_t-g_{t-1}\left|\right)}},$$

which analyzes the nearest domain more precisely and broadly and helps to reach the global minimum of the loss function. The DiffPNM can be described as follows:

$${\theta }_t={\theta }_{t-1}-{\displaystyle \frac{{\alpha }_t{\xi }_t{\hat{m}}_t}{(\sqrt{{\hat{v}}_t}+ϵ)\sqrt{(1+{\beta }_0^2)+{\beta }_0^2}}},$$

(11)

where

$$m_t={\beta }_1^2{m}_{t-1}+(1-{\beta }_1^2)g_t,$$

$${\hat{m}}_t={\displaystyle \frac{(1+{\beta }_0)m_t-{\beta }_0{m}_{t-1}}{1-{\beta }_1^t}},$$

$$v_t={\beta }_2{v}_{t-1}+(1-{\beta }_2)g_t^2,$$

$$v_{max}=max(v_t,v_{max}),$$

$${\hat{v}}_t={\displaystyle \frac{v_{max}}{1-{\beta }_2^t}}.$$

This optimization algorithm contains the additional moment ${\beta }_0$, which helps to build a more accurate direction towards a global minimum of the loss function.

Theorem 2.

Let $\alpha >0$, ${\beta }_0,{\beta }_1,{\beta }_2$ are moments, ϵ is a vanishing parameter, ${\xi }_t$ is the friction coefficient, $\eta ={\displaystyle \frac{{\beta }_1^2}{\sqrt{{\beta }_2}}}$, $\lambda \in (0,1)$, and $f_t$ is continuous convex function, whose $g_t$ and ${\theta }_t$ with ${\theta }^{*}$ satisfy the regret bound conditions. Then, the DiffGrad has the following regret-bound assessment:

$$R\left(T\right)\le {\displaystyle \frac{D^2\sqrt{T}}{2\alpha (1-{\beta }_0){(1-{\beta }_1)}^2}}\sum _{i=1}^d(1+exp(-\left|g_{1,i}\right|\left)\right)\sqrt{{\hat{v}}_{T,i}}$$

$$+{\displaystyle \frac{\alpha (1+{\beta }^{\prime })G_{\infty }^3{G}^{-2}}{(1-{\beta }_0){(1-{\beta }_1)}^2{(1-\eta )}^2\sqrt{1-{\beta }_2}}}\sum _{i=1}^d{\parallel g_{1:t,i}\parallel }_2$$

$$+{\displaystyle \frac{D_{\infty }^2{G}_{\infty }\sqrt{1-{\beta }_2}}{2\alpha }}\sum _{i=1}^d{\displaystyle \frac{{\beta }^{\prime }}{(1-{\beta }_0){(1-{\beta }_1)}^2{(1-\lambda )}^2}}.$$

(12)

The YogiPNM described in paper [16] shows the best results in solving the pattern recognition problem on skin lesion image datasets. Like DiffPNM, this optimization algorithm relies on the positive–negative moment estimation technique and adjusting the learning rate update. Such an optimizer has the transformed second moment $v_t$ by defining a more accurate gradient direction instead of considering that it does not contain the additional parameters. The YogiPNM is defined by the following formula:

$${\theta }_t={\theta }_{t-1}-{\displaystyle \frac{{\alpha }_t{\hat{m}}_t}{(\sqrt{{\hat{v}}_t}+ϵ)\sqrt{(1+{\beta }_0^2)+{\beta }_0^2}}},$$

(13)

where

$$m_t={\beta }_1^2{m}_{t-1}+(1-{\beta }_1^2)g_t,$$

$${\hat{m}}_t={\displaystyle \frac{(1+{\beta }_0)m_t-{\beta }_0{m}_{t-1}}{1-{\beta }_1^t}},$$

$$v_t={\beta }_2{v}_{t-1}+(1-{\beta }_2)sign(u_{t-1}-g_t^2)g_t^2,$$

$$v_{max}=max(v_t,v_{max}),$$

$${\hat{v}}_t={\displaystyle \frac{v_{max}}{1-{\beta }_2^t}}.$$

Theorem 3.

Let $\alpha >0$, ${\beta }_0,{\beta }_1,{\beta }_2$ be moments, ϵ a vanishing parameter, $\eta ={\displaystyle \frac{{\beta }_1^2}{\sqrt{{\beta }_2}}}$, $\lambda \in (0,1)$, and $f_t$ the continuous convex function, whose $g_t$ and ${\theta }_t$ with ${\theta }^{*}$ satisfy the regret bound conditions. Then, the YogiPNM has the following regret-bound assessment:

$$R\left(T\right)\le {\displaystyle \frac{D^2\sqrt{T}}{2\alpha (1-{\beta }_0){(1-{\beta }_1)}^2}}\sum _{i=1}^d\sqrt{{\hat{v}}_{T,i}}$$

$$+{\displaystyle \frac{\alpha (1+{\beta }^{\prime })G_{\infty }^3{G}^{-2}}{(1-{\beta }_0){(1-{\beta }_1)}^2{(1-\eta )}^2\sqrt{1-{\beta }_2}}}\sum _{i=1}^d{\parallel g_{1:t,i}\parallel }_2$$

$$+{\displaystyle \frac{D_{\infty }^2{G}_{\infty }\sqrt{1-{\beta }_2}}{2\alpha }}\sum _{i=1}^d{\displaystyle \frac{{\beta }^{\prime }}{(1-{\beta }_0){(1-{\beta }_1)}^2{(1-\lambda )}^2}}.$$

(14)

Like YogiPNM, PNMBelief [16] shows good results in skin disease classification problems. Its main feature is the learning rate adaptation considering the “belief” in the current gradient direction. There is a difference between PNMBelief and YogiPNM in parameters $v_t$ and $s_t$, which are defined as exponential moving averages of $g_t^2$ and ${(g_t-m_t)}^2$, respectively. Considering $s_t$ as the prediction of the gradient at the next time step, if the observed gradient greatly deviates from the prediction, then the algorithm distrusts the current observation and takes a small step; if the observed gradient is close to the prediction, it can be trusted and a large step can be taken.

$${\theta }_t={\theta }_{t-1}-{\displaystyle \frac{{\alpha }_t{\hat{m}}_t}{(\sqrt{{\hat{s}}_t}+ϵ)\sqrt{(1+{\beta }_0^2)+{\beta }_0^2}}},$$

(15)

where

$$m_t={\beta }_1^2{m}_{t-1}+(1-{\beta }_1^2)g_t,$$

$${\hat{m}}_t={\displaystyle \frac{(1+{\beta }_0)m_t-{\beta }_0{m}_{t-1}}{1-{\beta }_1^t}},$$

$$s_t={\beta }_2{s}_{t-1}+(1-{\beta }_2){(g_t-m_t)}^2{g}_t^2,$$

$$s_{max}=max(s_t,s_{max}),$$

$${\hat{s}}_t={\displaystyle \frac{s_{max}}{1-{\beta }_2^t}}.$$

Theorem 4.

Let $\alpha >0$, ${\beta }_0,{\beta }_1,{\beta }_2$ be moments, ϵ a vanishing parameter, $\eta ={\displaystyle \frac{{\beta }_1^2}{\sqrt{{\beta }_2}}}$, $\lambda \in (0,1)$, and $f_t$ the continuous convex function, whose $g_t$ and ${\theta }_t$ with ${\theta }^{*}$ satisfy the regret bound conditions. Then, the PNMBelief has the following regret-bound assessment:

$$R\left(T\right)\le {\displaystyle \frac{D^2\sqrt{T}}{2\alpha (1-{\beta }_0){(1-{\beta }_1)}^2}}\sum _{i=1}^d\sqrt{{\hat{v}}_{T,i}}$$

$$+{\displaystyle \frac{\alpha (1+{\beta }^{\prime })G_{\infty }^3{G}^{-2}\sqrt{1+logT}}{(1-{\beta }_0){(1-{\beta }_1)}^2{(1-\eta )}^2\sqrt{1-{\beta }_2}}}\sum _{i=1}^d{\parallel g_{1:t,i}\parallel }_2$$

$$+{\displaystyle \frac{D_{\infty }^2{G}_{\infty }\sqrt{1-{\beta }_2}}{2\alpha }}\sum _{i=1}^d{\displaystyle \frac{{\beta }^{\prime }}{(1-{\beta }_0){(1-{\beta }_1)}^2{(1-\lambda )}^2}}.$$

(16)

Positive–negative momentum optimization algorithms showed their ability to increase the quality of solving pattern recognition problems in multi-modal [16] and ensemble neural networks [15]. Considering the properties of DiffPNM, YogiPNM, and PNMBelief in an ensemble network containing factorized convolutional layers, we can decrease the precision loss while solving the object detection problem.

3. Proposed Yolov8-CP-Astrid and Yolov8-Tucker-2 Neural Networks

The Yolov8 architecture consists of many convolutional layers in the backbone, neck, and head blocks, as shown in Figure 1. These three blocks contain the Conv, SPPF, Bootleneck, C2f, and detect modules. The kernel in each Yolov8 convolution layer is represented by the fourth-order tensor $\underline{\mathbf{W}}\in {\mathbb{R}}^{T\times C\times D_1\times D_2}$, where T and C are the number of output and input channels, and $D_1$ and $D_2$ are the size of the filter. In many deep convolutional neural networks, kernels process much redundant information. Therefore, we approximate the kernel tensors by lower-rank structures obtained from various tensor decompositions. To accelerate the training process, we decompose these layers into blocks of sub-layers of a smaller shape. The decomposition of the Conv2d layer is made using the CP-Astrid [24] and Tucker-2 schemes [25], shown in Figure 4 and Figure 5.

In the Yolov8 network, we factorize the Conv2d convolutional layers by CP-Astrid and Tucker-2 decompositions. Next, we substitute the compressed Conv2d layer in Conv, Bootleneck, SPPF, C2f, and Detect blocks from Figure 1. One may use CP-Lebedev and Tucker-2-CP in Yolov8, but these factorization models reduce many more computations that negatively impact the preservation of precision (accuracy). Considering the Figure 4, the full tensor $\underline{\mathbf{X}}\in {\mathbb{R}}^{T\times C\times D_1\times D_2}$ is Conv2d layer. In the case of CP decomposition, we apply Astrid’s scheme to $\underline{\mathbf{X}}\in {\mathbb{R}}^{T\times C\times D_1\times D_2}$ and receive three sublayers ${\underline{\mathbf{X}}}_1\in {\mathbb{R}}^{R\times C\times 1\times 1}$, ${\underline{\mathbf{X}}}_2\in {\mathbb{R}}^{R\times R\times D_1\times D_2}$, and ${\underline{\mathbf{X}}}_3\in {\mathbb{R}}^{T\times R\times 1\times 1}$. One should note that the tensor $\underline{{\mathbf{X}}_2}\in {\mathbb{R}}^{R\times R\times D_1\times D_2}$ is the tensor product of ${\underline{\mathbf{X}}}_2^{\left(1\right)}\in {\mathbb{R}}^{R\times R\times D_1\times 1}$ and ${\underline{\mathbf{X}}}_2^{\left(2\right)}\in {\mathbb{R}}^{R\times R\times 1\times D_2}$. In [26], the authors noticed this fact and proposed the CP-Lebedev decomposition, which is based on factorization of Conv2d tensor $\underline{\mathbf{X}}$ into ${\underline{\mathbf{X}}}_1\in {\mathbb{R}}^{R\times C\times 1\times 1}$, ${\underline{\mathbf{X}}}_2^{\left(1\right)}\in {\mathbb{R}}^{R\times R\times D_1\times 1}$, ${\underline{\mathbf{X}}}_2^{\left(2\right)}\in {\mathbb{R}}^{R\times R\times 1\times D_2}$, and ${\underline{\mathbf{X}}}_3\in {\mathbb{R}}^{T\times R\times 1\times 1}$. In the case of Tucker decomposition, we use the Tucker-2 layer decomposition model, which divides $\underline{\mathbf{X}}\in {\mathbb{R}}^{T\times C\times D_1\times D_2}$ into ${\underline{\mathbf{X}}}_1\in {\mathbb{R}}^{R1\times C\times 1\times 1}$, ${\underline{\mathbf{X}}}_2\in {\mathbb{R}}^{R2\times R1\times D_1\times D_2}$, and ${\underline{\mathbf{X}}}_3\in {\mathbb{R}}^{T\times R2\times 1\times 1}$. Using the CP-Astrid scheme to the sublayer ${\underline{\mathbf{X}}}_2\in {\mathbb{R}}^{R2\times R1\times D_1\times D_2}$, we obtain ${\underline{\mathbf{X}}}_2^{\left(1\right)}\in {\mathbb{R}}^{R^{\prime }\times R1\times 1\times 1}$, ${\underline{\mathbf{X}}}_2^{\left(2\right)}\in {\mathbb{R}}^{R^{\prime }\times R^{\prime }\times D_1\times D_2}$, and ${\underline{\mathbf{X}}}_2^{\left(3\right)}\in {\mathbb{R}}^{R2\times R^{\prime }\times 1\times 1}$, which transforms the Tucker decomposition into Tucker-CP analog [27]. In practice, it is usually considered that $D=D_1=D_2$. Let $I_1$ and $I_2$ be the height and width of the input image, respectively. Suppose ${\hat{I}}_1$ and ${\hat{I}}_2$ are the reduced height and width after using the convolution operation. For clearness, we list the computational complexities of the layer factorizations in

Table 2. Computational complexity of convolutional layer factorization methods.

Layer Factorization	Computational Complexity	References
Full convolution	$O(CT{D}^2{I}_1{I}_2)$	-
CP-Astrid	$O(R(C{I}_1{I}_2+D{I}_1{I}_2+D{\hat{I}}_1{I}_2+T{\hat{I}}_1{\hat{I}}_2))$	[24]
CP-Lebedev	$O(R(C{I}_1{I}_2+D^2{I}_1{I}_2+T{\hat{I}}_1{\hat{I}}_2))$	[26]
Tucker-2	$O(C{R}_1{I}_1{I}_2+R^{\prime }(R_1{I}_1{I}_2+D^2{I}_1{I}_2+R_2{\hat{I}}_1{\hat{I}}_2)+T{R}_2{\hat{I}}_1{\hat{I}}_2)$	[25]
Tucker-2-CP	$O(R_1{I}_1{I}_2+R_{2}D(R_1{I}_1{I}_2+R_3{\hat{I}}_1{I}_2)+R_{3}T{\hat{I}}_1{\hat{I}}_2)$	[27]

.

Let us substitute the blocks of the Yolov8 architecture, such as Conv, SPPF, BootleNeckm C2f, and Detect, in convolutional layer decomposition models shown in Figure 6 and Figure 7. The convolution can be presented as the mapping $\underline{\mathbf{X}}\in {\mathbb{R}}^{T\times C\times I}\to \underline{\mathbf{Y}}\in {\mathbb{R}}^{T^{\prime }\times C^{\prime }\times O}$ by the following formula:

$$y_{t^{\prime },c^{\prime },o}=\sum _{t=1}^T\sum _{c=1}^C\sum _{i=1}^I{w}_{t-t^{\prime }+1,c-c^{\prime }+1,i,o}{x}_{t,c,i},$$

where T is the height, C is the width, and I is the number of channels of the input data. Analogically, $T^{\prime }$ is the height, $C^{\prime }$ is the width, and O is the number of channels of the output data. Next, we decompose the tensor $\underline{\mathbf{W}}$ via the Tucker-2 model and obtain the following equation:

$$w_{t,c,i,o}=\sum _{r_1=1}^{R_2}\sum _{r_2=1}^{R_2}\sum _{r_3=1}^{R_3}{g}_{r_1,r_2,r_3}{b}_{t,r_1}^{\left(1\right)}{b}_{c,r_2}^{\left(2\right)}{b}_{o,r_3}^{\left(3\right)},$$

(17)

where $\left[g_{r_1,r_2,r_3}\right]=\underline{\mathbf{G}}$ is the core tensor of shape $(R_1\times R_2\times R_3)$ and $\left[b_{t,r_1}^{\left(1\right)}\right]={\mathbf{B}}^{\left(\mathbf{1}\right)}\in {\mathbb{R}}^{T\times R_1}$, $\left[b_{c,r_2}^{\left(2\right)}\right]={\mathbf{B}}^{\left(\mathbf{2}\right)}\in {\mathbb{R}}^{C\times R_2}$, and $\left[b_{o,r_3}^{\left(3\right)}\right]={\mathbf{B}}^{\left(\mathbf{3}\right)}\in {\mathbb{R}}^{O\times R_3}$ are the factor matrices. Next, we incorporate the convolutional layer decomposition (17) into the Yolov8 architecture (Figure 1). We will denote such an architecture as the Yolov8-Tucker-2 neural network.

In the case of CP-Astrid decomposition, we have the following decomposition of $\underline{\mathbf{W}}$:

$$w_{t,c,i,o}=\sum _{r_1=1}^R\sum _{r_2=1}^R\sum _{r_3=1}^R{\lambda }_{r_1,r_2,r_3}{b}_{t,r_1}^{\left(1\right)}{b}_{c,r_2}^{\left(2\right)}{b}_{o,r_3}^{\left(3\right)},$$

(18)

where the core tensor $\underline{\Lambda }=diag\left(\underline{\mathbf{G}}\right)$. Next, we use the convolutional layer decomposition at each module of the Yolov8 architecture. We call this model Yolov8-CP-Astrid.

The proposed Yolov8-Tucker-2 and Yolov8-CP-Astrid architectures are designed to accelerate the solution of the object detection problem. However, it is necessary to reduce the precision and recall losses caused by simplifying the convolutional layers at each Yolov8 module. We address this issue with positive–negative momentum optimizers DiffPNM, YogiPNM, and PNMBelief. The majority of modern neural networks utilize conventional optimization algorithms, such as SGD and Adam. These approaches are still relevant for loss function minimization, which can be found in the backpropagation process. However, these optimization algorithms do not guarantee the achievement of the global minimum of the loss function. Therefore, it is necessary to use DiffPNM, YogiPNM, and PNMBelief, which operate with positive–negative momentum estimation. This technique has proven its superiority over existing basic optimization algorithms in [15] and enables us to achieve the global minimum for a smaller number of epochs (iteration), which speeds up the training process.

4. Results

In this section, we examine the training process of the compressed Yolov8 neural network, containing positive–negative momentum optimizers and tensor decomposition models. The training process is investigated while solving the object detection problem in images from the DIOR dataset.

For the program implementation of the Yolov8 architecture and positive–negative momentum optimizers, we use the PyTorch 1.11 library of Python 3.9.3. For the realization of CP and Tucker decompositions, we use the TensorLy library [28].The splits were originally 50% train and 50% test sample. We made a few changes by keeping the 50% train for only training and splitting the test dataset to 20–80% for validation and testing parts. The number of epochs equals 20. The learning rate $\alpha =0.001$, the moments ${\beta }_0=0.9,{\beta }_1=0.999,{\beta }_2=0.999$, $ϵ=1\times {10}^{-8}$, and the weight decay is 0. For better reinforcement, we use weight decoupling and the amsgrad procedure. The training process is implemented in NVIDIA Tesla T4, PCIe 3.0, 16 GiB GDDR6, 256 bit, and GPU 585 MHz. The rank of CP and Tucker decomposition is 100. In

Table 3. The results of object detection problem solving by conventional and decomposed Yolov8n models. Bold font indicates the best results.

Neural Network	Optimizer	Box Loss	Cls Loss	Dfl Loss	P (%)	R (%)	Time (s)	FPS
Yolov8n	SGD	1.34	0.721	1.167	81.945	59.774	24,119.94	69.98
	Adam	1.131	0.743	1.033	82.045	60.653	24,177.19	70.14
	DiffGrad	0.978	0.611	0.992	84.586	62.109	24,273.14	71.32
	Yogi	0.984	0.640	0.986	84.416	61.025	24,192.47	71.65
	DiffPNM	0.894	0.428	0.930	86.305	63.841	24301.57	72.57
	YogiPNM	0.917	0.455	0.914	86.26	63.517	24,283.95	72.32
	PNMBelief	0.934	0.520	0.946	86.111	63.299	24,503.03	72.14
Proposed	SGD	2.893	1.523	1.922	73.108	55.489	11,464.56	61.64
	Adam	2.344	1.128	1.455	79.463	58.041	11,523.63	62.46
	DiffGrad	1.455	1.161	1.385	81.402	59.144	11,554.03	62.88
	Yogi	1.573	1.294	1.422	81.318	59.097	11,537.84	62.72
Yolov8n-CP-Astrid	DiffPNM	1.254	0.946	1.188	81.593	59.704	11540.17	63.32
	YogiPNM	1.017	1.005	1.225	82.050	60.109	11,501.01	63.58
	PNMBelief	1.308	1.120	1.427	81.356	59.110	11,576.46	63.19
Proposed	SGD	2.485	1.382	1.603	74.581	58.853	13,442.75	64.27
	Adam	2.144	1.176	1.367	81.833	59.647	13,849.85	65.11
	DiffGrad	1.346	0.870	1.184	82.113	60.309	13,884.55	65.73
	Yogi	1.372	0.845	1.150	82.124	60.428	138,762.30	65.79
Yolov8n-Tucker-2	DiffPNM	1.099	0.814	1.040	82.885	60.881	14,510.85	66.12
	YogiPNM	1.142	0.895	0.998	82.843	60.811	14,399.25	65.92
	PNMBelief	1.224	1.104	1.209	82.045	60.292	14,737.85	65.84

, we present a list of the results of solving the object detection problem in images from the DIOR dataset [29] by full tensor format and compressed Yolov8 models with SGD, Adam, DiffPNM, YogiPNM, and PNMBelief.

Considering the results shown in Table 3, one can see that the training of conventional Yolov8 models requires 24,119.94–24,503.03 s. In this case, the DiffPNM achieves the highest precision and recall values $86.305\%$ and $63.841\%$ with the least values of box, class, and distribution focal losses, which have the values $0.894$, $0.428$, $0.930$, respectively. For 24,283.95 s, the YogiPNM achieves precision and recall with values $86.262\%$ and $63.517\%$, respectively. The box, class, and distribution focal losses are $0.917$, $0.520$, and $0.946$. The PNMBelief lets the Yolov8 model solve the object detection problem in images from the DIOR dataset with lower precision and recall.

The SGD and Adam achieved the worst precision and recall, regardless of the smaller time costs. In the case of training the Yolov8-CP-Astrid model, the best precision and recall were achieved by YogiPNM with values $82.050\%$ and $60.109\%$ for 11,501.01 s. The DiffPNM and PNMBelief demonstrated the second and third results in precision and recall, respectively. The SGD and Adam achieved less precision and recall value because of the inferior loss function minimization. For the Yolov8-Tucker-2 model, the DiffPNM received the best precision and recall values, $82.885\%$ and $60.881\%$, respectively. The YogiPNM achieved almost the same results in precision and recall as DiffPNM, $82.843\%$ and $60.811\%$, respectively. The precision and recall achieved by PNMBelief are $82.045\%$ and $60.292\%$. From Figure 8 and Figure 9, one can see that the Yolov8-Tucker-2 with DiffPNM obtained almost the same results as Yolov8 with DiffPNM. However, the proposed neural network does not have distinct train stations and finds false vehicles on 21964 images. In image 21972, the proposed Yolov8-Tucker-2 does not find chimneys and accepts the false building as a train station. In image 21966, the proposed models detected the same bridge twice and discovered a false harbor. In other cases, the proposed Yolov8-Tucker-2 network detects the objects correctly.

Next, in Figure 10 and Figure 11, we demonstrate the per epochs via train and validation bounded box, class and distribution focal loss functions, precision, recall, mAP50(B), and mAP50-95(B).

Figure 12 demonstrates normalized confusion matrices obtained by testing Yolov8 with DIffPNM and Yolov8-Tucker-2 using DiffPNM for identifying detected objects in images from the DIOR dataset.

We considered expanding the study by solving the object detection problem on the VisDrone 2020 dataset, which contains 10 classes of objects. However, the images in this dataset often contain small objects, such as people, bicycles, and motorcycles, which are challenging to detect using considered approaches.

Analyzing the results given in

Table 4. The results of object detection by conventional and decomposed Yolov8n models. The bold font highlights best results.

Neural Network	Optimizer	Box Loss	Cls Loss	Dfl Loss	P (%)	R (%)	Time (s)	FPS
Yolov8n	SGD	1.433	0.964	1.309	77.753	28.192	9433.42	63.19
	Adam	1.295	0.922	1.214	78.603	29.348	9511.09	65.39
	DiffGrad	1.243	0.912	1.224	79.053	29.614	9589.48	66.42
	Yogi	1.250	0.918	1.217	78.044	29.582	9514.60	65.70
	DiffPNM	1.084	0.844	1.159	79.372	30.198	9610.80	68.23
	YogiPNM	1.126	0.884	1.190	79.344	30.084	9583.48	68.14
	PNMBelief	1.224	0.901	1.231	79.162	29.842	9587.94	67.64
Proposed	SGD	2.593	1.782	1.997	71.249	24.814	5806.05	55.14
	Adam	2.665	1.857	2.045	71.040	24.576	5842.63	56.05
	DiffGrad	2.375	1.686	1.863	71.704	25.430	5865.94	56.47
	Yogi	2.349	1.634	1.799	71.859	25.900	5851.10	56.55
Yolov8n-CP-Astrid	DiffPNM	2.074	1.576	1.704	72.571	26.463	5894.03	57.74
	YogiPNM	2.099	1.621	1.750	72.493	26.395	5884.74	57.10
	PNMBelief	2.154	1.690	1.803	72.111	26.245	5896.48	56.83
Proposed	SGD	2.328	1.652	1.740	73.746	26.704	6078.64	57.78
	Adam	2.362	1.670	1.748	73.583	26.585	6104.55	58.84
	DiffGrad	2.075	1.582	1.689	73.815	27.020	6154.81	59.53
	Yogi	2.136	1.648	1.703	73.784	26.913	6131.66	59.42
Yolov8n-Tucker-2	DiffPNM	1.928	1.594	1.644	74.306	27.822	6224.34	60.75
	YogiPNM	1.894	1.570	1.625	74.788	27.914	6166.30	60.32
	PNMBelief	1.983	1.609	1.651	74.156	27.417	6189.94	60.30

, one can notice that Yolov8 requires 9433.42–9587.94 s for training. The DiffPNM achieves the highest precision and recall values $79.372\%$ and $30.084\%$ with the lowest values of box, class, and distribution focal losses, which have the values $1.084$, $0.844$, $1.159$, respectively. Next, the YogiPNM receives precision and recall, $79.344\%$ and $30.084\%$, respectively. The box, class, and distribution focal losses have values of $1.126$, $0.884$, and $1.190$. The rest of the optimizers show results with greater values of losses and lower values of precision and recall. The Yolov8-CP-Astrid model with DiffPNM gives precision and recall with values $72.571\%$ and $36.463\%$ for $5894.03$ s. The YogiPNM and PNMBelief have the second and third results in precision and recall, respectively. The rest of the state-of-the-art optimizers provide less precision and recall values. In the case of the Yolov8-Tucker-2 model, the YogiPNM received the best precision and recall values, $74.788\%$ and $27.914\%$, respectively. The second and third results in precision and recall were achieved by DiffPNM and PNMBeleif. Other optimizers demonstrate the results with lower precision and recall. Several images with object detection frames are shown in Figure 13 and Figure 14.

Considering Figure 13 and Figure 14, one can see that the Yolov8-Tucker-2 with DiffPNM achieved almost the same results as Yolov8 with DiffPNM. However, the neural network was not able to distinguish between small vehicles and people, bicycles and pedestrians, and trucks and vans.

Next, in Figure 15 and Figure 16, we demonstrate the per epochs via train and validation bounded box, class and distribution focal loss functions, precision, recall, mAP50(B), and mAP50-95(B).

Figure 17 shows normalized confusion matrices resulting from testing Yolov8 with DIffPNM and Yolov8-Tucker-2 with DiffPNM, respectively, for identifying detected objects in images from the VisDrone 2020 dataset.

The proposed Yolov-CP-Astrid and Yolov8-Tucker-2 neural network with DiffPNM, YogiPNM, and PNMBeleif achieved better results than the conventional Yolov8 with SGD and Adam. The proposed neural network architectures are faster than the SOTA model by 44–52% and increase the precision and recall by 0.84–0.94 and 0.228–1.07 percentage points, respectively.

The proposed Yolov8n with tensor decompositions and advanced optimization algorithms is able to solve objected detection problems on DIOR and VisDrone-2020 datasets as successfully as state-of-the-art models do. Our model shows mAP $68.14\%$, which is greater than Bayes R-CNN [30], with the following backbones: ResNet-50, ResNext-50, ShuffleNet, MobileNet V3-L. However, RegNet and MRENet show mAP values of $71.77\%$ and $73.91\%$, respectively. The R-CNN model with comprehensive learning techniques attained the mAP of $74.37\%$ on the DIOR [31]. An accelerated Yolov8n by tensor decompositions outperforms the EfficientDet in mAP on DIOR dataset [32]. The majority of considered state-of-the-art neural networks do not solve the object detection problem on the DIOR dataset better than accelerated Yolov8n with a positive–negative momentum optimizer. Meanwhile, RegNet and MRENet with tensor decompositions can preserve the accuracy of solving object detection problems. Considering the models of this object detection problem in VisDrone 2020, many studies contain the training using Yolo-based models [33,34]. However, Yolov8n has many modifications that show greater precision and mAP, such as Yolov8s, Yolov8m, Yolov8s, and Yolov8x. As the next possible research focus, it could be useful to apply tensor decompositions and advanced optimization algorithms to later Yolo-based and other perspective ANN models.

5. Discussion

Based on the results given in Table 3, it can be claimed that the tensor decomposition technique accelerates the training of neural networks while solving the object detection problem on the DIOR dataset.

Table 5. Evolution of neural networks for solving object detection problem.

Neural Network	Key Idea	Year
R-CNN [35]	Region-based object segmentation	2017
AttentionNet [36]	Quantized weak directions,	2017
	ensemble of iterative predictions
TMESNN [37]	Tensor multi-mode	2018
	expansion sum nuclear norm
Yolov8 [6]	Backbone, neck, and head blocks	2023
IPTM [38]	Infrared patch-tensor model	2024
	with Tensor multi-mode
	expansion sum nuclear norm
Proposed Yolov8-CP-Astrid	CP-Astrid decomposition of	2024
	backbone, neck, and head blocks
Proposed Yolov8-Tucker-2	Tucker-2 decomposition of	2024
	backbone, neck, and head blocks

represents the evolution of artificial neural networks with application to object detection problem.

The use of such advanced optimization algorithms allows the minimization of the precision and recall losses, which happen because of the convolutional layer decomposition in Conv, SPPF, BootleNeck, C2f, and Detect in Yolov8.

The comparison of conventional Yolov8 with SGD or Adam with Yolov8-CP-Astrid or Yolov8v-Tucker-2 with positive–negative momentum approaches shows that these advanced architectures outperform existing ones in speed, precision, and recall. One of the known shortcomings is that the proposed model fails to correctly detect objects like train stations, chimneys, and buildings in the DIOR dataset. The same problem can be found in the VisDrone 2020 dataset, where the proposed model sometimes confuses people with small vehicles, and vans can be confused with trucks. This issue can be explained by analyzing the entire figure via conventional convolutional layers. The advanced optimizers increased the precision and recall of Yolov8n and its accelerated models by CP and Tucker decompositions. The solution to this problem is the utilization of a transformer neural network, based on the self-attention layer. Also, the ensemble learning techniques can enhance the final precision of solving object detection problems on multiclass datasets. This fact makes the foundation for further exploring the combination of optimization algorithms and tensor decompositions. For example, the utilization of hierarchical Tucker [20], ADA-Tucker [39], singular value [40], and tensor-train [21] decomposition in neural networks can simplify its structure with minimal loss of final precision. Moreover, it is possible to reinforce such models with optimization algorithms, such as fractional-order [41], information-geometric [42], hybrid [43], and other metaheuristic [44] approaches.

6. Conclusions

Further studies can be devoted to compressing not just the deep convolutional but physics-informed, complex-valued, quantum, and graph neural networks. Along with the CP-Astrid and Tucker-2 decomposition models, we will use the tensor-train, hierarchical Tucker, and singular value decomposition, with the intent to accelerate the neural network training with minimal loss of accuracy. We also plan to modify positive–negative approaches using fractional derivatives or combination with population-based approaches. By involving the Riemann–Liouville, Caputo, and Grunwald–Letnikov fractional rational derivatives [45], the optimization process can take a lower number of epochs, which may also reduce the training time. Similarly, the hybrid approaches, like AdaSwarm [46], can significantly improve the quality of neural network performance. The use of information-geometric optimization algorithms, natural [47], and mirror [48] gradient descent can integrate the classical tensor decomposition into quantum machine learning [49].