## 1 Introduction
Given two images \(I_{1}\) and \(I_{2}\), and a model \(M\), parametrized by a set of weight and biases \(\theta\). The SCNN produces the distance between feature vectors according to Equation (2), where \(f(I_{1})\) and \(f(I_{2})\) are the outputs produced from the convolutional layers of the SCNN. \[M(\theta,I_{1},I_{2})=||f(I_{1})-f(I_{2})||_{1} \tag{2}\]  
where,  
* \(\theta\) represents the SCNN weights and biases;
* \(I_{1}\) and \(I_{2}\) represent the goal and state images, respectively;
* \(f\) denotes the convolutional feature encodings obtained from the SCNN. Let \(Y=0\) when inputs \(I_{1}\) and \(I_{2}\) represent the same class and \(Y=1\) otherwise. The SCNN attempts to learn a function such that \(M(\theta,I_{1},I_{2})\) is small when \(Y=0\) and large when \(Y=1\). Navigation in an environment is a fundamental capability of intelligent organisms [7]. This study aims to demonstrate how a mobile agent equipped with an onboard camera and a pre-trained Siamese convolutional neural network, can navigate from an arbitrary start state to a goal state. We assume that the goal/goal state is provided to the agent beforehand. In Equation (1) the sequence of rewards \(r\) is generated by starting at a state \(s\in S\) and following policy \(\pi\) until the episode terminates [13]. This value corresponds to the expected gain according to the specified criterion [8]. In this article we explore the use of a SCNN model to estimate a reward function that is able to guide a mobile robot equipped with an onboard camera from a start state to a goal state. For mapless navigation, a reward function defined by the distance between the agent and its goal is \(r_{t}=\frac{1}{dist(s_{t},G)}\). Substituting the new definition of reward into the definition of a states value, we obtain a value estimate that takes into account the distance between the agent and its goal (Equation (3)). \[V^{\pi}(s_{t})=\sum_{i=0}^{\infty}\gamma^{i}\frac{1}{dist(s_{t},G)}_{t+i} \tag{3}\]  
where,  
* \(\pi\) represents a policy; a function mapping states to actions;
* \(V^{\pi}(s_{t})\) denotes the agents value estimate; how much reward can be expected from starting in state \(s_{t}\) and following policy \(\pi\);
* \(dist(s_{t},G)\) represents the Euclidean distance between the current state and the goal state. Equation (3) provides a value estimate that is large when the distance between the agent and its goal is small, and small as the agent moves further away from its goal; this definition of value assumes that the distance between the agent and its goal \(dist(s_{t},G)\) can be found at every timestep \(t\). In this work, we demonstrate modifying the value estimate by replacing the reward estimate with the similarity estimate produced in Equation (2) to obtain Equation (4). This provides us with a 'distance' estimate that does not rely on localization techniques, and it is accessible at every \(t\) and can guide our agent towards its goal \(G\). \[V^{\pi}(s_{t})=\sum_{i=0}^{\infty}\gamma^{i}\frac{1}{M(\theta,s_{t},G)}_{t+i} \tag{4}\]  
The main technical contributions of this paper can be summarized as follows:  
* Identify a reformulation of an agent's value function that takes into account the distance between the agent and its goal;
* Propose the use of a SCNN to estimate the distance between an agent and its goal;
* We demonstrate that the distance function can used to guide the agent towards its goal;
* We demonstrate that given a relatively small sample size for training, the Siamese convolutional neural network is able to outperform state-of-the-art convolutional neural networks pre-trained on large samples with complex architectures. In this study, we make use of ResNet18 and a KNN baseline model for comparison. The rest of the article is structured as follows: Section 2 provides an overview of the recent related work from the mapless navigation literature. In Section 3, a discussion of the materials and methods used in this study is presented.