## 4 Problem Definition  
Let us suppose that we have \(N\) different Wi-Fi CSI data source distributions such that \(D=\{D_{1},D_{2},D_{3},\ldots,D_{N}\}\), and their probability distributions are represented as \(\{P_{i}(X_{s_{i}},Y_{s_{i}})\}_{i=1}^{N}\). These source distributions have distinct features in terms of sensed environmental characteristics taken from \(N\) heterogeneous spaces for which source distributions are not uniform, i.e., \(D_{1}\
eq D_{2}\
eq D_{3}\ldots\
eq D_{N}\). Each source domain is equipped with labels in such a way that \(T_{s}=\{(X_{s_{i}},Y_{s_{i}})\}_{i=1}^{N}\), where \(X_{s_{i}}=\{x_{s_{i}}^{j}\}_{j=1}^{M_{s}}\) denotes the \(j\)-\(th\) sample of the \(i\)-\(th\) source domain, having a total of \(M_{s}\) data samples in the underlying source domain and \(Y_{s_{i}}=\{y_{s_{i}}^{j}\}_{j=1}^{M_{s}}\) is the corresponding ground-truth label. A transfer learning task is to match these diverse source distributions to a target distribution \(P_{t}(X_{t},Y_{t})\), with domain data \(T_{t}=\{(X_{t},Y_{t})\}\) where \(X_{t}=\{x_{t}^{j}\}_{j=1}^{M_{t}}\) having no ground-truth labels \(\{y_{t}^{j}\}_{j=1}^{M_{t}}\). This target domain is distinct from all the source domains, i.e., \(D_{t}\
eq\{D_{1},D_{2},D_{3},\ldots,D_{N}\}\). Suppose that these diverse source domains are mapped into a common feature space \(D_{s}\) and align with the target domain \(D_{t}\) using traditional domain generalization methods. Such adversarial learning would mainly focus on common domain-invariant representations coming from all domains, which is not an optimal solution to generalize these diverse domains because of systems' inability to align myriad domain-invariant multi-source data features in a common feature space. This is likely to lead to poor performance in the domain-shifting Wi-Fi sensing tasks.  
This problem definition leads to the research question addressed in this paper: how best to align wireless sensor data for multi-feature domain distributions continuously to a new domain with no label observations so that the system can maintain high accuracy regarding heterogeneous domain discrepancies. A reasonable solution is an unsupervised network with multiple subnetworks, each used for mapping a pair of source and target domains into specific feature spaces and matching their distributions. This would entail minimizing the distance between domain-specific feature spaces via measuring the difference between two probability distributions, then utilizing domain-specific classifiers to classify each domain label before finally aligning the domain-specific classifiers' outputs for the target domain via domain discrepancy minimization techniques near the domain-specific decision boundary [67]. Domain discrepancy minimization refers to the process of reducing the differences between a source domain and a target domain, each belonging to distinct data distributions, to improve model generalization across them. Additionally, reverting to point-wise surrogates of the source and target distribution distances and similarities might be effective in multi-domain generalization problems [78]. The main objective of this paper is to apply multi-feature unsupervised domain adaptation techniques discussed above to cross unaligned domain-specific distributions in device-free human activity classification so that the model can leverage full benefits from a multi-source environment for an optimal solution of the problem, to investigate whether the multi-source unsupervised domain adaptation (MUDA) setup successfully matches diverse source distributions to a target domain, and to outmatch the classical single-source unsupervised domain adaptation (SUDA) methods used earlier as multi-source combined solutions.