Abstract Existing Source-Free Domain Adaptation (SFDA) methods typically adopt the feature distribution alignment paradigm via mining auxiliary information (eg., pseudo-labelling, source domain data generation). However, they are largely limited due to that the auxiliary information is usually error-prone whilst lacking effective error-mitigation mechanisms. To overcome this fundamental limitation, in this paper we propose a novel Target Prediction Distribution Searching (TPDS) paradigm. Theoretically, we prove that in case of sufficient small distribution shift, the domain transfer error could be well bounded. To satisfy this condition, we introduce a flow of proxy distributions that facilitates the bridging of typically large distribution shift from the source domain to the target domain. This results in a progressive searching on the geodesic path where adjacent proxy distributions are regularized to have small shift so that the overall errors can be minimized. To account for the sequential correlation between proxy distributions, we develop a new pairwise alignment with category consistency algorithm for minimizing the adaptation errors. Specifically, a manifold geometry guided cross-distribution neighbour search is designed to detect the data pairs supporting the Wasserstein distance based shift measurement. Mutual information maximization is then adopted over these pairs for shift regularization. Extensive experiments on five challenging SFDA benchmarks show that our TPDS achieves new state-of-the-art performance. The code and datasets are available at https://github.com/tntek/TPDS .

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Let $K_{r+1}$ be a complete graph with $r+1$ vertices. The $r$-uniform hypergraph $K_{r+1}^+$ is obtained from $K_{r+1}$ by enlarging each edge of $K_{r+1}$ with $r-2$ new vertices disjoint from $V(K_{r+1})$ such that distinct edges of $K_{r+1}$ are enlarged by distinct vertices. Let $H$ be a $K_{r+1}^+$-free linear $r$-uniform hypergraph with $n$ vertices. In this paper, we prove that when $n$ is sufficiently large, the spectral radius $\rho (H)$ of the adjacency tensor of $H$ is no more than $\frac{n}{r}$, i.e., $\rho (H)\leq \frac{n}{r}$, with equality if and only if $r|n$ and $H$ is a transversal design, where the transversal design is the balanced $r$-partite $r$-uniform hypergraph such that each pair of vertices from distinct parts is contained in one hyperedge exactly. An immediate corollary of this result is that $ex_r^{lin}(n,K_{r+1}^+)= \frac{n^2}{r^2}$ for sufficiently large $n$ and $r|n$, where $ex_r^{lin}(n,K_{r+1}^+)$ is the maximum number of edges of an $n$-vertex $K_{r+1}^+$-free linear $r$-uniform hypergraph, i.e., the linear Turán number of $K_{r+1}^+$.