The Cluster Hypothesis in Information Retrieval

Semi-Supervised Learning in Network-Structured Data via Total Variation Minimization

“…Another consequence of the duality result [41, Thm. 31.3] is a powerful characterization of the solutions of the primal (16) and dual problem (17). In particular, a pair of vectorŝ x ∈ R N ,ŷ ∈ R E are solutions to the primal (16) and dual problem (17), respectively, if and only if…”

“…Given any solutionŷ ∈ R E to the dual problem (17), any solutionx to the primal problem (16) and, in turn, to TV minimization (12) must be such that conditions (22) are satisfied. The optimality condition (22) is the launching point for a primal-dual method for solving (12) (see Section IV).…”

“…In particular, y solves (17) if and only if the flow f (y) , defined edge-wise by f (y) e = W e y e , solves (26).…”

“…For the optimal flowf define the dual vector y = f e /W e . According to Proposition 1, y is a solution to the dual problem (17). For this particular (optimal) dual vector y, any solutionx to TV minimization has to satisfy the optimality condition (22).…”

“…Indeed, many SSL methods rely on a cluster assumption: labels of close-by data points are similar [5], [10]- [12]. This assumption is at the heart of many successful methods in graph signal processing [13], imaging [14], trend filtering [15], anomaly detection [16], information retrieval [17], and social networks [4]. We implement this Parts of this work have been presented in the conference paper [1].…”