Nonlinear Independent Component Analysis for Continuous-Time Signals

A common problem in science and engineering is that an observed quantity, X, is determined by an unobserved source, S, which one is interested in. Denoting by f the deterministic relationship between X and S, one thus arrives at the equation (1) X f(S) where X is known but both the relation f and the source S are unknown. The premise that the data X is determined by its source S reflects in the assumption that f is a deterministic function, while the premise that S can be completely inferred from X -- i.e. that no information be lost in the process of going from S to X -- is reflected in the assumption that the function f is one-to-one; for simplicity, it is typically also assumed that f is onto. Any function f of this kind will be referred to as a mixing transformation. The central challenge, known as the problem of Blind Source Separation (BSS), then becomes to infer -- or'identify' -- the hidden source S from the given data X: Under which assumptions is it possible to recover the source data S in (1) if only (2) its mixture X is observed? To what extent can such a recovery be achieved and how can it be performed in practice? It is clear that without additional assumptions, the above problem of inference (2) is severely underdetermined: If X and equation (1) is the only information available but both f and S are unknown, then we may generally find infinitely many possible'explanations' ( S, f) for X which all satisfy (1) but are not otherwise meaningfully related to the true explanation (S, f) underlying the data. In many cases, however, this'indeterminacy of S given X with f unknown' can be controlled by imposing certain statistical conditions on the source S . The following simple example illustrates this situation.