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We propose a method that infers whether linear relations between twohigh-dimensional variables X and Y are due to a causal influence from X to Y orfrom Y to X. The earlier proposed so-called Trace Method is extended to theregime where the dimension of the observed variables exceeds the sample size.Based on previous work, we postulate conditions that characterize a causalrelation between X and Y. Moreover, we describe a statistical test and arguethat both causal directions are typically rejected if there is a common cause.A full theoretical analysis is presented for the deterministic case but ourapproach seems to be valid for the noisy case, too, for which we additionallypresent an approach based on a sparsity constraint. The discussed method yieldspromising results for both simulated and real world data.

Testing whether linear equations are causal: A free probability theory approach