Fisher　discriminant　analysis　(FDA)　is　a　popular　method　for　supervised　dimensionality　reduction.　FDA　seeks　for　an　embedding　transformation　such　that　the　ratio　of　the　between-class　scatter　to　the　within-class　scatter　is　maximized.　Labeled　data,　however,　often　consume　much　time　and　are　expensive　to　obtain,　as　they　require　the　efforts　of　human　annotators.　In　order　to　cope　with　the　problem　of　effectively　combining　unlabeled　data　with　labeled　data　to　find　the　embedding　transformation,　we　propose　a　novel　method,　called　subspace　semi-supervised　Fisher　discriminant　analysis　(SSFDA),　for　semi-supervised　dimensionality　reduction.　SSFDA　aims　to　find　an　embedding　transformation　that　respects　the　discriminant　structure　inferred　from　the　labeled　data　and　the　intrinsic　geometrical　structure　inferred　from　both　the　labeled　and　unlabeled　data.　We　also　show　that　SSFDA　can　be　extended　to　nonlinear　dimensionality　reduction　scenarios　by　applying　the　kernel　trick.　The　experimental　results　on　face　recognition　demonstrate　the　effectiveness　of　our　proposed　algorithm.　©　2009　Acta　Automatica　Sinica.　All　rights　reserved.