Table 3 Three feasible approximate techniques for matrix-variate two-sample test

(a) Multivariate test per probe

If probes are independent, we make a multivariate test on each \([condition, \ sample]\) matrix slice per probe. Such a multivariate test can be implemented by Algorithm 1 or one of the methods given in Table 2

(b) LDA-based multivariate test

we make the map \(s_u(f)= \mathbf{s}^T {\mathop{ {\beta}}\limits^{\rightarrow}}\) onto \({\mathop{ {\beta}}\limits^{\rightarrow}}\) per probe f with \({\mathop{ {\beta}}\limits^{\rightarrow}}\) obtained by either FDA or learning-based methods in Table 2, and make a multivariate test on \([ s_u(f_1), \cdots , s_u(f_g)]^T\) to consider multiple probes \(f_1, \ldots , f_g\) jointly

(c) Bilinear MDA-based test

We make the matrix-variate discriminative analysis (MDA) (see Eq.(33) & Eq.(34) in Ref. Xu (2015a)) to obtain \(\mathbf{v}, {\mathop{ {\beta}}\limits^{\rightarrow}}\), based on which we test \({\mathop{ {\beta}}\limits^{\rightarrow}}= \mathbf{0}\) by a multivariate test on \(\mathbf{s}_v= \mathbf{v}^T \mathbf{S}\) given \(\mathbf{v}\) fixed and test \(\mathbf{v}= \mathbf{0}\) by a multivariate test on \(\mathbf{s}_{\mathop{ {\beta}}\limits^{\rightarrow}}= {\mathop{ {\beta}}\limits^{\rightarrow}}^T \mathbf{S} ^T\) given \({\mathop{ {\beta}}\limits^{\rightarrow}}\) fixed

Alternatively, we may make test on the scalar statistics \(s_{v{\mathop{ {\beta}}\limits^{\rightarrow}}}= \mathbf{v}^T \mathbf{S} {\mathop{ {\beta}}\limits^{\rightarrow}}\)