Biometric Authentication: A Machine Learning Approach

3.4. Doubly-Stochastic EM

This section presents an EM-based algorithm for problems that possesses partial data with multiple clusters. The algorithm is referred to as as a doubly-stochastic EM. To facilitate the derivation, adopt the following notations:

• X = {x_t ∊ R^D; t = 1,..., T } is a sequence of partial-data.

• Z = {z_t ∊ C; t = 1, . . . , T} is the set of hidden-states.

• C = {C⁽¹⁾, . . . _,C(^J)}, where J is the number of hidden-states.

• Г = {γ⁽¹⁾⁾, . . . , γ^(K)} is the set of values that x_t can attain, where K is the number of possible values for x_t.

Also define two sets of indicator variables as:

and

Using these notations and those defined in