approach is to treat Q as part of the model parameters and to consider it as a usual estimation problem. This is often difficult from the computational aspect in that Q is a discrete matrix living on a high dimensional space, in particular, Q ∈ { 0 , 1 } $^{J}$×$^{K}$. Even with a reasonably small number of items and a few attributes, this space is often too large to explore thoroughly by any exist- ing numerical method as the dimension grows exponentially fast with both J and K . Estimators developed based on this idea, even though theoretically sound, often suffer from substantial com- putational overhead. One of such instances is the maximum likelihood estimator of Q . Generally speaking, optimizing a discrete and nonlinear function over { 0 , 1 } $^{J}$×K is computationally intensive and sometimes infeasible. This approach does not take advantage of the special structures of the Q -matrix and further of the likelihood function. 

A different approach is to cast the Q -matrix estimation in the context of variable selection. Consider the item response function f$_{j}$ ( y | θ, α ). If both the response Y j and the latent variable α were observed, then the estimation of Q is a regular variable selection problem. In most situations, f$_{j}$ takes the form of a generalized linear model, in which the responses to items are the dependent variables, the attributes play the role of covariates, and the item parameters θ are the regression coefficients. Thus, the Q -matrix estimation is equivalent to a variable selection problem. However, in the context of latent class models, the covariates α ’s are all missing and therefore the task is, rigorously speaking, to select latent variables. Chen et al. [3] took this viewpoint and developed estimation methods for the Q -matrix via regularized likelihood. 

The last approach is similar to the previous one, but is more generic and is the primary focus of the current analysis. The introduction of the Q -matrix suggests that a single item usually does not provide information to differentiate all dimensions of the attribute profile. In particular, q$_{jk}$ = 0 means that item j is irrelevant to attribute k . Under the setting of latent class models (not necessar- ily possessing a specific parameterization), this corresponds to an item-specific partial information structure . Each particular attribute profile α in the DCM parameterization corresponds to one latent class. If an item does not differentiate all dimensions of α , then some distinct attribute 

