The 3579 Counting Rule in Sparse Factor Analysis

Problem Statement

At the moment, the package has a single exported function called counting_rule_holds with one input variable: a binary matrix δ ∈ {0, 1}^m × r. This binary matrix δ should be the sparsity pattern of the factor loading matrix β ∈ ℝ^m × r in sparse factor analysis: y_i = βf_i + ϵ_i, where y_i ∈ ℝ^m is the ith vector of observations, f_i ∈ ℝ^r is the ith vector of latent factors, ϵ_i ∈ ℝ^m is the ith vector of idiosyncratic errors, and i = 1, ..., n. In sparse factor analysis, the factor loading matrix β may be sparse, i.e., β may only have a small number of non-zero entries. Importantly, β may have (estimated) structural zeros, and the sparsity pattern is δ = I(β ≠ 0), where I() is the indicator function. For mathematical tractability, we assume orthogonal factors, i.e., cov(f_i) = I_r identity, and homoskedasticity for the observation series, i.e., cov(ϵ_i) is diagonal.

In this setup, the covariance matrix cov(ϵ_i) of the idiosyncratic errors may not be uniquely identified if there are too many zeros in β (and thus in δ). The 3579 counting rule is a sufficient condition for the uniqueness of the covariance matrix cov(ϵ_i). More information on the 3579 counting rule can be found in the paper “Cover It Up! Bipartite Graphs Uncover Identifiability in Sparse Factor Analysis” by Darjus Hosszejni and Sylvia Frühwirth-Schnatter at https://arxiv.org/abs/2211.00671.

The 3579 Counting Rule

Def. Let δ ∈ {0, 1}^m × r be a binary matrix. Then, the 3579 counting rule is the following condition: for all q = 1, ..., r, all submatrices of δ consisting of q columns of δ have at least 2q + 1 non-zero entries.

For example, for δ₁ below the counting rule does not hold because there is a q (namely, q = 1), such that there is a submatrix of δ₁ consisting of 1 column of δ₁ that has only 2q = 2 non-zero entries: the middle column. For δ₂, however, the counting rule holds because there is no q such that there is a submatrix of δ₂ consisting of q columns of δ₂ that has only 2q non-zero entries.

$$\delta_1=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{pmatrix}, \quad\quad\quad \delta_2=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$$

The counting_rule_holds Function

We can check whether the 3579 counting rule holds for a given binary matrix delta using the counting_rule_holds function in the sparvaride package.

library(sparvaride)

We define two matrices as above in R:

delta1 <-
  matrix(c(1, 0, 0,
           0, 1, 0,
           0, 0, 1,
           1, 1, 1,
           1, 0, 1,
           1, 0, 1,
           1, 0, 1),
         nrow = 7, ncol = 3,
         byrow = TRUE)
delta2 <-
  matrix(c(1, 0, 0,
           0, 1, 0,
           0, 0, 1,
           1, 1, 1,
           1, 0, 1,
           1, 1, 1,
           1, 0, 1),
         nrow = 7, ncol = 3,
         byrow = TRUE)

Then, we call the counting_rule_holds function on these matrices:

counting_rule_holds(delta1)
#> [1] FALSE
counting_rule_holds(delta2)
#> [1] TRUE

Citation

For citing our work, please check the citation function in R:

citation("sparvaride")
#> To cite sparvaride in publications use:
#> 
#>   Hosszejni D, Frühwirth-Schnatter S (2022). "Cover It Up! Bipartite
#>   Graphs Uncover Identifiability in Sparse Factor Analysis."
#>   doi:10.48550/arXiv.2211.00671
#>   <https://doi.org/10.48550/arXiv.2211.00671>, arXiv: 2211.00671.
#> 
#> A BibTeX entry for LaTeX users is
#> 
#>   @Unpublished{,
#>     title = {Cover It Up! Bipartite Graphs Uncover Identifiability in Sparse Factor Analysis},
#>     author = {Darjus Hosszejni and Sylvia Frühwirth-Schnatter},
#>     year = {2022},
#>     note = {arXiv: 2211.00671},
#>     doi = {10.48550/arXiv.2211.00671},
#>   }