# Confusion on Math Notation Definition

I attempt to understand the formulation of dictionary learning for this paper:

Part 2: Confusion on math notation definitions (Conflict with my understanding)

(Part 1 is in the other post)

From both papers that were published in reputable venues, they defined the dictionary learning as follow:

Given the original feature representation $$X = [x_1, ..., x_N] ∈ R^{M×N}$$ , dictionary learning aims to learn a set of latent concepts or feature patterns, $$D = [d_1,..., d_D] ∈ R^{M×D}$$ and a latent sparse representation $$A = [α_1, ..., α_N] ∈ R^{D×N}$$, with the following empirical cost:

$$\min_D \frac1N \sum_{n=1}^N \ell(x_n, D), s.t., ||d_k||_{\ell_{2}} \le 1, \forall k = 1, ... , D$$

And in (2), they defined:

Let $$X = [x_1, x_2, . . . , x_N ] ∈ Rn×N$$ be the collection of $$N$$ (normalized) training samples that are assumed to be statistically independent. Using the example in part 1, $$N$$ here is the observation which means the matrix looks like this:

         $$o_1$$ $$o_2$$ $$o_3$$ $$o_4$$
$$p_1$$      1     2     3     4
$$p_2$$      2     3     4     5
$$p_3$$      3     4     5     6
$$p_4$$      4     5     6     7
$$p_5$$      1     2     2     3
label      1     1     0     0


But since this is the dictionary learning problem, instead of a conventional machine learning one, it is okay to represent the matrix in any form. So basically, if I want it to be my conventional way, I can just transpose the matrix to the one I familiar, from $$R^{5×4}$$ transpose to $$R^{4×5}$$, right?

Their formulation:

$$X_{M\times N} = D_{M \times D} \times A_{D \times N}$$

To replicate their experiments in my own setting, the formulation becomes:

$$X_{N \times M} = A_{N \times D} \times D_{D \times M}$$

Given that:

$$X^T = (DA)^T = A^T \times D^T$$

$$X = [x_1, ..., x_N] ∈ R^{N×M}$$, with $$N$$ represents the number of observations, $$M$$ represents the number of features / predictors.

Right?

• Can please put your main question in the title? – nbro Sep 8 '20 at 12:15