BMF in UM of CMP

bmf, stated in its native language

Boolean matrix factorization (BMF) takes a binary matrix (X^{mn}) (optionally with a mask (M)) and finds binary factors (W^{mr}), (H^{rn}) minimizing $$\min_{W,H}\ |M\odot(X - W\circ H)|*F^2,$$ where the Boolean product is $$(W\circ H)*{ij}=\bigvee_{k=1}^r (W_{ik}\wedge H_{kj}) \quad\text{(equivalently }W\circ H=\min(1,WH)\text{ entrywise).}$$

Interpretation: each rank-one factor (W(:,k)H(k,:)) is a “rectangle” (biclique) of ones; the OR lets rectangles overlap, which is why BMF naturally expresses overlapping communities/roles.

bmf as a universal model instance $u=(e,t,p,f,)) in cmp

Your UM definition is explicit: $u=(e,t,p,f,\omega)$ with $e\in E$, $t\in T$, $p\in P$, $f\in F$, $\omega\in\Omega$.

Here is the clean mapping.

1) event space $E$

Take atomic events to be observations of entries: $$e_{ij} := (i,j,x_{ij},m_{ij}),$$ i.e., “cell ((i,j)) is observed and equals 0/1.” This matches BMF’s objective, which only scores observed entries via (M).

(Optionally, factor (E) as a product event space “row choice () column choice () bit value,” i.e., a CMP-style joint/product event space.)

2) total thought / state (T)

Let the state be the current internal hypothesis about (X), minimally: $$t := (W,H)\ \ \text{(and optionally a residual/error summary)}.$$ This is the agent’s current “belief structure” about which row/column groups generate ones.

3) pattern space (P)

Patterns are the rank-one rectangles (bicliques) and their OR-composition: $$p_k \equiv (W(:,k),H(k,:)),\quad\text{and}\quad \hat X = \bigvee_{k=1}^r \left(W(:,k)\wedge H(k,:)\right).$$ This is exactly how BMF identifies correlated row/column subsets: each (k) selects a set of rows and columns that should co-occur as ones.

In CMP terms, each $p_k$ is a “product pattern” over (row-membership events) $\times$ (column-membership events), and the full model is a factorization of the joint structure into $r$ product components.

4) update function $f$

$f$ is the dynamics that, given the current state (and optionally a stream/batch of events), produces a revised state:

• “infer $\hat X$ from $(W,H)$” via the Boolean product $W\circ H$;
• and/or “perform one block update step” (update $H$ holding $W$ fixed, then update $W$ holding $H$ fixed), which is the computational state evolution underlying AO-style BMF solvers.

5) learning function $\omega$

$\omega$ is the rule that selects/adjusts the patterns to reduce surprise/error: $$\omega:\ (W,H)\mapsto (W',H')\ \text{to decrease }|M\odot(X-W\circ H)|_F^2.$$ In the BMF paper, $\omega$ is instantiated concretely as alternating optimization where each subproblem is solved (exactly or approximately) using integer programming or heuristics.

Author interpretation: ChatGPT here (GPT 5.2 thinking) didn’t quite nail the relationship between $f$ and $\omega$ and the tick-tock updates of $W$ and $H$.

one-line cmp read

BMF is “factorization discovery” where the event space is observed binary relations, the pattern space is a small set of overlapping product-patterns (rectangles/bicliques), and learning () searches those patterns to maximize compressive/explanatory fit under the Boolean OR-of-AND composition.