Learning a threshold from a matrix

Question

I have a 2D matrix, and I want to learn a threshold from this matrix. All items in the matrix bigger than the threshold will become 1s, and all smaller than the threshold will become 0s. Then, I have a differentiable loss function to evaluate the resulting 0-1 matrix. What methods are typically used to learn the threshold? Do we put the matrix across a network, or are there any non-neural-network methods?

Answer 1

You have some kind of loss function $L(y,\hat y)$ that takes in some kind of truth value, $y$, and the $0/1$ values from your matrix, $\hat y$.

Implicitly, this is a function of the truth values, $y$, of the threshold, $t$, and of the original, unrounded values in your matrix, $m$.

Your goal is to minimize the loss, so minimize $L(y, t, m)$. Depending on the complexity, you might choose to do this by hand (calculus) or perhaps on a computer by looping over possible thresholds. If the matrix values all are between $0$ and $1$, for instance, you might calculate the loss at every $0.01$ increment.

This is a fairly straightforward computation, since the loss function really only depends on one variable, $t$, as the truth labels $y$ and unrounded values $m$ are fixed.

Answer 2

The hierarchy of methods for thresholds are:

1. Hand-coded thresholds
2. Learned thresholds with a decision tree
3. Learned thresholds with an ensemble of trees (e.g., Random Forest or XGBoost)
4. Learned thresholds with neural networks