The formula $W_j^v \in \mathbb{R}^{d \times \frac{d}{\tau}}$ defines the value weight matrix, $W_j^v$. This matrix consists of real numbers and has dimensions of $d$ rows and $\frac{d}{\tau}$ columns. In the context of attention mechanisms, this matrix is used to transform the input values.

Value Weight Matrix $W_j^v$

A matrix $\mathbf{W}_h$ can be formed by concatenating a series of $M$ matrices or vectors, denoted as $\mathbf{W}_h^1, \mathbf{W}_h^2, \dots, \mathbf{W}_h^M$. The mathematical representation of this operation is $\mathbf{W}_h = [\mathbf{W}_h^1 \mathbf{W}_h^2 \dots \mathbf{W}_h^M]$.

Concatenated Weight Matrix Formula

The expression `⟨g₁⟩, ..., ⟨gₖ⟩` represents a sequence of k mathematical terms. Each term, `⟨gᵢ⟩`, is enclosed in angle brackets. This notation is context-dependent. For instance, in probability and statistics, `⟨gᵢ⟩` often denotes the expectation value of a random variable `gᵢ`. In group theory, it can represent the cyclic subgroup generated by the element `gᵢ`.

Sequence of Mathematical Terms in Angle Brackets

University of Michigan - Ann Arbor

1. Introduce formal sensitivity analysis for linear SCMs and reduce it to a non-zero constrained identification problem.
2. Introduce graphical process PUSH-FORWARD to simplify identification - which can they be solved by a number of algorithms
3. Develop graph-based constrained identification solving algorithm that returns sensitivity curve.

Contributions of Paper

Pearl, Judea, et al. “Sensitivity Analysis of Linear Structural Causal Models.”

http://proceedings.mlr.press/v97/cinelli19a/cinelli19a.pdf

Sensitivity Analysis of Linear Structural Causal Models

We define $G_O$ as model of the the original hypothesis. $G_A$ is the model made to account for confounders. $G_B$ is another variation of $G_O$.

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