From 54b50ed88a1bfdcd4764b7f0b3dc1fcdb2905d71 Mon Sep 17 00:00:00 2001 From: Chin-Yun Yu Date: Mon, 15 Apr 2024 13:54:03 +0000 Subject: [PATCH] Update math notation for initial condition gradients in README.md --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index ecb7e59..fbf0b0e 100644 --- a/README.md +++ b/README.md @@ -63,7 +63,7 @@ $$ ### Gradients for the initial condition $`y_t|_{t \leq 0}`$ The initial conditions provide an entry point at $t=0$ for filtering, as we cannot evaluate $t=-\infty$. -Let us assume $A_{t, :}|_{t \leq 0} = 0$ so $y_t|_{t \leq 0} = x_t|_{t \leq 0}$, which also means $\frac{\partial \mathcal{L}}{y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{x_t}|_{t \leq 0}$. +Let us assume $`A_{t, :}|_{t \leq 0} = 0`$ so $`y_t|_{t \leq 0} = x_t|_{t \leq 0}`$, which also means $`\frac{\partial \mathcal{L}}{y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{x_t}|_{t \leq 0}`$. Thus, the initial condition gradients are $$ @@ -73,7 +73,7 @@ $$ $$ In practice, we pad $N$ and $N \times N$ zeros to the beginning of $\frac{\partial \mathcal{L}}{\partial \bf y}$ and $\mathbf{A}$ before evaluating $\frac{\partial \mathcal{L}}{\partial \bf x}$. -The first $M$ outputs are the gradients to $y_t|_{t \leq 0}$ and the rest are to $x_t|_{t > 0}$. +The first $M$ outputs are the gradients to $`y_t|_{t \leq 0}`$ and the rest are to $`x_t|_{t > 0}`$. ### Time-invariant filtering