Equations (3.2)--(3.3) - Global gradient flows

Notation: $n = \#\{\text{training examples}\}$, $d$ is the input dimension, $k$ the feature dimension, $X:\mathbb R^d\to\mathbb R^n$ the data map, $Y$ the labels, $B:\mathbb R^d\to\mathbb R^k$ the features, and $v\in\mathbb R^k$ the head. Adjoint means Euclidean transpose. The loss is $\widehat L(v,B)=\|XB^\top v-Y\|^2$, with no normalization. The probability model, when present, is explicitly specified below; deterministic flow statements involve no random data assumption.

For every triple of natural numbers $n,d,k$, every continuous real-linear map $X:\mathbb R^d\to\mathbb R^n$, every $Y\in\mathbb R^n$, every $v_0\in\mathbb R^k$, and every continuous real-linear map $B_0:\mathbb R^d\to\mathbb R^k$, three assertions hold together. First, there exist functions $a:\mathbb R\to\mathbb R^k$ and $F:\mathbb R\to\mathcal L(\mathbb R^d,\mathbb R^k)$ with $a(0)=v_0$ and $F(0)=B_0$ such that for every real $t\geq0$ their derivatives within $[0,\infty)$ are $\dot a(t)=-2F(t)X^*(XF(t)^*a(t)-Y)$ and $\dot F(t)=\bigl[x\mapsto-2\langle X^*(XF(t)^*a(t)-Y),x\rangle a(t)\bigr]$, the latter being a derivative in the space of continuous linear maps. Second, for every two pairs $(a_1,F_1)$ and $(a_2,F_2)$ with those initial conditions and differential equations, $a_1(t)=a_2(t)$ and $F_1(t)=F_2(t)$ for every real $t\geq0$. Third, there exists $b:\mathbb R\to\mathbb R^k$ with $b(0)=v_0$ and derivative within $[0,\infty)$ equal to $-2B_0X^*(XB_0^*b(t)-Y)$ for every real $t\geq0$. Here $\mathcal L$ denotes continuous real-linear maps and $^*$ the Euclidean adjoint. The derivatives at zero are within the half-line. Values at negative times have no restrictions, and uniqueness of $b$ is not asserted. All dimensions may be zero, and no rank, normalization, or consistency hypotheses are imposed.

Formalization note: Formal ODE bridge for the source-backed parent; global existence and FT uniqueness are analytic obligations implicit in the source flow notation, not a separately numbered paper theorem. Source: Kumar, Raghunathan, Jones, Ma, and Liang, Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution, ICLR 2022, https://arxiv.org/pdf/2202.10054v1. Section 3.1, PDF p. 6, equations (3.2)--(3.3). Source-backed parent: Section 3.4, PDF p. 10, Proposition 3.7, equations (3.10)--(3.11); Appendix A.7, PDF pp. 45--47.