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Remember that $$\text{Data Structure}=\text{Organization}+\text{Storage}+\text{Operation},$$ base on which we develop algorithms to solve problems. Organization & Logical Structure The Organization, equivalently, the logical structure of data, is the intrinsic property of data, which determines the relationship between data elements and the operations on them, and it is independent from the storage. A simple example is the hash table, which realizes the mapping $$f:\text{key} \mapsto \text{value}$$ by storing entries at locations $\{\text{Hash}(\text{key})\}$. The hash function provides only a storage structure; it does not impose any logical relationship between elements. Now consider the case where keys are taken from $([n])$ and values from a given data type ...

Matrix Elementary Matrix The standard basis of $\mathbb F^n$ is $\{e_i\}_{1\leq i\leq n}$, where $e_i$ is the vector with $1$ in the $i$-th entry and $0$ elsewhere. It holds $$e_i^\top Ae_j=a_{ij},$$ This can be a test of a linear map $A:\mathbb F^n\to \mathbb F^n$ by the standard basis of $\mathbb F^n$. In total, $$A=\sum_{i,j=1}^n a_{ij}e_ie^\top_j=\sum_{i,j=1}^n a_{ij}E_{ij},$$We introduce a kind of basis of $M_n(\mathbb F)=\mathbb F^{n\times n}$ as $\{E_{ij}\}_{1\leq i,j\leq n}$, where $E_{ij}$ is the matrix with $1$ in the $(i,j)$-th entry and $0$ elsewhere, that is so-called elementary matrix, satisfying ...

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Let $V,U$ be $\mathbb F$-vector spaces, and $\varphi:V\to U$ be a linear map. Then the adjoint operator $\varphi^*:U^*\to V^*$ is defined by $$\langle \varphi^*(f),v\rangle=\langle f,\varphi(v)\rangle,\quad \forall f\in U^*,v\in V.$$A bilinear form $B:V\times U\to \mathbb F$ has matrix representation $G=(B(e_i,f_j))_{m\times n}$, where $\{e_i\}_{i=1}^m$ and $\{f_j\}_{j=1}^n$ are bases of $V$ and $U$, respectively, if $U,V$ are finite-dimensional. The rank of $B$ is defined as the rank of the matrix $G$, which is independent of the choice of bases. There exist bases $\{e_i\}_{i=1}^m$ and $\{f_j\}_{j=1}^n$ such that $B(e_i,f_j)=\delta_{ij}$ for $1\leq i,j\leq r$, and $B(e_i,f_j)=0$ otherwise, where $r=\mathrm{rank}(B)$. ...

Laplace Theorem Let $|A|$ be the determinant of order $n$. Fix $m$ rows, $1\leq m\leq n$, then we have $$|A|=\sum_{1\leq k_1...

Mollifier in L^1

Residue theorem, Jordan lemma, Variation of parameters, distribution theory

Residue theorem, Jordan lemma, Variation of parameters, distribution theory

对 HuggingFace Siglip2 模型实现 modeling_siglip2.py 的完整逐类分析。 覆盖 14 个核心类、200+ 变量/参数，每行关键代码均有功能解释和设计目的说明。 auto-generated from modular_siglip2.py — 这是最终生成的代码，非手动编辑。

从 modeling_siglip2.py 源码出发，完整推导 Siglip2 的 Sigmoid Contrastive Loss。 与 CLIP 的 Softmax CE Loss 逐行对比，解释可学习温度与偏置的设计动机。