Sum-of-squares: proofs, beliefs, and algorithms — Boaz Barak and David Steurer

# Notation

## Hiding constants

Unless explicitly stated otherwise, $$O(\cdot)$$-notation hides absolute multiplicative constants. Concretely, every occurrence of $$O(x)$$ is a placeholder for some function $$f(x)$$ that satisfies $$\forall x\in \R.\, \abs{f(x)}\le C\abs{x}$$ for some absolute constant $$C>0$$. Similarly, $$\Omega(x)$$ is a placeholder for a function $$g(x)$$ that satisfies $$\forall x\in \R.\, \abs{g(x)} \ge \abs{x}/C$$ for some absolute constant $$C>0$$.

## Vectors

All vectors are column vectors unless specified otherwise. In particular, the notation $$(a,b,c)$$ is short hand for a column vector with entries $$a,b,c\in \R$$,. We denote the coordinate basis of $$\R^n$$ by $$\set{e_i}_{i\in [n]}$$. For a vector $$v\in\R^n$$, we let $$\transpose v$$ be the corresponding row vector.

## Inner products and norms

For vectors $$u,v\in \R^n$$ with $$u=(u_1,\ldots,u_n)$$ and $$v=(v_1,\ldots,v_n)$$, we define the inner product of $$u$$ and $$v$$, unless specified otherwise, $\iprod{u,v}=\transpose u v=\sum_{i=1}^n u_i \cdot v_i\,.$ The (Euclidean) norm of a vector $$v$$ is $$\norm{v}=\iprod{v,v}^{1/2}$$. For $$p\ge 1$$, we define the $$\ell^p$$-norm of $$v$$, $\norm{v}_p = \Paren{\sum_{i=1}^n \abs{v_i}^p}^{1/p}\,.$ For $$p=\infty$$, we take the limit, so that $\norm{v}_\infty =\max_{i\in [n]} \abs{v_i}\,.$

## Kronecker product

For two matrices $$A$$ and $$B$$, their Kronecker product is the matrix A$$\otimes B$$ with entries $$(A\otimes B)_{ii',jj'} = A_{i,j} B_{i',j'}$$. This operation also applies to row and column vectors (viewed as matrices with only one column or one row). We use the notation $$A^{\otimes k}=A\otimes \cdots \otimes A$$ ($$k$$-times) for the $$k$$-fold tensor power of a matrix $$A$$.

## Matrices

For matrices with more than two indices, we separate row and column indices by a comma. For example if $$A$$ is a linear combination of matrices of the form $$e_i \transpose{(e_j \otimes e_k)}$$, we denote the entries of $$A$$ by $$A_{i,jk}$$. (Note that this convention is consistent with the above notation for Kronecker products.)

## Traces

The trace is cyclic, that is, for all matrices $$A\in \R^{m\times n}$$ and $$B\in \R^{n\times m}$$, $\Tr AB = \Tr BA \,.$ A consequence of this property is that for $$x,y\in \R^{n}$$ and $$A\in \R^{n\times n}$$, $\Tr A x \transpose y = \Tr \transpose y A x = \iprod{y, A x}\,.$

## Polynomials

Let $$\R[x]$$ be the set of polynomials with real coefficients in variables $$x=(x_1,\ldots,x_n)$$. For $$d\in \N$$, let $$\R[x]_{\le d}$$ be the set of polynomials of degree at most $$d$$.