Existence of positive tensor rank decompositions for unital symmetric real tensors

Proof of Conjecture 4.7 in finite dimension.

Let $P\in\operatorname{Sym}^3(\mathbb R^N)$, $N\ge1$, with unit $\alpha$, meaning $P(\alpha,u,v)=\langle u,v\rangle$. We construct $P=\sum_i\phi_i^{\otimes3}$ with $\langle\alpha,\phi_i\rangle>0$, the positivity condition in Obster, equations (4.9)–(4.10). Minimal length is unnecessary (footnote 11).

Choose orthonormal coordinates with $\alpha=re_0$, $r>0$. Then $Q=rP$ has unit $e_0$. If $Q=\sum_i\eta_i^{\otimes3}$ with $(\eta_i)_0>0$, then $\phi_i=r^{-1/3}\eta_i$ decomposes $P$, with $\alpha\cdot\phi_i=r^{2/3}(\eta_i)_0>0$. Hence assume $\alpha=e_0$.

Put $d=N-1$. The unit fixes $P_{000}=1$, $P_{00a}=0$, and $P_{0ab}=\delta_{ab}$ for $a,b>0$; write the remaining block as $T$. It suffices to construct positive weighted atoms satisfying

$$\sum_iw_i=1,\qquad \sum_iw_ix_i=0,\qquad \sum_iw_ix_ix_i^T=I_d,\qquad \sum_iw_ix_i^{\otimes3}=T,$$

because $\phi_i=w_i^{1/3}(1,x_i)$ then works. For $d=0$, use one atom.

By polarization, write $T=\sum_{j=1}^m c_jv_j^{\otimes3}$, allowing $m=0$. Choose

$$\varepsilon=\frac1{2(1+m+\sum_j\|v_j\|^2)},\qquad C=I_d-\varepsilon\sum_jv_jv_j^T.$$

Thus $C\succ0$ and $m\varepsilon<1/2$. For each $j$, set

$$s_j=c_j/\varepsilon,\quad a_j=1+\max(s_j,0),\quad b_j=1+\max(-s_j,0).$$

Place weights

$$\frac{\varepsilon}{a_j(a_j+b_j)}\ \text{at }a_jv_j,\qquad \frac{\varepsilon}{b_j(a_j+b_j)}\ \text{at }-b_jv_j.$$

These are positive. This pair has mass $\varepsilon/(a_jb_j)$, first moment zero, second moment $\varepsilon v_jv_j^T$, and third moment

$$\varepsilon(a_j-b_j)v_j^{\otimes3}=c_jv_j^{\otimes3}.$$

Together the pairs therefore have mass $M\le m\varepsilon<1/2$, mean zero, second moment $I_d-C$, and third moment $T$.

Diagonalize $C=\sum_{k=1}^d\lambda_ku_ku_k^T$, with $\lambda_k>0$ and orthonormal $u_k$. Choose $A>0$ with $\sum_k\lambda_k/A^2<1-M$. Add atoms $\pm Au_k$, each with weight $\lambda_k/(2A^2)$. They supply second moment $C$, with zero first and third moments. Put the remaining positive mass at the origin. All required moments now hold. ∎

The existence statement also follows from established moment theory; see Blekherman–Fialkow (2020), Corollary 2.2. The construction above is self-contained.

Developed with OpenAI Codex and reviewed by Claude Opus 5.