Covariance domination conjecture for the uniform prior over the ℓ1 ball

Counterexample in every dimension. Let $C_d=\{w\in\mathbb R^d:\|w\|_1\le1\}$, and let $W$ be uniform on $C_d$. For $U=|W_1|$, the exact marginal density and transverse conditional second moments are

$$f_U(u)=d(1-u)^{d-1},\qquad \mathbb E[W_j^2\mid U=u]=\frac{2(1-u)^2}{d(d+1)} \quad(0<u<1,\ j\ge2).$$

In particular,

$$\operatorname{Var}_{\mathrm{Unif}(C_d)}(W_j) =\frac{2}{(d+1)(d+2)}.$$

In Conjecture 1 of the original source, take any $d\ge2$, one observation, and its allowed parameters

$$x_1=e_1,\quad r_1=-1,\quad 0<\alpha<1,\quad \psi(t)=t^2/2,\quad c=1,\quad \xi_1=0.$$

Direct substitution into source equations (6)–(7) gives the reverse conditional density

$$p(w\mid0)\propto e^{-\alpha w_1^2}\mathbf1_{\{\|w\|_1\le1\}}.$$

Both $u\mapsto(1-u)^2$ and $u\mapsto e^{-\alpha u^2}$ strictly decrease on $(0,1)$. The strict covariance identity

$$\operatorname{Cov}(f(U),g(U)) =\frac12\mathbb E[(f(U)-f(U'))(g(U)-g(U'))]>0$$

for an independent copy $U'$ therefore gives, for every $j\ge2$,

$$\operatorname{Var}(W_j\mid0) =\frac{2}{d(d+1)} \frac{\mathbb E[(1-U)^2e^{-\alpha U^2}]} {\mathbb E[e^{-\alpha U^2}]} > \frac{2}{(d+1)(d+2)} =\operatorname{Var}_{\mathrm{Unif}(C_d)}(W_j).$$

Thus the asserted covariance domination fails strictly in every dimension $d\ge2$. All stated source restrictions hold, including $d>\alpha cn\|r\|_\infty$. For $d=2,\alpha=1/2$, an exact alternating-series certificate even yields

$$\operatorname{Var}(W_2\mid0)\ge\frac16+\frac{73}{10080}.$$

The failure persists with a full-rank design and a strictly concave conditional log density: taking two observations $x_1=e_1,x_2=e_2$, residuals $-1,-1/100$, and the same $\alpha,\psi,c$ gives density proportional to

$$\exp(-w_1^2/2-w_2^2/200)\mathbf1_{C_2}(w),$$

with transverse variance at least $1/6+12847/2016000>1/6$.

The latest published, truncated version also fails under its full hypotheses. The authors retain the analogous auxiliary assertion as equation (4.44) of their 2026 published follow-up, but its reverse conditional now has an additional truncation-normalization factor. To account for that factor and satisfy every hypothesis of its Theorem 1, choose

$$K=2,\ N=1,\ d=100000,\ \delta=1/20,\quad \beta=2\log(8000000),$$ $$x_1=e_1,\quad y_1=0,\quad \psi(t)=t,\quad V=a_0=a_1=a_2=1,\quad c_1=c_2=1/2.$$

Then $C_N=1$, $A_3=96\sqrt e$, and $\rho=\beta$. The required size restrictions hold:

$$K[\log(2Kd)+\log(1/\delta)]=\beta N,$$ $$A_3(\beta N)^2 <96\cdot2\cdot32^2 =196608<200000=Kd,$$

and $\beta N>2,\ K=2,\ H(1/20)<1/2$.

The truncation region is $B=[-L,L]^2$ for some $L>0$. For $T_u\sim N(u,\rho^{-1})$, put

$$P(u)=\Pr(|T_u|\le L), \qquad s(u)=\frac{e^{-\rho u^2/2}}{P(u)}.$$

Differentiating the Gaussian integral gives

$$(\log s)'(u) =-\rho\,\mathbb E[T_u\mid |T_u|\le L]<0 \quad(u>0);$$

the final strict sign follows by pairing $t$ with $-t$. Writing $u_i=w_{i,1}$, the joint conditional density after integrating transverse coordinates is proportional to

$$h(u_1)h(u_2)e^{-\beta(u_1+u_2)^2/8}, \qquad h(u)=(1-|u|)^{d-1}s(u)\mathbf1_{\{|u|<1\}}.$$

Hence the first neuron's marginal, relative to its uniform prior, is weighted by

$$F(u)=s(u)(h*g_\beta)(u), \qquad g_\beta(t)=e^{-\beta t^2/8}.$$

Convolution preserves the class of even functions decreasing on $[0,\infty)$, as follows immediately from the layer-cake representation by centered intervals. Since $s$ is strictly decreasing there, so is $F$. Applying the same strict slice-covariance argument above shows

$$\operatorname{Var}_{p^*(\,\cdot\,\mid0)}(w_{1,j}) > \operatorname{Var}_{\mathrm{Unif}(C_d)}(w_{1,j}) \quad(j\ge2).$$

Consequently, the latest published equation (4.44) fails even under every stated size and truncation hypothesis.

A separate variance error. Both papers print $d/((d+1)^2(d+2))$ for a signed uniform coordinate. That expression is instead $\operatorname{Var}(|W_j|)$; the correct signed variance is $2/((d+1)(d+2))$, also implied by the published paper's own Appendix equation (7.54).

These counterexamples refute the auxiliary covariance-domination assertions, not the main theorem of the later paper, which uses a different bound.

Sources: original Conjecture 1; final published article, equation (4.44).