Arimoto information combining convexity and concavity

C. Hirche, Rényi Bounds on Information Combining, ISIT 2020, arXiv:2004.14408v1, §IV-A, Conjecture IV.6:

There exists a value 1 < α̂ < 1.5783, such that k_α^A(k_α^A⁻¹(x) ∗ k_α^A⁻¹(y)) (47) is convex for 0 < α < 1 and concave for 1 < α ≤ α̂.

Equation (23) defines

$$ K_\alpha^A(X\mid Y)=e^{((1-\alpha)/\alpha)H_\alpha^A(X\mid Y)}. $$

For binary $X\sim{p,1-p}$, $k_\alpha^A(p)=K_\alpha^A(X)$, hence

$$ k_\alpha^A(p)=(p^\alpha+(1-p)^\alpha)^{1/\alpha},\qquad p\star q=p+q-2pq. $$

Write $k_\alpha=k_\alpha^A$ below. Lemma IV.3, cited from Hsu–Asoodeh–Salamatian–Calmon, gives convexity for $\alpha\ge2$; Lemma IV.4 shows neither convexity nor concavity for $\alpha\in(1.58,1.97)$.

The inverse is the restriction to $[0,1/2]$, with range $I_\alpha=[\min(1,\delta_\alpha),\max(1,\delta_\alpha)]$, where $\delta_\alpha=2^{(1-\alpha)/\alpha}$. Orders and exponents are real. For each permitted order and every fixed $y\in I_\alpha$, the claim is the full two-point ConvexOn or ConcaveOn inequality on the closed interval $I_\alpha$. The specified witness is $\hat\alpha=3/2$.

This scalar inequality supplies the separate curvature needed for binary Arimoto conditional Rényi information combining. The exact inverse branch, real orders and boundary inputs are part of the assertion.

The sign certificate at three halves requires an analytic proof over the entire bias interval. A numerical sign test does not establish that uniform claim. Rioul–Béguinot, HAL hal-05074668v1, Proposition 1 and Remark 6, stated the threshold $\hat\alpha=3/2$ from a numerical sign check without proving the sign throughout the parameter domain. See the source note.

Set $\psi_\alpha(t)=k_\alpha((1-t)/2)$ and $E_\alpha(t)=t\psi_\alpha''(t)/\psi_\alpha'(t)$ on $0<t<1$. For $u=(1+t)/2$, $w=(1-t)/2$, $S=u^\alpha+w^\alpha$ and $D=u^{\alpha-1}-w^{\alpha-1}$, differentiation gives

$$ \psi_\alpha'={1\over2}S^{1/\alpha-1}D,\qquad \psi_\alpha''={\alpha-1\over4}S^{1/\alpha-2}(uw)^{\alpha-2}. $$

All bases are strictly positive. The slope has the sign of $\alpha-1$. For $b=\alpha-1$, $t=\tanh s$ and $s>0$,

$$ \mathcal E_b(s)=E_\alpha(\tanh s) ={b\sinh s\cosh^2s\over\sinh(bs)\cosh((b+1)s)}>0. $$

Its derivative divided by itself is

$$ L_b(s)=\coth s+2\tanh s-b\coth(bs)-(b+1)\tanh((b+1)s). $$

For fixed $s>0$, both $f_s(r)=r\coth(rs)$ and $g_s(r)=r\tanh(rs)$ strictly increase on $r>0$. In the first case the derivative numerator is $\sinh z\cosh z-z>0$, since its value at zero is zero and its derivative is $2\sinh^2z>0$ for $z>0$. The second derivative formula is $g_s'(r)=\tanh(rs)+rs/\cosh^2(rs)>0$.

If $-1<b<0$, put $q=-b\in(0,1)$. Oddness gives $b\coth(bs)=q\coth(qs)$, so $L_b=f_s(1)-f_s(q)+2g_s(1)-g_s(b+1)>0$. If $0<b\le1/2$, the same comparisons give $L_b\ge L_{1/2}$. At that exact endpoint,

$$ \mathcal E_{1/2}(s)={\cosh^2s\over2\cosh s-1},\qquad \mathcal E_{1/2}'(s)= {2\cosh s(\cosh s-1)\sinh s\over(2\cosh s-1)^2}>0. $$

Thus $E_\alpha$ increases for both claimed order ranges. The identity $p\star q=(1-tv)/2$, for $p=(1-t)/2$ and $q=(1-v)/2$, gives $G_\alpha(\psi_\alpha(t),\psi_\alpha(v))=\psi_\alpha(tv)$. The transported first derivative is $R(t)=v\psi_\alpha'(vt)/\psi_\alpha'(t)$, and

$$ R'(t)={v\psi_\alpha'(vt)\over t\psi_\alpha'(t)} (E_\alpha(vt)-E_\alpha(t))\le0. $$

The prefactor is positive. For $\alpha<1$ the inverse bias decreases, giving a nondecreasing derivative in the coordinate $x$ and hence convexity. For $\alpha>1$ it increases, giving concavity. Continuity extends these inequalities to the closed interval. Bias $v=0$ gives a constant slice; $v=1$ gives the identity slice.

The formal proof reuses the Cachin slope rather than rederiving the real-power second derivative. With $c=(1-\alpha)/\alpha$ and the shared positive bias slope $f(t)=-\frac{d}{dt}h_\alpha((1-t)/2)$, $\psi_\alpha=\exp(c h_\alpha)$ and $\psi_\alpha'=-c\psi_\alpha f$. The shared certificate $f'=fF/t$ gives $E_\alpha=F-ctf=A/(1+tT)$. The identity $T=\tanh(bs)$ and uniqueness of derivatives identify this expression with $\mathcal E_b$. Also, $k_\alpha^{-1}(x)=h_\alpha^{-1}(\log(x)/c)$, so continuity of the inverse comes directly from the existing inverse.

| Needed interface | Reused source | Use | | --- | --- | --- | | Convolution and bias product | BinaryRenyiCombining.bconv, bconv_bias | The combining reduction | | Positive power sum and monotone probability entropy | HircheCachinCombiningConcavity.powerSum_pos, h_strictMonoOn, h_image | The exponential change of coordinate and exact range | | Specified entropy inverse | hInv_h, hInv_continuousOn | The Arimoto probability inverse and its continuity | | Bias derivative and positive slope | psi_hasDerivAt, slope_pos | The Arimoto bias derivative and its sign | | Real-power elasticity | slope_hasDerivAt, slope_eq, T_comp_hasDerivAt | The Arimoto elasticity and its hyperbolic identity | | Closed-interval curvature | Mathlib MonotoneOn.convexOn_of_deriv, AntitoneOn.concaveOn_of_deriv | Continuous endpoint extension |

A violation consists of an allowed real order, a permitted fixed $y$, two points of $I_\alpha$ and a convex weight violating the corresponding inequality. Source fidelity and bibliographic priority are separate from mathematical correctness of this statement.

The settling declaration is result : claim in D5/S3/Entropy/RenyiCombining/HircheArimotoCombining.lean. Its statement includes all real orders in both ranges and both coordinate endpoints. The hyperbolic certificate is ArimotoHyperbolic.strictMonoOn_elasticity.

Conjecture IV.6 is PROVED with $\hat\alpha=3/2$ by result : claim. Since $1<3/2<1.5783$, this witness meets the source's strict numerical bound. The conjecture asks for the existence of such a threshold; it does not assert concavity throughout $(3/2,1.5783)$.

The following full separate-curvature classification on positive orders is computed, with its strictness and additional order ranges OPEN as formalization targets:

| Order | Internal slices | | --- | --- | | $0<\alpha<1$ | Strictly convex | | $1<\alpha\le3/2$ | Strictly concave | | $3/2<\alpha<2$ | Neither convex nor concave | | $\alpha\ge2$ | Strictly convex; this is the regime of Hirche's Lemma IV.3 |

Here “internal” means the fixed second bias belongs to $(0,1)$. Endpoint slices are constant or identity and satisfy the non-strict inequalities. At $\alpha=1$ the original coordinate is constant and the specified inverse does not exist.

For the computed sharpness calculation, let $Q(z)=\sinh z/z$ with $Q(0)=1$. Its convergent even Taylor expansion is $Q(z)=1+z^2/6+O(z^4)$. Therefore

$$ \mathcal E_b(s)={Q(s)\cosh^2s\over Q(bs)\cosh(\alpha s)},\qquad E_\alpha(t)=1-{(2\alpha-3)(\alpha+1)\over3}t^2+O_\alpha(t^4). $$

For $\alpha>3/2$ and every fixed $0<v<1$, the difference $E_\alpha(vt)-E_\alpha(t)$ is strictly positive for sufficiently small positive $t$. In particular, $v=1/2$ gives the slice $y=k_\alpha(1/4)$ and positive second derivative, excluding concavity. For $3/2<\alpha<2$, the explicit formula for $\psi''/\psi'$ shows $E_\alpha(t)\to+\infty$ as $t\to1^-$, whereas $E_\alpha(vt)$ has a finite limit. The second derivative is then negative near that endpoint, excluding convexity as well. For $b\ge1$, parameter comparison gives $L_b\le L_1=2(\tanh s-\tanh(2s))<0$, so elasticity decreases and internal slices are strictly convex. Sharpness for $\alpha>3/2$, including failure of concavity at $y=k_\alpha(1/4)$ for $\alpha\in(3/2,1.5783)$, is computed / OPEN: these sharpness, strictness and remaining-range arguments are not formalized.

Citation indexes and subsequent literature have not been exhaustively checked; bibliographic priority and the absence of a later analytic proof are not established.