Bounds on the Number of Modes of a Gaussian Mixture Density
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866913131880513536 |
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| author | Nguyen, Hien Duy |
| author_facet | Nguyen, Hien Duy |
| contents | We derive explicit upper bounds for the number of nondegenerate critical points of a $k$-component Gaussian mixture density in $\mathbb{R}^d$, and the number of modes when the modal set is finite, together with lower bounds. By normalizing the critical-point equations by a reference component, for $k\ge2$ we get the direct Pfaffian bound \[ U_{\mathrm{het}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+2\min(d,k-1)+1\right)^{k-1}. \] For the same parameter range, an exact elimination augmented by an algebraic reciprocal variable gives the alternative bound \[ U_{\mathrm{aug}}(d,k)= 2^{\binom{k-1}{2}}(d+1)\left((2k-1)d+2k-1\right)^{k-1}. \] Thus, for $k\ge2$, the best critical-point bound is their minimum. A Morse-theoretic argument improves the corresponding finite-mode upper bound to \[ \left\lfloor \frac{\min\{U_{\mathrm{het}}(d,k),U_{\mathrm{aug}}(d,k)\}+1}{2}\right\rfloor. \] In the homoscedastic case, for $k\ge2$, the direct bound improves to \[ U_{\mathrm{hom}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+\min(d,k-1)+1\right)^{k-1}, \] an affine-rank reduction replaces $d$ by the affine rank of the component means, and an augmented homoscedastic reduction gives the dimension-free bound \[ U_{\mathrm{aug,hom}}(k)=2^{\binom{k-1}{2}+1}(2k)^{k-1}. \] On the lower-bound side, for $d,k\ge 2$ we obtain \[ L_{\mathrm{bin}}(d,k)=k+\max_{2\le r\le \min(d,k)}\binom{k}{r}, \] together with a padding-product family that in particular implies the linear lower bound $d+k-1$, and a seed-closure principle that packages product and padding constructions. We further give explicit bounds for the number of connected components of the critical set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15531 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounds on the Number of Modes of a Gaussian Mixture Density Nguyen, Hien Duy Statistics Theory Combinatorics We derive explicit upper bounds for the number of nondegenerate critical points of a $k$-component Gaussian mixture density in $\mathbb{R}^d$, and the number of modes when the modal set is finite, together with lower bounds. By normalizing the critical-point equations by a reference component, for $k\ge2$ we get the direct Pfaffian bound \[ U_{\mathrm{het}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+2\min(d,k-1)+1\right)^{k-1}. \] For the same parameter range, an exact elimination augmented by an algebraic reciprocal variable gives the alternative bound \[ U_{\mathrm{aug}}(d,k)= 2^{\binom{k-1}{2}}(d+1)\left((2k-1)d+2k-1\right)^{k-1}. \] Thus, for $k\ge2$, the best critical-point bound is their minimum. A Morse-theoretic argument improves the corresponding finite-mode upper bound to \[ \left\lfloor \frac{\min\{U_{\mathrm{het}}(d,k),U_{\mathrm{aug}}(d,k)\}+1}{2}\right\rfloor. \] In the homoscedastic case, for $k\ge2$, the direct bound improves to \[ U_{\mathrm{hom}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+\min(d,k-1)+1\right)^{k-1}, \] an affine-rank reduction replaces $d$ by the affine rank of the component means, and an augmented homoscedastic reduction gives the dimension-free bound \[ U_{\mathrm{aug,hom}}(k)=2^{\binom{k-1}{2}+1}(2k)^{k-1}. \] On the lower-bound side, for $d,k\ge 2$ we obtain \[ L_{\mathrm{bin}}(d,k)=k+\max_{2\le r\le \min(d,k)}\binom{k}{r}, \] together with a padding-product family that in particular implies the linear lower bound $d+k-1$, and a seed-closure principle that packages product and padding constructions. We further give explicit bounds for the number of connected components of the critical set. |
| title | Bounds on the Number of Modes of a Gaussian Mixture Density |
| topic | Statistics Theory Combinatorics |
| url | https://arxiv.org/abs/2605.15531 |