We address the problem of reconstructing a set of K binary patterns <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{\Xi }= (\varvec{\xi }^1,..., \varvec{\xi }^K) \in \{-1, +1 \}^{N \times K}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mn>1</mn> </msup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msup> <mrow> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mi>K</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>N</mi> <mo>×</mo> <mi>K</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, starting from their covariance matrix <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{\Xi }\varvec{\Xi }^T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <msup> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <mi>T</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and from a set of mixtures of the form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{\sigma }_M= \text {sign}(\sum _{\mu \in \mathcal {S}_M \subseteq \{1,..., K\}} \varvec{\xi }^{\mu })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">σ</mi> </mrow> <mi>M</mi> </msub> <mo>=</mo> <mtext>sign</mtext> <mrow> <mo stretchy="false">(</mo> <msub> <mo>∑</mo> <mrow> <mi>μ</mi> <mo>∈</mo> <msub> <mi mathvariant="script">S</mi> <mi>M</mi> </msub> <mo>⊆</mo> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>K</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> <msup> <mrow> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mi>μ</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left\lvert S_M\right\rvert =M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="|" open="|"> <msub> <mi>S</mi> <mi>M</mi> </msub> </mfenced> <mo>=</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation>. To this aim, we engage a modular Hebbian network, specified by the interaction matrix <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{J}^H = \frac{1}{N}\varvec{\Xi }\varvec{\Xi }^T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="bold-italic">J</mi> </mrow> <mi>H</mi> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <msup> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <mi>T</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\varvec{\sigma }_M}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">σ</mi> </mrow> <mi>M</mi> </msub> </math></EquationSource> </InlineEquation> is taken as the initial state, repeated in each of the constituting L modules and the stable state resulting from a Gibbs evolution rule is taken as an L-tuple of candidate patterns. Finally, by properly handling <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{J}^H$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-italic">J</mi> </mrow> <mi>H</mi> </msup> </math></EquationSource> </InlineEquation> we can derive the projector matrix <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varvec{J}^K = \varvec{\Xi }(\varvec{\Xi }^T \varvec{\Xi })^{-1} \varvec{\Xi }^T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="bold-italic">J</mi> </mrow> <mi>K</mi> </msup> <mo>=</mo> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <mi>T</mi> </msup> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mi mathvariant="bold">Ξ</mi> </mrow> <mi>T</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> by which we can determine whether these candidate patterns are a good estimate for the ground patterns.

Pattern disentanglement and reconstruction by modular Hebbian networks

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2026年7月28日

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Elena Agliari,
Shahed Akter,
Andrea Alessandrelli,
Alberto Fachechi

摘要

We address the problem of reconstructing a set of K binary patterns $\varvec{\Xi }= (\varvec{\xi }^1,..., \varvec{\xi }^K) \in \{-1, +1 \}^{N \times K}$ $Ξ = (ξ^{1}, . . ., ξ^{K}) \in {- 1, + 1}^{N \times K}$ , starting from their covariance matrix $\varvec{\Xi }\varvec{\Xi }^T$ $Ξ Ξ^{T}$ and from a set of mixtures of the form $\varvec{\sigma }_M= \text {sign}(\sum _{\mu \in \mathcal {S}_M \subseteq \{1,..., K\}} \varvec{\xi }^{\mu })$ $σ_{M} = sign (\sum_{μ \in S_{M} \subseteq {1, . . ., K}} ξ^{μ})$ , with $\left\lvert S_M\right\rvert =M$ $|S_{M}| = M$ . To this aim, we engage a modular Hebbian network, specified by the interaction matrix $\varvec{J}^H = \frac{1}{N}\varvec{\Xi }\varvec{\Xi }^T$ $J^{H} = \frac{1}{N} Ξ Ξ^{T}$ , where ${\varvec{\sigma }_M}$ $σ_{M}$ is taken as the initial state, repeated in each of the constituting L modules and the stable state resulting from a Gibbs evolution rule is taken as an L-tuple of candidate patterns. Finally, by properly handling $\varvec{J}^H$ $J^{H}$ we can derive the projector matrix $\varvec{J}^K = \varvec{\Xi }(\varvec{\Xi }^T \varvec{\Xi })^{-1} \varvec{\Xi }^T$ $J^{K} = Ξ {(Ξ^{T} Ξ)}^{- 1} Ξ^{T}$ by which we can determine whether these candidate patterns are a good estimate for the ground patterns.