1. Gabatarwa

Wannan cikakken binciken takarda yana bincika matsalar gyaran ingantaccen alamar bincike akan filin Stiefel daga mahangar ka'idoji da na lissafi. Babbar matsalar da aka magance ita ce haɓaka aikin alamar bincike wanda aka ayyana shi azaman f_α(X) = alamar(X^TAX + X^TD) / [alamar(X^TBX)]^α, inda X yana cikin filin Stiefel O_n×k = {X ∈ R^n×k : X^TX = I_k}. Matakan A da B matakan ma'auni ne na n×n tare da B kasancewa tabbatacce rabin kuma yana da matsayi fiye da n-k, D matrix ne na n×k, kuma ma'auni α yana tsakanin 0 da 1. Sharuɗɗan matsayi(B) > n-k yana tabbatar da cewa maƙasudin yana ci gaba da zama tabbatacce ga duk X mai yuwuwa.

Tsarin gyaran filin Stiefel yana ba da ingantaccen tushen lissafi don magance wannan nau'in matsaloli, wanda ke da muhimmiyar tasiri a fagage da yawa na kimiyar bayanai da koyon inji. Binciken ya kafa sharuɗɗan da ake buƙata a cikin sifar matsalolin ƙimar darajar mara layi tare da dogaro da eigenvectors kuma ya haɓaka algorithms na lissafi masu haɗawa bisa maimaita filin da ya dace (SCF).

1.1 Ayyukan Da Suka Gabata

Takardar ta gano kuma ta bincika manyan lokuta uku na musamman waɗanda aka yi nazari sosai a cikin wallafe-wallafen da suka gabata:

Nazarin Rarrabuwa na Layi na Fisher

Tare da D = 0 da α = 1, matsalar ta ragu zuwa max_X∈On×k alamar(X^TAX) / alamar(X^TBX), wanda ke tasowa a cikin nazarin rarrabuwa na layi na Fisher don koyo mai kulawa. Hanyoyin da suka gabata sun canza wannan zuwa matsalar gano sifili: warware φ(λ) = 0 inda φ(λ) := max_X∈On×k alamar(X^T(A - λB)X). An tabbatar da cewa aikin φ(λ) ba ya raguwa kuma yawanci yana da sifili guda ɗaya, wanda za'a iya samu ta amfani da hanyar Newton. Sharuɗɗan Karush-Kuhn-Tucker (KKT) suna haifar da matsalar ƙimar darajar mara layi (NEPv): H(X)X = XΛ, inda H(X) matrix ne mai ma'ana mai darajar aikin X kuma Λ = X^TH(X)X.

Nazarin Alaƙa na Al'ada na Orthogonal

Tare da A = 0 da α = 1/2, matsalar ta zama max_X∈On×k alamar(X^TD) / √alamar(X^TBX), wanda ke fitowa a cikin nazarin alaƙa na al'ada na orthogonal (OCCA). Wannan tsari yana aiki azaman ƙwayoyin tsarin maimaite mai canzawa. Sharuɗɗan KKT na wannan yanayin ba sa ɗaukar sifar NEPv nan da nan amma ana iya canza su daidai zuwa ɗaya, yana ba da damar warwarewa ta hanyar maimaita SCF tare da ingantaccen aikin bayan-gudanarwa.

Matsalar Procrustes Maras Daidaituwa

Lokaci na uku na musamman yana haɗuwa da matsalar Procrustes maras daidaituwa, ko da yake ba a cika bayyana dalla-dalla a cikin abin da aka ba da shi ba. Duk ukun lokutan sun nuna fa'idar fa'ida na tsarin gyaran ingantaccen alamar bincike a cikin nau'ikan tsarin koyo na ƙididdiga daban-daban.

2. Tsarin Matsala

An ayyana matsalar gyaran ingantaccen alamar bincike gabaɗaya a hukumance kamar haka:

Matsala (1.1a): max_X^TX=Ik f_α(X)

Inda: f_α(X) = [alamar(X^TAX + X^TD)] / [alamar(X^TBX)]^α

Ma'auni sun gamsar: 1 ≤ k < n, I_k shine matrix na ainihi k×k, A, B ∈ R^n×n suna da ma'auni tare da B tabbatacce rabin kuma matsayi(B) > n-k, D ∈ R^n×k, matrix variable X ∈ R^n×k, da ma'auni 0 ≤ α ≤ 1.

Takardar kuma ta lura cewa wani yanayi mai yuwuwa mafi girma tare da ƙarin akai c a cikin maƙasudi ana iya sake tsara shi azaman yanayi na musamman na Matsala (1.1) ta hanyar sarrafa algebra, yana nuna cikakkiyar tsarin da aka tsara.

3. Tushen Ka'idoji

Binciken ya kafa sakamako na ka'idoji na asali da yawa:

Sharuɗɗan Da Ake Bukata

An samo muhimman sharuɗɗan ingantacciyar matsala don matsalar gyaran ingantaccen alamar bincike azaman matsalolin ƙimar darajar mara layi tare da dogaro da eigenvector (NEPv). Don yanayin musamman na Fisher's LDA (α=1, D=0), NEPv yana ɗaukar sifar H(X)X = XΛ, inda H(X) = A - λ(X)B da λ(X) = alamar(X^TAX)/alamar(X^TBX).

Wanzuwa da Keɓantacce

Don Matsala (1.3) (yanayin LDA na Fisher), an tabbatar da cewa babu masu haɓaka gida—sai kawai masu haɓaka duniya suke wanzu. Wannan muhimmin kaddara yana tabbatar da cewa kowace algorithm mai haɗawa za ta kai ga mafita mafi kyau a duniya.

Fassarar Geometric

Ingantawa yana faruwa akan filin Stiefel, wanda ke da ingantaccen tsarin geometric. An bincika haɗuwar algorithms dangane da filin Grassmann G_k(Rⁿ) (tarin duk fage masu girma k na Rⁿ), yana ba da hangen nesa na geometric akan tsarin ingantawa.

4. Hanyoyin Lissafi

Takardar ta ba da shawara kuma ta bincika maimaitawar filin da ya dace (SCF) don warware matsalar gyaran ingantaccen alamar bincike:

Algorithm na SCF

Asalin maimaitawar SCF don Matsala (1.3) shine: H(X_i-1)X_i = X_iΛ_i-1, farawa da fara # An dakatar domin gujewa iyakokin API