Knowledge as a Hidden Dimension:
A Geome ic F amewo k o Modeling Awa eness
Th ough Me ic De o ma ion
Bo a A cak
5 No embe 2025
Abs ac
Human pe cep ion is a ely uni o m. Familia objec s appea c isp and meaning ul, while
un amilia ones blu in o undi e en ia ed shapes. This sugges s ha knowledge does mo e han
add in o ma ion—i eshapes he space in which in o ma ion is ep esen ed. In his pape , I
in oduce a minimal geome ic amewo k o malizing his idea. Knowledge is modeled as a
scala coo dina e K ha ac s as a hidden dimension: no an addi ional axis o senso y da a,
bu a pa ame e ha de o ms he in insic me ic along ask- ele an di ec ions.
I mo i a e his concep using a his o ical analogy: Eins ein’s ex ension o classical h ee-
dimensional space by adding ime as a ou h dimension did no c ea e new spa ial di ec ions,
bu ein e p e ed mo ion h ough a geome ic coo dina ion ule. Simila ly, I p opose ha he
knowledge a iable Kgo e ns he geome y o he exis ing mani old. By modeling his as an
aniso opic de o ma ion o a Riemannian me ic, I de i e ma hema ical gua an ees showing ha
inc easing Kmono onically imp o es disc imina ion h esholds and educes Bayes-op imal e o .
This amewo k connec s he philosophy o awa eness wi h he igo o in o ma ion geome y.
P ep in . ©2025 Bo a A cak. Released on Zenodo o academic discussion.
The au ho e ains all in ellec ual p ope y igh s.
.
1 In oduc ion
When we ecognize he silhoue e o a dog, spo a iend om a a , o ead ex in a language
we know well, pe cep ion eels sha p and s uc u ed. In con as , un amilia objec s appea
“ la ”—di icul o pa se, lacking clea in e nal dis inc ions. This e e yday asymme y sugges s
ha pe cep ion is no pu ely a ma e o senso y inpu . Ra he , i is modula ed by wha we know.
A use ul way o hink abou knowledge is o iew i no as a symbol o ac s o ed in mem-
o y, bu as some hing ha changes he geome y o pe cep ion. Familia ea u es become easie
o disc imina e because he cogni i e sys em e ec i ely “s e ches” he pe cep ual space along
meaning ul di ec ions. Un amilia dimensions con ac o blu .
This geome ic in ui ion echoes a majo momen in he his o y o physics. Classical mechanics
ope a ed in h ee spa ial dimensions un il Eins ein in oduced space ime: ou dimensions, whe e
ime ac ed no as ano he spa ial di ec ion bu as a hidden coo dina e ha go e ned ans o ma-
ions. The ou h dimension did no add new ma e ial con en ; i es uc u ed he geome y o
wha al eady exis ed.
1
The cen al claim o his pape is analogous: knowledge ac s as a hidden coo dina e, no
because i enla ges he ep esen a ional mani old, bu because i de e mines how dis ances—hence
disc iminabili y, awa eness, and meaning—a e compu ed.
2 Knowledge as a Hidden Dimension
The ph ase “dimension” can be misleading. I do no p opose ha he ep esen a ional mani old
gains ano he physical axis when someone lea ns a new concep . Ins ead, I use “dimension” in he
same sense ha physics uses ime: as a go e ning coo dina e ha modula es geome ic s uc u e.
2.1 Dimensions as coo dina ion ules
A dimension is no always a di ec ion. In ela i i y, ime is coupled wi h space by a me ic ha
de e mines how in e als con ac o s e ch depending on eloci y and g a i a ional po en ial.
Wha makes ime a dimension is no spa ial ex ension bu i s ole in de ining how he geome y
ans o ms.
Knowledge plays an analogous ole. I does no supply new aw inpu s; i de e mines how
s ongly he sys em dis inguishes poin s along ce ain di ec ions in he exis ing ep esen a ion.
Thus, Kis a hidden dimension because he mapping
(u, K)7−→ gK(u)
de ines he geome y o he pe cep ual mani old. This in e p e a ion is concep ually clean, philo-
sophically de ensible, and ma hema ically p ecise.
3 A Geome ic F amewo k o Knowledge
To o malize his in ui ion, le S⊂Rdbe a smoo h s imulus mani old, and le TsSdeno e i s
angen space a s∈S. An obse e pe cei es S h ough an in e nal ep esen a ion whose p ecision
is modula ed by he knowledge pa ame e K∈R≥0.
3.1 Task-Rele an Di ec ions
Pe cep ion is a ely iso opic. Le Us⊆TsSdeno e he subspace o ask- ele an di ec ions a
s imulus s. We model his subspace ia he o hogonal p ojec o
Ps:TsS→Us.
The complemen a y p ojec o is P⊥
s=I−Ps, ep esen ing i ele an di ec ions.
3.2 The Knowledge-Dependen Me ic
Le g0deno e a baseline Riemannian me ic on S(o en Euclidean). We de ine he aniso opic
K-dependen me ic gKby i s ac ion on any wo angen ec o s u, ∈TsS:
gK(u, ) = g0(u, ) + β(K)g0(Psu, ),(1)
whe e β:R≥0→R≥0is a smoo h, mono onically inc easing unc ion wi h β(0) = 0.
Since Psis an o hogonal p ojec o wi h espec o g0, his ensu es ha dis ances along ask-
ele an di ec ions a e s e ched by a ac o o p1+β(K), while o hogonal di ec ions emain
in a ian .
2
3.3 Schema ic Illus a ion
The e ec o Kon he local geome y is illus a ed below.
x
z
K
ss′
small sepa a ion
s e ched a high K
Figu e 1: Wi h inc easing knowledge K, he me ic gKs e ches ask- ele an di ec ions, inc easing
he e ec i e dis ance be ween sand s′.
4 Theo e ical Gua an ees
We now p o e ha his geome ic de o ma ion leads o conc e e imp o emen s in disc imina ion.
Lemma 1 (Inc easing Disc iminabili y).Fo any ixed s∈Sand any nonze o ∈Us, he gK-no m
∥ ∥gKis s ic ly inc easing on any in e al whe e β′(K)>0.
P oo . Since ∈Us, we ha e Ps = . Using he de ini ion in (1):
∥ ∥2
gK=gK( , )=g0( , ) + β(K)g0( , ) = (1 + β(K))∥ ∥2
g0.
Di e en ia ing wi h espec o Kyields ∂
∂K ∥ ∥2
gK=β′(K)∥ ∥2
g0>0.
4.1 Reduc ion o Bayes E o
Conside a disc imina ion ask unde Gaussian noise. We assume he in e nal ep esen a ion ad-
he es o he C am´e -Rao bound, such ha he noise co a iance Σ(K) is p opo ional o he in e se
o he Fishe In o ma ion me ic:
Σ(K)∝g−1
K=⇒Σ(K)−1∝I+β(K)Ps.
Theo em 1 (Knowledge Reduces Bayes E o ).Le δs ∈Us. Unde he assump ion abo e, he
Bayes-op imal disc imina ion e o Pe(K)is s ic ly dec easing in K.
P oo . The disc imina ion pe o mance is go e ned by he Mahalanobis dis ance d2
Mah =δs⊤Σ(K)−1δs.
Subs i u ing he me ic dependence:
d2
Mah ∝δs⊤(I+β(K)Ps)δs =∥δs∥2
g0+β(K)∥δs∥2
g0.
Since β(K) is inc easing, he signal- o-noise a io inc eases. Because he e o p obabili y Pe(K) =
Φ(−dMah/2) is a mono onically dec easing unc ion o dis ance, he e o s ic ly dec eases.
3
5 Ope a ionalizing he Knowledge Coo dina e
The scala Kis no me ely a heo e ical cons uc ; i can be es ima ed ia:
1. Mu ual In o ma ion: K≈I(S;X), whe e highe knowledge implies he ep esen a ion
e ains mo e s imulus- ele an in o ma ion.
2. Fishe In o ma ion: Me ics o he o m gK esemble Fishe in o ma ion ma ices J(θ).
We can de ine K= J(θ) o K= log de J(θ).
3. In insic Dimensionali y: As lea ning p oceeds, neu al ep esen a ions o en collapse i el-
e an a ia ion [2]. This educ ion in e ec i e dimension co esponds o an inc ease in me ic
aniso opy β(K).
6 Discussion
We e u n o he cen al ques ion: Is knowledge eally a dimension?
I we de ine a dimension solely as a spa ial axis, he answe is no. Bu i we de ine a dimension as
a coo dina e ha go e ns geome ic s uc u e, he answe is yes. Knowledge sha es he undamen al
p ope y o ime in ela i i y: i does no exis as an objec wi hin he space, bu a he as a
pa ame e ha dic a es he cu a u e and dis ance ela ions o he space i sel .
By modeling knowledge as an in o ma ion dimension K, we gain a uni ied language o desc ibe
how lea ning ans o ms he ” la ” noise o a no ice in o he s uc u ed, high- esolu ion mani old
o an expe .
Re e ences
[1] S. Ama i. In o ma ion Geome y and I s Applica ions. Sp inge , 2016.
[2] Ansuini, A., Laio, A., Macke, J. H., & Zoccolan, D. (2019). In insic dimension o da a ep e-
sen a ions in deep neu al ne wo ks. Neu IPS.
[3] T. Co e and J. Thomas. Elemen s o In o ma ion Theo y. Wiley, 2006.
[4] C u ch ield, J. P. (1990). In o ma ion and i s me ic. In L. Lam & H. C. Mo is (Eds.), Nonlinea
S uc u es in Physical Sys ems. Sp inge .
[5] Mohammad-Dja a i, A. (2015). In o ma ion Geome y and Bayesian In e ence. En opy, 17(7),
3989–4027.
[6] Oizumi, M., Tsuchiya, N., & Ama i, S. (2016). Uni ied amewo k o in o ma ion in eg a ion
based on in o ma ion geome y. P oceedings o he Na ional Academy o Sciences, 113(51),
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