Dimensional ReductionIdea : data in high - D space400 neurons ⇒ 400 D

Lives on low - D manifold

*← ID manifold

¥720manifold

Idea : I latent facts (small # I

that drove the data (activity

PCA : General idea .. model dsfa

as coming from high - D'"a

problem

i

ID : pcxi-ztq.e-x-ET.EE:7:Ltahe mean = O

ZD &P(xn) =PCx) PCD

¥I×ND - P(x, , ka . . . , xn) = PCX

. ) Pcxz) - . - Paw)res

=d-⇐Thoros. .oI

e- E¥¥

" "EI:O..)

= he' 'zxTcI

c-'

x. = ex . - - - '" 'E .

. .÷)f÷)= E ¥! he

-EEC't

t¥÷ .✓ = E

- '

✓~

- -

= E TEi Ej = Sij -

-

⇒ e-'="

-

er er r

e,

er 'M )E-

'stir;) e- It's )

KIKIE"-

- Efird= # E. Mtd

I -- E'reme → E'me .- nie # (F) =L (Y)( t -

- J⇐ E

-'

MEE-'

y-

LEF E -- 'e

old -1 new

NF -- E

- 'ME I -- E

NEE old#now 044 then.> old

now → old

Ei - Ej= Sij E orthogonalE-' '

= ET EET=ETE=I

Back to Gaussian

PCH -_ke¥'±map toe.

c-so- Ico"Geir.

c- '→ o- 'c-'

Ox÷÷÷÷÷÷÷÷:* to

¥-7I=he¥EE'E

Cij -_ (xix; > = Sijo'

C = SEXT>8=0450--0- 'Lexi > O

= (o-'k¥0 >= LEET>

Conclusion : Gaussian distribution inarbitrary orthug basis

is tee-IC- '

I= PH) e -' e-Em

where C -- GET>

But : there's a special basisin which C is diagonal- eigenvector basis of C

& in that basis fxixj3-sijo.ieSo diag entries of C d re variances

& C is symmetric cxxtyt = xxt⇒ always has a complete

orthonormal basis of eigeruco'sw/ real eigenvalues

Recall : Gaussian is the Max entropydistribution w/ given meandvariance on to ,d)

ID Msx entropy constraining Lf 7,t.se?xHL(PCxH---fdxPCIxlnPCxI

+ do ffdx PG) - I )-) t y

, ( fdx Pcxsf,Cx) - G.GD)+ Xz (fax PG) fzcx) - Azt . . .

SLJp = - In PCX) - I t to t d , f. (x) thefts

= O t - - -

PG) = eI@If.Cx) t feast.. -

f. (x ) = X fz (x ) = CX - Cx> It

e-theCx-K)) 't text )- Cx-Xod'

=) e Tor

PCA -- mean & CoV⇒ principal axes = eigvec 's ofcov

④ Zero - mean data→ E

(2) Find C= Leet)

(3) PCI = Eigvec ofC w/ most var

PC L =' ' -c

' ' w 2"frost was

etc .

¥fPc 2→ Pick # that have X7o of variance

⑦"var

Relationship to SVD

reunionsTinie

M -

.NXT T

MMT : NXN covariance -

neuron neuron CoV

(a. a;) --Imm 'T;

supMTM : Txt time - time covariance

M= US VT UUT--Ip p T VVT -_I

NXN diag TXT

NXT

MMT -_ us ✓ TVSTUTNki¥TxN

marquee. I = uns.int Lies)- -

eigenvectors = columns of✓

eigenvalues = Sf

M -

- USVT

Mtm -- VSTUTUSVTTXT Txw ¥nxT Txt

= V S2 VT

eigenvectors are columns ofV

w/ eigenvalues Sf

Neuron PCA = eigenvectors of MMT

of UTime PCA = I ' u MTM

u r

m -- § Ekiti-

nm"

M = Nx # trials X # stain x # time)

Variants on PCA

Demised PCA (DPCA)kobdk

.. --

I machens

Find components w/2016

most variance about some b.Spectordata

Data: neurons x time x stimuli x decisionsN t s d

x trialsNx (SDK) Itsuki k

ItsD= # tsdk>k

X tsd = (Xtsdk>kI -- Gtsd) tsdIt = Lxtsd - E)sd

so, I#a - E)ed(Etsd - E)st

Its = (Etsd - I -It - Is -Ed>d

¥.¥÷: :{ ¥Etsd = Itsd - I - It

- Is -Ea -Its- F

-Eod -Xsd

E- tsdk = I tsdk - Itsd

Its ⇐ Is +Its"stimulus team"

Itd ⇐ EdtIed' ' decision term

"

I tsd ⇐ Isdtxtsd ' 'stain -decinteraction"

Itsuki E t Itt Ies tItdtItsd+ Etsdk

X ⇒ Itsulk ke sdk -I

X.E

N x k TSD

NXT unique values

Xts repealed KSD times

X = Xt t Xest Xed t Xtsdtxnoise= Ep Xp t Xnoise

( Xa Xf ) = 0 for a # b

XXT = Ct t Ces t Ced t CtsdtcnoiseNXKSTD TestDX N

= § Co, t noise