dimensional · 2020-03-27 · dimensional reduction idea: data in high-d space400 neurons 㱺400d...
TRANSCRIPT
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) = [email protected]) 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