From 35e3f27a3588bff4089fcfc69bf46dd1f76e3dde Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Tue, 3 Feb 2026 16:00:20 +0100 Subject: [PATCH 01/12] add rwa note --- chapters/Lezione-6.typ | 163 +++++++++++++++++++++++++++++++++++++++++ main.typ | 3 + 2 files changed, 166 insertions(+) create mode 100644 chapters/Lezione-6.typ diff --git a/chapters/Lezione-6.typ b/chapters/Lezione-6.typ new file mode 100644 index 0000000..a91c78d --- /dev/null +++ b/chapters/Lezione-6.typ @@ -0,0 +1,163 @@ +#import "../template.typ": * + += Lezione-6 + +In pythorch l'unità base sono i tensori. N-Data -> N indicies +Sono ricchi rispetto a i dati numpy. Contiene dei dati e ogni dato ha un tipo: +- Data +- Type +- Device (CPU o GPU) +Inoltre ci sono una serie di proprietà che servono per il machine learning. + +Vengono usati per modellare tutta la parte _nuemerica_ : +- input +- output +- parametri + +Tendenzialmente dato un dato di qualsiasi natura dovremmo andare a conventrilo in un tensore. Il tensore è una sorta di "wrap". + +Un numpyarray quando diventa un tensore condivide la memoria (sono collegati per risparmiare memoria, puntatore alla stessa locazione di memoria). + +Dal punto di vista delle dimensioni un tensore si legge seguendo le parentesi ``` [[[1,2,3],[4,5,6]]]``` è un tensore tri-dimensionale, la shape è `[1,3,3]` + +== Tenosre + +Per inizializzare un tensore usiamo: +```py + import torch + data = [[1,2],[3,4]] + x_data = torch.tensor(data) + print(x_data) +``` +#nota()[ + Il tipo viene dedotto automaticamente dall'interprete. +] + +Inoltre è possibile costruire un tensore da un vettore numpy e viceversa +```py + np_array=np.array(data) + x_np = torh.from_numpy(np_array) +``` +Il tipo di dato e la shape vengono ereditati dal vettore numpy. + +Riassumendo un tensore ha: +- Shape +- Datatype +- Device + +#attenzione()[ + Pythorch di default usa i `float32` di default mentre numpy usa i `float64`. Se ereditiamo da numpy ci potrebbe essere un errore di conversione, bisgona riconvertire corretamente i dati. +] + +=== Operazioni + +Per quanto riguardo il calcolo matriciale ad esempio avviene con l'operatore `@`. Mentre l'operazione element-wise tra due vettor si può usare `*` come una normale moltiplicazione. + +//aggiugnere esempio +Prodotto matriciale tra due tensori di batch diversi: +``` + a = torch.randn(2,4,5,4) + b = torch.randn(2,1,4,3) +``` + +Le prime due dimensioni prendono il nome di batch $(2,4)$ e $(2,1)$ il cuore di questi due batch sono delle matrici di dimensioni $(5,4)$ e $(4,3)$. Le dimensioni dei batch non sono compatibili, faciamo *broadcasting*. Mentre la terza e la quarta dimensione sono il risultato di un prodotto matriciale. +``` +a = torch.randn(2,4,5,4) #[B1, B2, 5, 4] +b = torch.randn(2,1,4,3) #[B1, 1, 4, 3] + +x = torch.matmul(a,b)#[] +``` +//aggiungere immagine +In questo prodotto avviene la replicazione, il missmatch viene coperto dal boradcasting. Ovvero replico la struttura dati con struttura minore tante volte quanto serve per arrivare alla dimensione dell'altro operando. + +il matmul fissa le dimensioni `[...,n,m]*[...,n,m]`. Il batch finale avrà una dimensione $2,4$ mentre sulle due dimensioni avviene il prodotto matriciale così come lo conosciamo. La shape finale di `c.shape== (2,4,5,3)`. + +//aggiungere esempio broadcasting con somma + +Una somam di due vettori 1*4 e 4*1 è una matrice 4*4. + +#nota()[ + Un vettore 1*n e n*1 non sono la stessa cosa, c'è un operazione di trasposizione in mezzo. +] + +Un modo coinciso per fare compoment-wise product è `torch.ensum`. Prende come parametro una stringa sche descrive cosa deve essere moltiplicato e le operazioni: +#esempio()[ + ``` + A = torch.randn(2,3) + B = torch.rand(3,4) + C = torch.einsum(ik,kj->ij) # Dove k è la dimensione comune, sommiamo lungo k + ``` + //aggiungere mapping formula + + - Nel caso di prodotto matrice bvettor è `ij,j -> i` + - Nel caso di prodotto element-wise è `i,j,i,j->i,j` + - Nel caso di matrice a più dimensioni otteniamo: `nij,njk -> nik`. Vado a moltiplicare solamente una dimensione delle matrici, scelgo una dimensione e moltiplico lungo quel asse i vettori. +] + +La mean e la std di un immagine. In questo caso sto facendo la STD sul 3 ovvero il numero di canali, la media per il rosso, blue e verde `x.mean(dim=(0,2,3))` + +#attenzione()[ + La scrittura `tensor.add_(5)` serve per le operazioni in place. Molto importante perchè sto modificando il dato su cui lavoro ma non vado ad espandere il dato di nuovo +] + +== Normalizzazione + +La batch normalization può essere fatta attraverso : +$(B,C,H,W)$ dove: +- B = batch size +- C = numero di canali +- H,W = dimensioni spazioli + +Nei modelli deep i canali sono molto importanti in quanto continuano a cambiare gli embedding in spazi vettoriali di dimensione B. + +Per ogni canale C vado a normalizzare, calcolo media e varianza su tutte le rimanenti variabili. Nella formula $c$ rimane sempre fisso: +$ + mu_c = 1 / ("BHW") sum_(b,h,w) x_(b,c,h,w) \ + sigma^2_c = 1 / ("BHW") sum_(b,h,w) x_(b,c,h,w) - mu_c +$ +Il valore vinale è che ho cambiato il valore del dato $x$ da $hat(x)$. $epsilon$ è un numero molto piccolo per rimanere sopra 0 (?): +$ + hat(x)_(b,c,h,w) = (x_(b,c,h,w)-mu_c)/(sqrt(sigma^2_c + epsilon)) +$ +Tutto questo lo fa il moduli `batchNorm2d`. + +== Tensor Reshaping + +I data sono memorizati in maniera lineare in memoria. `view` e `reshape` cambiano la dimensione dei dati senza copiarli. + +Quando creiamod dei tensori creiamo dei blocchi contigui. + +```py + x.arrange(12) # x è un tentore da 0 a 12 + x.view(3,4) # stessi dati di x riorganizzati in 3*4 +``` +#nota()[ + `View` lavora solamente su dati contigui. + ```py + x = torch.arange(12).view(3,4) + y = x.trnspose(0,1) + z = y.view(-1) #inferisce lui le dimensioni + ``` + La matrice viene trasposta ed è per quello che non trova più la contiguita, diventa una matrice $4*3$, di conseguenza ho una riga in pi che non ci sta nello spazio allcoato precedentemente. +] + +`reshape` è molto più flessibile, può implicare una ricopiatura dei dati se necessario. + + + + + + +#esempio()[ + Dato un batch di immagini, ovvero un tensore con $(B,3,H,W)$ dove $B$ numero di immagini, $3$ canali e $H e W$ dimensione dell'immagine. + Vogliamo computare la meida tra tutti i canali $R,G,B$. Dove: + $ + x -> hat(x) = (x-mu)/sigma + $ + Dove prima abbatto i picchi dividendo per la deviazioen tramite $sigma$ poi vado a traslare sottraendo $mu$ (centro) rispetto alla media. + + Alla fine verifico che la media sia circa zero e la deviazione sia crica 1. +] + + + diff --git a/main.typ b/main.typ index 03046f3..e72b847 100644 --- a/main.typ +++ b/main.typ @@ -36,3 +36,6 @@ #include "chapters/Lezione2-Gerarchia-Thread.typ" #include "chapters/Lezione3-Warp.typ" #include "chapters/Lezione4.typ" + + +#include "chapters/Lezione-6.typ" From 3a88edbebace3520fba8b8be8229fbfdf4e2d8f7 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Tue, 3 Feb 2026 16:06:38 +0100 Subject: [PATCH 02/12] update --- chapters/Lezione-6.typ | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/chapters/Lezione-6.typ b/chapters/Lezione-6.typ index a91c78d..315e088 100644 --- a/chapters/Lezione-6.typ +++ b/chapters/Lezione-6.typ @@ -138,7 +138,14 @@ Quando creiamod dei tensori creiamo dei blocchi contigui. y = x.trnspose(0,1) z = y.view(-1) #inferisce lui le dimensioni ``` - La matrice viene trasposta ed è per quello che non trova più la contiguita, diventa una matrice $4*3$, di conseguenza ho una riga in pi che non ci sta nello spazio allcoato precedentemente. + La matrice viene trasposta ed è per quello che non trova più la contiguita. + + Il concetto di contiguità significa: "I numeri che sono vicini nella matrice (logica) sono vicini anche nella striscia di memoria RAM (fisica)? + + - Matrice originale ($3 times 4$): La prima riga è `0, 1, 2, 3`. In memoria sono seduti uno accanto all'altro? Sì (Contiguo). + - Matrice trasposta ($4 times 3$): La prima riga ora è composta dai numeri `0, 4, 8`. In memoria sono seduti vicini? No! Tra lo 0 e il 4 ci sono sedute altre persone (1, 2, 3) che appartengono ad altre righe della nuova matrice. + + Perché view(-1) si rompe? Il comando view(-1) è "stupido": dice "Dammi tutti i dati in fila, dall'inizio alla fine della memoria". Se lo fai sulla trasposta, view andrebbe a leggere la memoria fisica nell'ordine originale: 0, 1, 2, 3....Ma la tua matrice trasposta logicamente dovrebbe iniziare con 0, 4, 8....C'è un disaccordo tra l'ordine fisico e quello logico. PyTorch se ne accorge e ti dice: "Non posso darti una vista piatta (view) perché se leggo la memoria di fila ti do i numeri nell'ordine sbagliato rispetto alla tua trasposta". ] `reshape` è molto più flessibile, può implicare una ricopiatura dei dati se necessario. From e5aa39f3ed2f240047f61aee358a51c1ce2ccf0c Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Tue, 3 Feb 2026 17:06:34 +0100 Subject: [PATCH 03/12] updat e --- chapters/Lezione-6.typ | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/chapters/Lezione-6.typ b/chapters/Lezione-6.typ index 315e088..30585bd 100644 --- a/chapters/Lezione-6.typ +++ b/chapters/Lezione-6.typ @@ -166,5 +166,17 @@ Quando creiamod dei tensori creiamo dei blocchi contigui. Alla fine verifico che la media sia circa zero e la deviazione sia crica 1. ] +== Numpy e Tensor + +Attenzione condividono memoria. Può essere pericoloso, nel molemto in cui passo a un tensore da un numpy array e faccio operazione sul tensore allora modifico anche l'array numpy originale +//aggiugnere esempio + +`unsqueexe(dim)` = aggiunge nuove dimensioni di dimensione 1 alla posizione dim. +`squeeze(dim)` = operazione contraria, rimuove degli $1$ dalle dimensioni + +La concatenazione permette di concatenare tensori diversi su una certa dimensione. + +`torch.stack` + From e361a66bd4ed7f043af9796b1d283a06ea642f21 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Wed, 4 Feb 2026 14:19:23 +0100 Subject: [PATCH 04/12] rename --- chapters/{Lezione-6.typ => Lezione-7.typ} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename chapters/{Lezione-6.typ => Lezione-7.typ} (100%) diff --git a/chapters/Lezione-6.typ b/chapters/Lezione-7.typ similarity index 100% rename from chapters/Lezione-6.typ rename to chapters/Lezione-7.typ From b9a8f8c1c8fe1f373390aefc27cd252af772b6df Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Wed, 4 Feb 2026 15:33:01 +0100 Subject: [PATCH 05/12] update --- main.typ | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/main.typ b/main.typ index e72b847..8b6d544 100644 --- a/main.typ +++ b/main.typ @@ -38,4 +38,4 @@ #include "chapters/Lezione4.typ" -#include "chapters/Lezione-6.typ" +#include "chapters/Lezione-7.typ" From 005b1806a4b60a0b631a1144f20aa57da003b1fa Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Fri, 6 Feb 2026 10:44:35 +0100 Subject: [PATCH 06/12] update --- chapters/Lezione-7.typ | 2 +- main.typ | 2 ++ 2 files changed, 3 insertions(+), 1 deletion(-) diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index 30585bd..fc0cf29 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -1,6 +1,6 @@ #import "../template.typ": * -= Lezione-6 += Pythorch In pythorch l'unità base sono i tensori. N-Data -> N indicies Sono ricchi rispetto a i dati numpy. Contiene dei dati e ogni dato ha un tipo: diff --git a/main.typ b/main.typ index 8b6d544..1597910 100644 --- a/main.typ +++ b/main.typ @@ -32,10 +32,12 @@ // YOUR CONTENT STARTS HERE // ============================================================================ // +#part("Cuda Nimba") #include "chapters/Lezione1-Introduzione.typ" #include "chapters/Lezione2-Gerarchia-Thread.typ" #include "chapters/Lezione3-Warp.typ" #include "chapters/Lezione4.typ" +#part("Pythorc") #include "chapters/Lezione-7.typ" From a14813eeb962bfda0f7e36fa50b02446eab21d8e Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Fri, 6 Feb 2026 13:00:59 +0100 Subject: [PATCH 07/12] update --- chapters/Lezione-7.typ | 76 +++++++++++++++++++++++------------------- 1 file changed, 42 insertions(+), 34 deletions(-) diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index fc0cf29..853a033 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -1,54 +1,62 @@ #import "../template.typ": * -= Pythorch += PyTorch -In pythorch l'unità base sono i tensori. N-Data -> N indicies -Sono ricchi rispetto a i dati numpy. Contiene dei dati e ogni dato ha un tipo: -- Data -- Type -- Device (CPU o GPU) -Inoltre ci sono una serie di proprietà che servono per il machine learning. +Si tratta di una libreria che permette di andare a costruire modelli di deep learning. La maggior parte della libreria è scritta in `C++/Cuda`. -Vengono usati per modellare tutta la parte _nuemerica_ : -- input -- output -- parametri +== Tensori -Tendenzialmente dato un dato di qualsiasi natura dovremmo andare a conventrilo in un tensore. Il tensore è una sorta di "wrap". +I tensori sono l'*unità base* di PyTorch. Si tratta di una generalizzazione di una scalare, vettore o matrice (una sorta di _wrap_). Aggiunge una serie di operazioni in più.\ +Un tensore può avere diverse dimensioni: +- $0D -> "Scalari"$ +- $1D -> "Vettori"$ +- $2D -> "Matrici"$ +- $3D+ -> "Tensori a grande dimensioni"$ -Un numpyarray quando diventa un tensore condivide la memoria (sono collegati per risparmiare memoria, puntatore alla stessa locazione di memoria). +In particolare un tensore, contiene: +- *Dati*, valori che incapsula +- *Type* (`dtype`), i dati che contiene avranno un certo tipico. Il tipo determina: precisione, utilizzo di memoria, operazioni valide +- *Device*, se risiede su `CPU` o `GPU`. Possono quindi essere eseguiti sulla GPU e possono sfruttare l'accelerazione hardware -Dal punto di vista delle dimensioni un tensore si legge seguendo le parentesi ``` [[[1,2,3],[4,5,6]]]``` è un tensore tri-dimensionale, la shape è `[1,3,3]` - -== Tenosre +Vengono usati per modellare tutta la parte _nuemerica_ del nostro modello: input, output e iperparametri. -Per inizializzare un tensore usiamo: +Per inizializzare un tensore da dei dati, possiamo: ```py import torch data = [[1,2],[3,4]] x_data = torch.tensor(data) print(x_data) ``` + #nota()[ Il tipo viene dedotto automaticamente dall'interprete. ] -Inoltre è possibile costruire un tensore da un vettore numpy e viceversa +Inoltre è possibile costruire un tensore da un array `numpy` e viceversa: ```py - np_array=np.array(data) - x_np = torh.from_numpy(np_array) +import numpy as np +data = [[1,2],[3,4]] +np_array = np.array(data) +x_np = torch.from_numpy(np_array) +print(x_np) +``` +Il vantaggio è che le prorpeità (`shape,datatype`) del vettore `numpy` vengono *ereditate* dal `tensor`, a meno che non vengano sovrascritte: +```py +x_ones = torch.ones_like(x_data) # [[1,1],[1,1]] +x_rand = torch.rand_like(x_data, dtype=torch.float) +#[[1.2,0.3],[0.23,2.3]] ``` -Il tipo di dato e la shape vengono ereditati dal vettore numpy. - -Riassumendo un tensore ha: -- Shape -- Datatype -- Device - #attenzione()[ - Pythorch di default usa i `float32` di default mentre numpy usa i `float64`. Se ereditiamo da numpy ci potrebbe essere un errore di conversione, bisgona riconvertire corretamente i dati. + Quando si converte un vettore `numpy` ad un `tensor` ci sono due aspetti da considerare: + - `PyTorch` di default usa i `float32`, mentre `numpy` usa i `float64`. Se ereditassimo direttamente da `numpy` ci potrebbe essere un errore di conversione, bisgona *riconvertire* corretamente i dati. + + - Un `numpy` array quando diventa un `tensor`, continua a *condividere la memoria* (puntatore alla stessa locazione di memoria). Se si effetuano dei cambiamenti sul `tensor`, influiscono anche sull'array `numpy`. ] +=== Shape + +Dal punto di vista delle dimensioni un tensore si legge seguendo le parentesi ``` [[[1,2,3],[4,5,6]]]``` è un tensore tri-dimensionale, la shape è `[1,3,3]` + === Operazioni Per quanto riguardo il calcolo matriciale ad esempio avviene con l'operatore `@`. Mentre l'operazione element-wise tra due vettor si può usare `*` come una normale moltiplicazione. @@ -140,11 +148,11 @@ Quando creiamod dei tensori creiamo dei blocchi contigui. ``` La matrice viene trasposta ed è per quello che non trova più la contiguita. - Il concetto di contiguità significa: "I numeri che sono vicini nella matrice (logica) sono vicini anche nella striscia di memoria RAM (fisica)? - + Il concetto di contiguità significa: "I numeri che sono vicini nella matrice (logica) sono vicini anche nella striscia di memoria RAM (fisica)? + - Matrice originale ($3 times 4$): La prima riga è `0, 1, 2, 3`. In memoria sono seduti uno accanto all'altro? Sì (Contiguo). - Matrice trasposta ($4 times 3$): La prima riga ora è composta dai numeri `0, 4, 8`. In memoria sono seduti vicini? No! Tra lo 0 e il 4 ci sono sedute altre persone (1, 2, 3) che appartengono ad altre righe della nuova matrice. - + Perché view(-1) si rompe? Il comando view(-1) è "stupido": dice "Dammi tutti i dati in fila, dall'inizio alla fine della memoria". Se lo fai sulla trasposta, view andrebbe a leggere la memoria fisica nell'ordine originale: 0, 1, 2, 3....Ma la tua matrice trasposta logicamente dovrebbe iniziare con 0, 4, 8....C'è un disaccordo tra l'ordine fisico e quello logico. PyTorch se ne accorge e ti dice: "Non posso darti una vista piatta (view) perché se leggo la memoria di fila ti do i numeri nell'ordine sbagliato rispetto alla tua trasposta". ] @@ -171,12 +179,12 @@ Quando creiamod dei tensori creiamo dei blocchi contigui. Attenzione condividono memoria. Può essere pericoloso, nel molemto in cui passo a un tensore da un numpy array e faccio operazione sul tensore allora modifico anche l'array numpy originale //aggiugnere esempio -`unsqueexe(dim)` = aggiunge nuove dimensioni di dimensione 1 alla posizione dim. +`unsqueexe(dim)` = aggiunge nuove dimensioni di dimensione 1 alla posizione dim. `squeeze(dim)` = operazione contraria, rimuove degli $1$ dalle dimensioni -La concatenazione permette di concatenare tensori diversi su una certa dimensione. +La concatenazione permette di concatenare tensori diversi su una certa dimensione. -`torch.stack` +`torch.stack` From 90f22a421fc617812d1213531be020df4ec7894a Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Fri, 6 Feb 2026 15:57:45 +0100 Subject: [PATCH 08/12] update --- assets/broadcasting-result.png | Bin 0 -> 592108 bytes assets/broadcasting.png | Bin 0 -> 580142 bytes chapters/Lezione-7.typ | 78 ++++++++++++++++++++++++++------- 3 files changed, 61 insertions(+), 17 deletions(-) create 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z@u8c8Bi}$wm4{Z@wT(L{xa<-$QwXA+Xg=ge5#V~>Xo~UPk;Ei1wAMgXgMM{k{(V`Hy{8L4%7af0J6P+K1I_gFLF!A^U5nR9wWYB&NnDKMQ}bHgO2j zS&Q%LAsNWmQob!ANoDu&;3HXi{RtEarJ|FNx9OIyOlo07b>Zyqk7kfu8*O zA4YIO8j%f4+sy~n&;n)WBFK5hWyr8P{zDMTj%7qn{jJb5zBdagU_gziGj3j~I zf*Tpg?@!nH(>ce|)5Q5{_sp3-w^+Sq!1QpaP7OZvq|G;xL>L(Po$|-Vd&f5zS%V^{>vFl!#<$DRG_?&6NCY9JEe| z6(p!XU)~g(iX5oG#Wv-rX{0QFB ziS5?z8@J-PCPS=w#ZNo9oYf$yl6@LrjHWnad9o z#q>mvwjrqMwk+u^Wu^GAN$0EfK?g!J)u8Na?ZNjk^S0MmYF;n2ix}zD92o)02*5{0 LL0!I9);#Qgle`wL literal 0 HcmV?d00001 diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index 853a033..4e904b9 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -55,30 +55,74 @@ x_rand = torch.rand_like(x_data, dtype=torch.float) === Shape -Dal punto di vista delle dimensioni un tensore si legge seguendo le parentesi ``` [[[1,2,3],[4,5,6]]]``` è un tensore tri-dimensionale, la shape è `[1,3,3]` +La dimensione di un `tensor` si leggono dall'esterno verso l'interno. Ad esempio: +```py + tensor( + [[ + [1,2,3], + [4,5,6], + [7,8,9] + ]] + ) + torch.size([1,3,3]) # dim 0,1,2 +``` +In questo caso è un tensore 3D, c'è una dimensione $3*3$. -=== Operazioni +== Operazioni -Per quanto riguardo il calcolo matriciale ad esempio avviene con l'operatore `@`. Mentre l'operazione element-wise tra due vettor si può usare `*` come una normale moltiplicazione. +Di default le operazioni tra tensori possono essere eseguite sia sulla `CPU` che sulla `GPU`, generalmente i tensori vengono allocati inizialmente sulla `CPU`. -//aggiugnere esempio -Prodotto matriciale tra due tensori di batch diversi: -``` - a = torch.randn(2,4,5,4) - b = torch.randn(2,1,4,3) -``` +Esistono inoltre degli operatori caricati, ad esempio: +- `@`: utilizzato per *operazioni di algebra lineare*, come il prodotto matriciale +- `*`: utilizzato per *element-wise product*, ad esempio prodotto degli elementi cella per cella di un vettore -Le prime due dimensioni prendono il nome di batch $(2,4)$ e $(2,1)$ il cuore di questi due batch sono delle matrici di dimensioni $(5,4)$ e $(4,3)$. Le dimensioni dei batch non sono compatibili, faciamo *broadcasting*. Mentre la terza e la quarta dimensione sono il risultato di un prodotto matriciale. -``` -a = torch.randn(2,4,5,4) #[B1, B2, 5, 4] -b = torch.randn(2,1,4,3) #[B1, 1, 4, 3] +=== Broadcasting -x = torch.matmul(a,b)#[] +Si tratta di un'operazione fatta in automatico da `PyTorch` quando i *batch* (prime due dimensioni) di due tensori non corrispondono: +```py +a = torch.randn(2, 4, 5, 4) # [B1, B2, 5, 4] +b = torch.randn(2, 1, 4, 3) # [B1, 1, 4, 3] +c = torch.matmul(a, b) +print("Shape of c:", c.shape) # [2, 4, 5, 3] ``` -//aggiungere immagine -In questo prodotto avviene la replicazione, il missmatch viene coperto dal boradcasting. Ovvero replico la struttura dati con struttura minore tante volte quanto serve per arrivare alla dimensione dell'altro operando. +Nell'esempio la dimensione dei batch di $a$ e $b$ non corrispondono, in particolare: +- $a "batch"(2,4)$ +- $b "batch"(2,1)$ +#figure( + image("../assets/broadcasting.png", width: 65%), + caption: [ + Rappresentazione dei tensori $a$ e $b$ (input) + ], +) +Siccome le dimensioni dei batch non corrispondono viene fatto *broadcasting* (in modo implicito). Il tensore con dimensione minore viene espanso tante volte quanto serve per arrivare alla dimensione dell'operando più grande, in modo da croprire così il missmatch.\ +Nell'esempio sorpa il la seconda dimensione del batch del tenosore $b$ viene posta a $4$. Il tensore $c$ risultante avrà una `c.shape == (2, 4, 5, 3)`, dove: +- $2 ->$ batch esterno +- $4 ->$ gruppo interno del batch +- $5 ->$ numero di righe per ogni matrice +- $3 ->$ numero di colonne per ogni matrice + +#figure( + image("../assets/broadcasting-result.png", width: 65%), + caption: [ + Rappresentazione dei tensori $a$ e $b$ (input) + ], +) + +#attenzione()[ + Non tutte le forme di tensori sono compatibili. Affinché il broadcasting funzioni, `PyTorch` confronta le dimensioni dei due tensori partendo da *destra verso sinistra* (dall'ultima dimensione alla prima). Due dimensioni sono *compatibili* se: + - Sono uguali + - Una delle due è $1$. + + Se un tensore ha meno dimensioni dell'altro, `PyTorch` aggiunge virtualmente delle dimensioni $1$ a sinistra. +] + + + + + + + -il matmul fissa le dimensioni `[...,n,m]*[...,n,m]`. Il batch finale avrà una dimensione $2,4$ mentre sulle due dimensioni avviene il prodotto matriciale così come lo conosciamo. La shape finale di `c.shape== (2,4,5,3)`. //aggiungere esempio broadcasting con somma From cfd36abc4d589e5811eb5a34cf01ec865f9800aa Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Fri, 6 Feb 2026 16:17:41 +0100 Subject: [PATCH 09/12] add img --- chapters/Lezione-7.typ | 128 ++++++++++++++++++++++++++++++++++++++--- 1 file changed, 121 insertions(+), 7 deletions(-) diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index 4e904b9..8a00740 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -116,21 +116,135 @@ Nell'esempio sorpa il la seconda dimensione del batch del tenosore $b$ viene pos Se un tensore ha meno dimensioni dell'altro, `PyTorch` aggiunge virtualmente delle dimensioni $1$ a sinistra. ] +#esempio()[ + Supponiamo di avere il seguente codice: + ```py + A = torch.tensor([[1.], [2.], [3.], [4.]]) + B = torch.tensor([[5., -5., 5., -5., 5.]]) + C = A+B + ``` + Dove $A$ è un vettore colonna $4 times 1$, mentre $B$ è un vettore riga $1 times 5$. In questo caso sia $A$ che $B$ vengono espansi ad una matrice $4 times 4$: + + #figure( + scale(80%)[ + #grid( + columns: (auto, auto, auto), + column-gutter: 1.5em, + row-gutter: 0.8em, + align: horizon, + + // Colonna sinistra: A e B + [ + #align(center)[ + #text(size: 8pt, weight: "bold", fill: blue)[*A*] + #table( + columns: 1, + stroke: 0.5pt + black, + align: center + horizon, + inset: 3pt, + [1], + [2], + [3], + [4], + ) + ] + + #v(0.8em) + + #align(center)[ + #text(size: 8pt, weight: "bold", fill: blue)[*B*] + #table( + columns: 5, + stroke: 0.5pt + black, + align: center + horizon, + inset: 3pt, + [5], [−5], [5], [−5], [5], + ) + ] + ], + + // Colonna centrale: frecce e replicazioni + [ + #v(0.3em) + #align(center)[ + #text(size: 7pt, fill: red, weight: "bold")[replicate] + #text(size: 10pt, fill: red)[⟷] + + #v(0.2em) + + #table( + columns: 5, + stroke: 0.5pt + black, + align: center + horizon, + inset: 3pt, + [1], [1], [1], [1], [1], + [2], [2], [2], [2], [2], + [3], [3], [3], [3], [3], + [4], [4], [4], [4], [4], + ) + ] + + #v(0.8em) + + #align(center)[ + #text(size: 7pt, fill: red, weight: "bold")[replicate] + #text(size: 10pt, fill: red)[⟺] + + #v(0.2em) + + #table( + columns: 5, + stroke: 0.5pt + black, + align: center + horizon, + inset: 3pt, + [5], [−5], [5], [−5], [5], + [5], [−5], [5], [−5], [5], + [5], [−5], [5], [−5], [5], + [5], [−5], [5], [−5], [5], + ) + ] + ], + + // Colonna destra: frecce e risultato + [ + #v(1.5em) + #align(center)[ + #text(size: 12pt)[↘] + + #v(0.3em) + + #table( + columns: 5, + stroke: 0.5pt + black, + align: center + horizon, + inset: 3pt, + [6], [−4], [6], [−4], [6], + [7], [−3], [7], [−3], [7], + [8], [−2], [8], [−2], [8], + [9], [−1], [9], [−1], [9], + ) + + #v(0.3em) + #text(size: 9pt, weight: "bold", fill: blue)[$C = A + B$] + + #v(1.5em) + #text(size: 14pt)[↗] + ] + ], + ) + ], + caption: [Broadcasting nella somma], + ) +] +=== Torch ensum -//aggiungere esempio broadcasting con somma - -Una somam di due vettori 1*4 e 4*1 è una matrice 4*4. - -#nota()[ - Un vettore 1*n e n*1 non sono la stessa cosa, c'è un operazione di trasposizione in mezzo. -] Un modo coinciso per fare compoment-wise product è `torch.ensum`. Prende come parametro una stringa sche descrive cosa deve essere moltiplicato e le operazioni: #esempio()[ From 4ff43d09a8f202278947e11dde38015029479c9f Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Fri, 6 Feb 2026 16:52:53 +0100 Subject: [PATCH 10/12] update --- chapters/Lezione-7.typ | 46 ++++++++++++++++++++++++++++++------------ 1 file changed, 33 insertions(+), 13 deletions(-) diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index 8a00740..7f8dc62 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -238,27 +238,47 @@ Nell'esempio sorpa il la seconda dimensione del batch del tenosore $b$ viene pos ] -=== Torch ensum +=== Torch Einsum +Si tratta di una notazione per effetuare in modo coinciso operazioni compoment-wise. `torch.einsum` prende come argomento una stringa che descrive: +- l'operazione che si vuole effettuare +- i tensori su cui si deve operare +- il tensore risultato +#nota()[ + L'operazione viene eseguita su tutti gli indici che non appaiono tra gli indici del risultato. +] + +*Prodotto tra matrici*: Date due matrici $P "e" Q$ il prodotto: +$ + M_(i,k) = sum_j P_(i,j)Q_(j,k) +$ +può essere scritta come `M = torch.einsum('ij,jk->ik', P, Q)` + +*Prodotto matrice vettore*: Dato un vettore $M$ e $v$: +$ + w_i = sum_j M_(i,j)v_j +$ +può essere scritta come `torch.einsum("ij,j->i", M, v)` + +*Prodotto element-wise*: Date due matrici $P "e" Q$: +$ + M_(i,j) = P_(i,j)*Q_(i,j) +$ +può essere scritta come `torch.einsum("ij,ij->ij", P, Q)` +*Prodotto Matriciale a batch*: Dati due tensori $3D$ $P "e" Q$: +$ + M_(n,i,k) = sum_j (P_(n,i,j)Q_(n,j,k)) +$ +può essere scritta come `torch.einsum("nij,njk->nijk", P, Q)` + +=== Media e Deviazione -Un modo coinciso per fare compoment-wise product è `torch.ensum`. Prende come parametro una stringa sche descrive cosa deve essere moltiplicato e le operazioni: -#esempio()[ - ``` - A = torch.randn(2,3) - B = torch.rand(3,4) - C = torch.einsum(ik,kj->ij) # Dove k è la dimensione comune, sommiamo lungo k - ``` - //aggiungere mapping formula - - Nel caso di prodotto matrice bvettor è `ij,j -> i` - - Nel caso di prodotto element-wise è `i,j,i,j->i,j` - - Nel caso di matrice a più dimensioni otteniamo: `nij,njk -> nik`. Vado a moltiplicare solamente una dimensione delle matrici, scelgo una dimensione e moltiplico lungo quel asse i vettori. -] La mean e la std di un immagine. In questo caso sto facendo la STD sul 3 ovvero il numero di canali, la media per il rosso, blue e verde `x.mean(dim=(0,2,3))` From 142091e960b344b35adcaf1d527b592ecde703cf Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Sat, 7 Feb 2026 12:42:45 +0100 Subject: [PATCH 11/12] update --- chapters/Lezione-7.typ | 75 +++++++++++++++++++++++++++++++++--------- 1 file changed, 59 insertions(+), 16 deletions(-) diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index 7f8dc62..e07b3d3 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -76,6 +76,11 @@ Esistono inoltre degli operatori caricati, ad esempio: - `@`: utilizzato per *operazioni di algebra lineare*, come il prodotto matriciale - `*`: utilizzato per *element-wise product*, ad esempio prodotto degli elementi cella per cella di un vettore +#attenzione()[ + La scrittura `tensor.add_(5)` serve per le operazioni *in place*. Esse modificano il tensore in maniera diretta. +] + + === Broadcasting Si tratta di un'operazione fatta in automatico da `PyTorch` quando i *batch* (prime due dimensioni) di due tensori non corrispondono: @@ -253,7 +258,7 @@ Si tratta di una notazione per effetuare in modo coinciso operazioni compoment-w $ M_(i,k) = sum_j P_(i,j)Q_(j,k) $ -può essere scritta come `M = torch.einsum('ij,jk->ik', P, Q)` +può essere scritta come `torch.einsum('ij,jk->ik', P, Q)` *Prodotto matrice vettore*: Dato un vettore $M$ e $v$: $ @@ -271,41 +276,79 @@ può essere scritta come `torch.einsum("ij,ij->ij", P, Q)` $ M_(n,i,k) = sum_j (P_(n,i,j)Q_(n,j,k)) $ -può essere scritta come `torch.einsum("nij,njk->nijk", P, Q)` +può essere scritta come `torch.einsum("nij,njk->nik", P, Q)` === Media e Deviazione +Calcolare la media e la deviazione standard sui tensori è molto comune. In particolare, vengono utilizzate per eseguire data normalization e batch normalization nei modelli di deep learning. Le funzioni sono due: +- `torch.mean()` +- `torch.std()` +In entrambe le funzioni è possibile specificare su quali dimensioni del tensore non si vuole fare la media. + +#table( + columns: 5, + [_Input_], [_Shape_], [_Media_], [_STD_], [_Risultato_], + [Vettore $1D$], [`[N]`], [`x.mean()`], [`x.std()`], [Scalare], + [Matrice $2D$], [`[M,N]`], [`x.mean(dim=0 or dim=1)`], [`x.std(dim=0 or dim=1)`], [Media per riga o colonna], + [Batch immagine], [`[B,3,H,W]`], [`x.mean(0,2,3)`], [`x.std(0,2,3)`], [Media per canale RGB], +) +== Normalizzazione dei batch +Nei modelli di deep learning, la normalizzazione dei batch ha un ruolo molto fondamentale. Tale tecnina viene usata per rendere l'*addestramento* delle reti neurali *più veloce* e *più stabile*. +Tale tecnica, va a risolvere il problema del $mr("internal covariate shift")$. Quando si va ad adestrare una rete neurale, i pesi del primo layer cambiano man mano, cambiando la distribuzione dei dati che arriva al secondo layer. Il secondo layer deve quindi _riadattarsi_ continuamente alla nuova distribuzione, rallentando l'apprendimento.\ +La soluzione prende il nome di $mg("batch norm")$. Essa va a normalizzare l'input di ogni layer nascosto per stabilizzare la distribuzione. Per ogni mini-batch di dati durante il training, il layer BatchNorm esegue $3$ passaggi: -La mean e la std di un immagine. In questo caso sto facendo la STD sul 3 ovvero il numero di canali, la media per il rosso, blue e verde `x.mean(dim=(0,2,3))` +- Calcola la media $mu$ e la varianza $sigma^2$ del batch corrente. -#attenzione()[ - La scrittura `tensor.add_(5)` serve per le operazioni in place. Molto importante perchè sto modificando il dato su cui lavoro ma non vado ad espandere il dato di nuovo +- *Normalizzazione*: Sottrae la media e divide per la deviazione standard (aggiungendo un numero piccolo $epsilon$ per evitare le divisione per zero): + $ + hat(x) = (x - mu)/(sqrt(sigma^2+epsilon)) + $ + I dati del batch avranno ora media $0$ e varianza $1$ + +- *Scale and shift*: I dati normalizzati vengono moltiplicati per un iper-parametro $gamma$ e un parametro $beta$ appreso dal modello: + $ + y = gamma hat(x)+ beta + $ + #nota()[ + Se venisse forzata media $0$ e varianza $1$, potremmo limitare la capacità della rete. $gamma$ e $beta$ permettono alla rete di imparare se le serve una distribuzione diversa (o addirittura di annullare la normalizzazione se necessario). + ] + +#informalmente()[ + I vantaggi introdotti sono due: + - *Velocità*: Permette di usare Learning Rate molto più alti (convergenza rapida). + + - *Stabilità*: Rende la rete meno sensibile all'inizializzazione casuale dei pesi. + + - *Regolarizzazione*: Introduce un leggero "rumore" (poiché le statistiche dipendono dal batch casuale), agendo come un leggero Dropout ed evitando l'overfitting. ] -== Normalizzazione +=== batchNorm2d -La batch normalization può essere fatta attraverso : -$(B,C,H,W)$ dove: -- B = batch size -- C = numero di canali -- H,W = dimensioni spazioli +In Pytorch `nn.BatchNorm2d` è l'implementazione del layer batch norm specifica per dati $4D$ (immagini), tipicamente usata nelle Reti Neurali Convoluzionali (CNN). -Nei modelli deep i canali sono molto importanti in quanto continuano a cambiare gli embedding in spazi vettoriali di dimensione B. +Il layer prende in input un tensore con la seguente shape $(B, C, H, W)$ dove: +- $B$ è la batch size +- $C$ è il numero di canali +- $H,W$ sono le dimensioni spaziali -Per ogni canale C vado a normalizzare, calcolo media e varianza su tutte le rimanenti variabili. Nella formula $c$ rimane sempre fisso: +Il layer esegue una *channel-wise batch normalization*. Per ogni canale $c$ calcola la media e la varianza su tutte le altre dimensioni ($c$ rimane fisso): $ mu_c = 1 / ("BHW") sum_(b,h,w) x_(b,c,h,w) \ - sigma^2_c = 1 / ("BHW") sum_(b,h,w) x_(b,c,h,w) - mu_c + sigma^2_c = 1 / ("BHW") sum_(b,h,w) (x_(b,c,h,w) - mu_c)^2 $ -Il valore vinale è che ho cambiato il valore del dato $x$ da $hat(x)$. $epsilon$ è un numero molto piccolo per rimanere sopra 0 (?): + +#nota()[ + I canali $C$ rappresentano le features estratte (embedding). La BatchNorm normalizza ogni feature indipendentemente dalle altre, ma coerentemente su tutto il batch $B$ e su tutta l'immagine spaziale. +] + +Successivamente viene applicata la normalizzazione al batch: $ hat(x)_(b,c,h,w) = (x_(b,c,h,w)-mu_c)/(sqrt(sigma^2_c + epsilon)) $ -Tutto questo lo fa il moduli `batchNorm2d`. == Tensor Reshaping From 7dc9ec3b27299f1bf07b5edc62c56ed4535b6f8b Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Sat, 7 Feb 2026 13:20:28 +0100 Subject: [PATCH 12/12] update --- chapters/Lezione-7.typ | 576 +++++++++++++++++++++++++++++++++++++++-- 1 file changed, 551 insertions(+), 25 deletions(-) diff --git a/chapters/Lezione-7.typ b/chapters/Lezione-7.typ index e07b3d3..1229c14 100644 --- a/chapters/Lezione-7.typ +++ b/chapters/Lezione-7.typ @@ -352,60 +352,586 @@ $ == Tensor Reshaping -I data sono memorizati in maniera lineare in memoria. `view` e `reshape` cambiano la dimensione dei dati senza copiarli. +Quando viene creato un tensore, i dati vengono memorizati in *maniera lineare* in memoria. Le funzioni `view` e `reshape` cambiano la dimensione dei dati *senza copiarli* (viene cambiata solamente la loro _visualizzazione_). -Quando creiamod dei tensori creiamo dei blocchi contigui. +=== Transpose +L'operazione `transpose()` o `.t()` *scambia due dimensioni* di un tensore. Per matrici 2D, scambia righe e colonne. + +#figure( + cetz.canvas({ + import cetz.draw: * + + // Primo esempio: 2x3 -> 3x2 + content((0, 3), [#text(10pt)[`x = torch.tensor([[1, 3, 0], [2, 4, 6]])`]]) + + // Matrice originale 2x3 + let colors = (green, yellow, red, blue, fuchsia, eastern) + let idx = 0 + for i in range(2) { + for j in range(3) { + rect((j * 0.8, -i * 0.8 + 2), (j * 0.8 + 0.8, -i * 0.8 + 2 - 0.8), fill: colors.at(idx), stroke: black) + idx += 1 + } + } + + // Freccia + content((3, 1.2), text(20pt)[→]) + + // Matrice trasposta 3x2 + content((5.5, 3), [#text(10pt)[`x.t()`]]) + let data = ((0, 3), (2, 4), (1, 5)) + idx = 0 + for i in range(3) { + for j in range(2) { + let color_idx = data.at(i).at(j) + rect( + (j * 0.8 + 4.5, -i * 0.8 + 2), + (j * 0.8 + 4.5 + 0.8, -i * 0.8 + 2 - 0.8), + fill: colors.at(color_idx), + stroke: black, + ) + } + } + }), + caption: [Transpose: scambio tra righe e colonne], +) + +==== Transpose su tensori multidimensionali + +Per tensori con più di 2 dimensioni, `transpose(dim0, dim1)` scambia le dimensioni specificate. Questo è utile per riorganizzare i dati mantenendo le altre dimensioni invariate. + +#figure( + cetz.canvas({ + import cetz.draw: * + + // Tensore originale 4D: shape (2, 2, 3, 1) visualizzato come matrici sovrapposte + content((0, 5), [#text(9pt)[`x = torch.tensor([[[[1, 2, 1],`]]) + content((0, 4.6), [#text(9pt)[` [[2, 1, 2]],`]]) + content((0, 4.2), [#text(9pt)[` [[[3, 0, 3],`]]) + content((0, 3.8), [#text(9pt)[` [[0, 3, 0]]]])`]]) + + // Primo livello (layer 0) + let colors1 = (green, blue, green, yellow, red, yellow) + let data1 = (1, 2, 1, 2, 1, 2) + for i in range(2) { + for j in range(3) { + rect( + (j * 0.5, -i * 0.5 + 3), + (j * 0.5 + 0.5, -i * 0.5 + 3 - 0.5), + fill: colors1.at(i * 3 + j), + stroke: black + 1pt, + ) + } + } + + // Secondo livello (layer 1) - leggermente offset per effetto 3D + let colors2 = (green, eastern, green, eastern, green, eastern) + for i in range(2) { + for j in range(3) { + rect( + (j * 0.5 + 0.15, -i * 0.5 + 3 - 0.15), + (j * 0.5 + 0.5 + 0.15, -i * 0.5 + 3 - 0.5 - 0.15), + fill: colors2.at(i * 3 + j), + stroke: black + 1pt, + ) + } + } + + // transpose(0, 1) - scambia dimensioni 0 e 1 + content((2.5, 2.3), text(14pt)[→]) + content((4.5, 5), [#text(9pt)[`x.transpose(0, 1)`]]) + + // Risultato transpose(0, 1) + let colors1_t01 = (green, green, blue, eastern, green, green) + for i in range(2) { + for j in range(3) { + rect( + (j * 0.5 + 3.5, -i * 0.5 + 3), + (j * 0.5 + 3.5 + 0.5, -i * 0.5 + 3 - 0.5), + fill: colors1_t01.at(i * 3 + j), + stroke: black + 1pt, + ) + } + } + + let colors2_t01 = (yellow, eastern, red, green, yellow, eastern) + for i in range(2) { + for j in range(3) { + rect( + (j * 0.5 + 3.5 + 0.15, -i * 0.5 + 3 - 0.15), + (j * 0.5 + 3.5 + 0.5 + 0.15, -i * 0.5 + 3 - 0.5 - 0.15), + fill: colors2_t01.at(i * 3 + j), + stroke: black + 1pt, + ) + } + } + + // transpose(0, 2) - scambia dimensioni 0 e 2 + content((2.5, 0.8), text(14pt)[→]) + content((4.5, 1.5), [#text(9pt)[`x.transpose(0, 2)`]]) + + // Risultato transpose(0, 2) + let colors1_t02 = (green, yellow, blue, red, green, yellow) + for i in range(2) { + for j in range(3) { + rect( + (j * 0.5 + 3.5, -i * 0.5 + 1.2), + (j * 0.5 + 3.5 + 0.5, -i * 0.5 + 1.2 - 0.5), + fill: colors1_t02.at(i * 3 + j), + stroke: black + 1pt, + ) + } + } + + let colors2_t02 = (green, eastern, green, eastern, green, eastern) + for i in range(2) { + for j in range(3) { + rect( + (j * 0.5 + 3.5 + 0.15, -i * 0.5 + 1.2 - 0.15), + (j * 0.5 + 3.5 + 0.5 + 0.15, -i * 0.5 + 1.2 - 0.5 - 0.15), + fill: colors2_t02.at(i * 3 + j), + stroke: black + 1pt, + ) + } + } + }), + caption: [Transpose su tensori 4D: scambio di dimensioni specifiche], +) + +#informalmente()[ + - `transpose(0, 1)`: scambia le *prime due dimensioni* (es. batch e canali). Utile quando si vuole riorganizzare l'ordine dei batch rispetto ai canali. + - `transpose(0, 2)`: scambia la *prima e terza dimensione* (es. batch e altezza). Permette di riordinare i dati spaziali rispetto al batch. + + La trasposizione è fondamentale in operazioni come la convoluzione trasposta o quando si passano dati tra layer con formati diversi (es. NCHW ↔ NHWC). +] + +#nota()[ + La trasposizione *non copia i dati* in memoria, ma cambia solo l'interpretazione degli indici. Questo può rendere il tensore *non contiguo* in memoria. +] + +=== View + +Esempio di utilizzo di `view`: ```py x.arrange(12) # x è un tentore da 0 a 12 x.view(3,4) # stessi dati di x riorganizzati in 3*4 + tensor([[0,1,2,3], + [4,5,6,7], + [8,9,10,11]]) ``` -#nota()[ - `View` lavora solamente su dati contigui. + +#figure( + cetz.canvas({ + import cetz.draw: * + + // Esempio 1: view(-1) - flattening + content((0, 3.5), [#text(10pt)[`x = torch.tensor([[1, 3, 0], [2, 4, 6]])`]]) + + // Matrice originale 2x3 + let colors = (green, yellow, red, blue, fuchsia, eastern) + let idx = 0 + for i in range(2) { + for j in range(3) { + rect((j * 0.6, -i * 0.6 + 3), (j * 0.6 + 0.6, -i * 0.6 + 3 - 0.6), fill: colors.at(idx), stroke: black) + idx += 1 + } + } + + // Freccia + content((2.5, 2.4), text(16pt)[→]) + + // Vettore 1D + content((5.5, 3.5), [#text(10pt)[`x.view(-1)`]]) + for i in range(6) { + rect((i * 0.6 + 3.5, 3), (i * 0.6 + 3.5 + 0.6, 2.4), fill: colors.at(i), stroke: black) + } + + // Esempio 2: view(3, -1) - rimodellamento + content((0, 1.5), [#text(10pt)[`x = torch.tensor([[1, 3, 0], [2, 4, 6]])`]]) + + // Matrice originale 2x3 + idx = 0 + for i in range(2) { + for j in range(3) { + rect((j * 0.6, -i * 0.6 + 1), (j * 0.6 + 0.6, -i * 0.6 + 1 - 0.6), fill: colors.at(idx), stroke: black) + idx += 1 + } + } + + // Freccia + content((2.5, 0.4), text(16pt)[→]) + + // Nuova forma 3x2 + content((5.5, 1.5), [#text(10pt)[`x.view(3, -1)`]]) + idx = 0 + for i in range(3) { + for j in range(2) { + rect( + (j * 0.6 + 3.5, -i * 0.6 + 1), + (j * 0.6 + 3.5 + 0.6, -i * 0.6 + 1 - 0.6), + fill: colors.at(idx), + stroke: black, + ) + idx += 1 + } + } + }), + caption: [View: riorganizzazione dei dati senza copia], +) + +#attenzione()[ + `View` effettua il reshape solamente di tensori contigui in memoria. ```py x = torch.arange(12).view(3,4) y = x.trnspose(0,1) z = y.view(-1) #inferisce lui le dimensioni ``` - La matrice viene trasposta ed è per quello che non trova più la contiguita. + La matrice viene trasposta (scambio righe colonne) ed è per quello che non trova più la contiguità. - Il concetto di contiguità significa: "I numeri che sono vicini nella matrice (logica) sono vicini anche nella striscia di memoria RAM (fisica)? + #informalmente()[ + Il concetto di contiguità significa: _I numeri che sono vicini nella matrice (logica) sono vicini anche nella striscia di memoria RAM (fisica)_. + ] + + - Matrice originale ($3 times 4$): La prima riga è `0, 1, 2, 3`, i dati in memoria sono contigui. + - Matrice trasposta ($4 times 3$): La prima riga ora è composta dai numeri `0, 4, 8`. Tuttavia, la rappresentazione _logica_ non corrisponde con quella _fisica_. Ad esempio, lo $0$ e il $4$ non sono contigui in memoria, ci sono di mezzo `1, 2, 3` che appartengono ad altre righe della nuova matrice. - - Matrice originale ($3 times 4$): La prima riga è `0, 1, 2, 3`. In memoria sono seduti uno accanto all'altro? Sì (Contiguo). - - Matrice trasposta ($4 times 3$): La prima riga ora è composta dai numeri `0, 4, 8`. In memoria sono seduti vicini? No! Tra lo 0 e il 4 ci sono sedute altre persone (1, 2, 3) che appartengono ad altre righe della nuova matrice. + `view(-1)` da errore in quanto la funzione restituisce i dati in fila, dall'inizio alla fine della memoria. Essendo eseguito sulla matrice trasposta, `view` andrebbe a leggere la memoria fisica nell'ordine originale: `0, 1, 2, 3....`. Tuttavia la matrice trasposta _logicamente_ dovrebbe iniziare con `0, 4, 8....`. C'è quindi un disaccordo tra l'ordine fisico e quello logico. - Perché view(-1) si rompe? Il comando view(-1) è "stupido": dice "Dammi tutti i dati in fila, dall'inizio alla fine della memoria". Se lo fai sulla trasposta, view andrebbe a leggere la memoria fisica nell'ordine originale: 0, 1, 2, 3....Ma la tua matrice trasposta logicamente dovrebbe iniziare con 0, 4, 8....C'è un disaccordo tra l'ordine fisico e quello logico. PyTorch se ne accorge e ti dice: "Non posso darti una vista piatta (view) perché se leggo la memoria di fila ti do i numeri nell'ordine sbagliato rispetto alla tua trasposta". + La soluzione è usare `reshape` in questo caso. ] -`reshape` è molto più flessibile, può implicare una ricopiatura dei dati se necessario. +=== Reshape +`reshape` è molto più flessibile. Tuttavia, può *implicare una ricopiatura dei dati se necessario*. +```py + x = torch.arange(12).view(3, 4) + y = x.t() # transpose -> non-contiguous + z = y.reshape(-1) # works; makes a copy if needed + print(z) + tensor([0,4,8,1,5,9,2,6,10,3,7,11]) +``` +=== Expand +L'operazione `expand()` *replica i dati* lungo le dimensioni specificate *senza allocare memoria aggiuntiva* (usa broadcasting). +#figure( + cetz.canvas({ + import cetz.draw: * + + content((0, 3.5), [#text(10pt)[`x = torch.tensor([[1, 3, 0], [2, 4, 6]])`]]) + + // Matrice originale 2x3 + let colors = (green, yellow, red, blue, fuchsia, eastern) + let idx = 0 + for i in range(2) { + for j in range(3) { + rect((j * 0.5, -i * 0.5 + 3), (j * 0.5 + 0.5, -i * 0.5 + 3 - 0.5), fill: colors.at(idx), stroke: black) + idx += 1 + } + } + + // Freccia + content((2.2, 2.6), text(16pt)[→]) + + // Espansione con .view(1, 2, 3).expand(3, 2, 3) + content((6.5, 3.5), [#text(10pt)[`x.view(1, 2, 3).expand(3, 2, 3)`]]) + + for d in range(3) { + let offset = d * 0.3 + idx = 0 + for i in range(2) { + for j in range(3) { + let alpha = if d == 0 { 100% } else { 70% } + rect( + (j * 0.5 + 3.5 + offset, -i * 0.5 + 3 - offset), + (j * 0.5 + 3.5 + 0.5 + offset, -i * 0.5 + 3 - 0.5 - offset), + fill: colors.at(idx).lighten(d * 20%), + stroke: black, + ) + idx += 1 + } + } + } + }), + caption: [Expand: replicazione dei dati mediante broadcasting], +) -#esempio()[ - Dato un batch di immagini, ovvero un tensore con $(B,3,H,W)$ dove $B$ numero di immagini, $3$ canali e $H e W$ dimensione dell'immagine. - Vogliamo computare la meida tra tutti i canali $R,G,B$. Dove: - $ - x -> hat(x) = (x-mu)/sigma - $ - Dove prima abbatto i picchi dividendo per la deviazioen tramite $sigma$ poi vado a traslare sottraendo $mu$ (centro) rispetto alla media. +#nota()[ + `expand()` *non alloca nuova memoria* ma semplicemente modifica gli stride del tensore per far sì che gli stessi dati vengano "visti" più volte. Questa operazione è molto efficiente rispetto a una vera copia dei dati. - Alla fine verifico che la media sia circa zero e la deviazione sia crica 1. + *Differenze chiave*: + - `view()` / `reshape()`: cambiano la _forma_ dei dati esistenti + - `transpose()`: scambia _dimensioni_ + - `expand()`: _replica virtualmente_ i dati lungo nuove dimensioni ] -== Numpy e Tensor +=== Add/Remove dim -Attenzione condividono memoria. Può essere pericoloso, nel molemto in cui passo a un tensore da un numpy array e faccio operazione sul tensore allora modifico anche l'array numpy originale -//aggiugnere esempio +Per tali scopi esistono due funzioni: +- *`unsqueexe(dim)`* = aggiunge una nuova dimensione di size $1$ alla posizione dim. Da tensore $1D$ a tensore $2D$: + ```py + x = torch.tensor([1, 2, 3, 4])# shape([4]) + x_unsqueezed = x.unsqueeze(0)# shape([1,4]) + ``` -`unsqueexe(dim)` = aggiunge nuove dimensioni di dimensione 1 alla posizione dim. -`squeeze(dim)` = operazione contraria, rimuove degli $1$ dalle dimensioni +- *`squeeze(dim)`* = operazione contraria, rimuove degli $1$ dalle dimensioni specificate: + ```py + y = torch.zeros(1, 3, 1, 5)# Size([1,3,1,5]) + y_squeezed = y.squeeze()# Size([3,5]) -La concatenazione permette di concatenare tensori diversi su una certa dimensione. + ``` +=== Concatenation / Splitting + +Le operazioni di concatenazione e splitting permettono di combinare o dividere tensori lungo dimensioni specifiche. + +==== torch.cat + +Concatena una sequenza di tensori *lungo una dimensione esistente*. Tutti i tensori devono avere la stessa forma eccetto nella dimensione di concatenazione. + +```py +x1 = torch.tensor([[1, 2], [3, 4]]) # Shape: [2, 2] +x2 = torch.tensor([[5, 6], [7, 8]]) # Shape: [2, 2] + +# Concatenazione lungo dim=0 (righe) +result = torch.cat([x1, x2], dim=0) +# tensor([[1, 2], +# [3, 4], +# [5, 6], +# [7, 8]]) # Shape: [4, 2] + +# Concatenazione lungo dim=1 (colonne) +result = torch.cat([x1, x2], dim=1) +# tensor([[1, 2, 5, 6], +# [3, 4, 7, 8]]) # Shape: [2, 4] +``` + +#figure( + cetz.canvas({ + import cetz.draw: * + + // Esempio cat dim=0 + content((0, 3.5), [#text(9pt)[`torch.cat([x1, x2], dim=0)`]]) + + // x1 + rect((0, 3), (0.6, 2.4), fill: green, stroke: black) + rect((0.6, 3), (1.2, 2.4), fill: yellow, stroke: black) + rect((0, 2.4), (0.6, 1.8), fill: blue, stroke: black) + rect((0.6, 2.4), (1.2, 1.8), fill: red, stroke: black) + + content((1.6, 2.4), text(16pt)[+]) + + // x2 + rect((2, 3), (2.6, 2.4), fill: fuchsia, stroke: black) + rect((2.6, 3), (3.2, 2.4), fill: orange, stroke: black) + rect((2, 2.4), (2.6, 1.8), fill: purple, stroke: black) + rect((2.6, 2.4), (3.2, 1.8), fill: eastern, stroke: black) + + content((3.6, 2.4), text(16pt)[=]) + + // Risultato + rect((4, 3), (4.6, 2.4), fill: green, stroke: black) + rect((4.6, 3), (5.2, 2.4), fill: yellow, stroke: black) + rect((4, 2.4), (4.6, 1.8), fill: blue, stroke: black) + rect((4.6, 2.4), (5.2, 1.8), fill: red, stroke: black) + rect((4, 1.8), (4.6, 1.2), fill: fuchsia, stroke: black) + rect((4.6, 1.8), (5.2, 1.2), fill: orange, stroke: black) + rect((4, 1.2), (4.6, 0.6), fill: purple, stroke: black) + rect((4.6, 1.2), (5.2, 0.6), fill: eastern, stroke: black) + + // Esempio cat dim=1 + content((0, 0), [#text(9pt)[`torch.cat([x1, x2], dim=1)`]]) + + // x1 + rect((0, -0.5), (0.6, -1.1), fill: green, stroke: black) + rect((0.6, -0.5), (1.2, -1.1), fill: yellow, stroke: black) + rect((0, -1.1), (0.6, -1.7), fill: blue, stroke: black) + rect((0.6, -1.1), (1.2, -1.7), fill: red, stroke: black) + + content((1.6, -1.1), text(16pt)[+]) + + // x2 + rect((2, -0.5), (2.6, -1.1), fill: fuchsia, stroke: black) + rect((2.6, -0.5), (3.2, -1.1), fill: orange, stroke: black) + rect((2, -1.1), (2.6, -1.7), fill: purple, stroke: black) + rect((2.6, -1.1), (3.2, -1.7), fill: eastern, stroke: black) + + content((3.6, -1.1), text(16pt)[=]) + + // Risultato + rect((4, -0.5), (4.6, -1.1), fill: green, stroke: black) + rect((4.6, -0.5), (5.2, -1.1), fill: yellow, stroke: black) + rect((5.2, -0.5), (5.8, -1.1), fill: fuchsia, stroke: black) + rect((5.8, -0.5), (6.4, -1.1), fill: orange, stroke: black) + rect((4, -1.1), (4.6, -1.7), fill: blue, stroke: black) + rect((4.6, -1.1), (5.2, -1.7), fill: red, stroke: black) + rect((5.2, -1.1), (5.8, -1.7), fill: purple, stroke: black) + rect((5.8, -1.1), (6.4, -1.7), fill: eastern, stroke: black) + }), + caption: [torch.cat: concatenazione lungo dimensioni esistenti], +) + +==== torch.stack + +Concatena una sequenza di tensori *creando una nuova dimensione*. Tutti i tensori devono avere esattamente la stessa forma. -`torch.stack` +```py +x1 = torch.tensor([[1, 2], [3, 4]]) # Shape: [2, 2] +x2 = torch.tensor([[5, 6], [7, 8]]) # Shape: [2, 2] + +# Stacking lungo dim=0 (crea una nuova dimensione) +result = torch.stack([x1, x2], dim=0) +# Shape: [2, 2, 2] +# tensor([[[1, 2], +# [3, 4]], +# [[5, 6], +# [7, 8]]]) + +# Stacking lungo dim=1 +result = torch.stack([x1, x2], dim=1) +# Shape: [2, 2, 2] +``` +#figure( + cetz.canvas({ + import cetz.draw: * + + content((0, 3), [#text(9pt)[`torch.stack([x1, x2], dim=0)`]]) + + // x1 e x2 come layer separati + content((0, 2.5), [#text(8pt)[x1]]) + rect((0, 2.2), (0.6, 1.6), fill: green, stroke: black) + rect((0.6, 2.2), (1.2, 1.6), fill: yellow, stroke: black) + rect((0, 1.6), (0.6, 1), fill: blue, stroke: black) + rect((0.6, 1.6), (1.2, 1), fill: red, stroke: black) + + content((1.6, 1.6), text(16pt)[→]) + + content((2.5, 2.5), [#text(8pt)[x2]]) + rect((2.5, 2.2), (3.1, 1.6), fill: fuchsia, stroke: black) + rect((3.1, 2.2), (3.7, 1.6), fill: orange, stroke: black) + rect((2.5, 1.6), (3.1, 1), fill: purple, stroke: black) + rect((3.1, 1.6), (3.7, 1), fill: eastern, stroke: black) + + content((4.2, 1.6), text(16pt)[→]) + + // Risultato: due layer impilati + content((5.5, 2.5), [#text(8pt)[result (3D)]]) + // Layer 1 (x1) + rect((5, 2.2), (5.6, 1.6), fill: green, stroke: black + 1.5pt) + rect((5.6, 2.2), (6.2, 1.6), fill: yellow, stroke: black + 1.5pt) + rect((5, 1.6), (5.6, 1), fill: blue, stroke: black + 1.5pt) + rect((5.6, 1.6), (6.2, 1), fill: red, stroke: black + 1.5pt) + + // Layer 2 (x2) - offset per effetto 3D + rect((5.2, 2), (5.8, 1.4), fill: fuchsia, stroke: black + 1.5pt) + rect((5.8, 2), (6.4, 1.4), fill: orange, stroke: black + 1.5pt) + rect((5.2, 1.4), (5.8, 0.8), fill: purple, stroke: black + 1.5pt) + rect((5.8, 1.4), (6.4, 0.8), fill: eastern, stroke: black + 1.5pt) + }), + caption: [torch.stack: crea una nuova dimensione impilando i tensori], +) + +#nota()[ + *Differenza tra `cat` e `stack`*: + - `torch.cat`: unisce tensori lungo una dimensione *esistente*, aumentando la size di quella dimensione + - `torch.stack`: crea una *nuova dimensione* e posiziona i tensori lungo quella dimensione +] + +==== torch.chunk + +Divide un tensore in un *numero specifico di chunk* (parti) lungo una dimensione. Se la divisione non è uniforme, l'ultimo chunk sarà più piccolo. + +```py +x = torch.arange(12).view(4, 3) +# tensor([[ 0, 1, 2], +# [ 3, 4, 5], +# [ 6, 7, 8], +# [ 9, 10, 11]]) + +# Divide in 2 chunk lungo dim=0 +chunks = torch.chunk(x, chunks=2, dim=0) +# (tensor([[0, 1, 2], +# [3, 4, 5]]), +# tensor([[ 6, 7, 8], +# [ 9, 10, 11]])) + +# Divide in 3 chunk lungo dim=1 +chunks = torch.chunk(x, chunks=3, dim=1) +# Ogni chunk ha shape [4, 1] +``` + +==== torch.split + +Divide un tensore in chunk di *dimensioni specificate*. Più flessibile di `chunk` perché permette di specificare la dimensione esatta di ogni parte. + +```py +x = torch.arange(12).view(4, 3) + +# Split con dimensione uniforme +splits = torch.split(x, split_size_or_sections=2, dim=0) +# (tensor([[0, 1, 2], +# [3, 4, 5]]), +# tensor([[ 6, 7, 8], +# [ 9, 10, 11]])) + +# Split con dimensioni diverse +splits = torch.split(x, split_size_or_sections=[1, 2, 1], dim=0) +# (tensor([[0, 1, 2]]), # size 1 +# tensor([[3, 4, 5], # size 2 +# [6, 7, 8]]), +# tensor([[9, 10, 11]])) # size 1 +``` + +#figure( + cetz.canvas({ + import cetz.draw: * + + content((0, 3), [#text(9pt)[`torch.split(x, [1, 2, 1], dim=0)`]]) + + // Tensore originale 4x3 + let colors = (green, yellow, red, blue, fuchsia, eastern, orange, purple, gray, maroon, olive, navy) + let idx = 0 + for i in range(4) { + for j in range(3) { + rect((j * 0.5, -i * 0.5 + 2), (j * 0.5 + 0.5, -i * 0.5 + 2 - 0.5), fill: colors.at(idx), stroke: black) + idx += 1 + } + } + + content((2, 0.75), text(16pt)[→]) + + // Risultati dello split + content((3, 3), [#text(8pt)[chunk 1]]) + for j in range(3) { + rect((j * 0.5 + 3, 2), (j * 0.5 + 3 + 0.5, 1.5), fill: colors.at(j), stroke: black) + } + + content((5, 3), [#text(8pt)[chunk 2]]) + idx = 3 + for i in range(2) { + for j in range(3) { + rect((j * 0.5 + 5, -i * 0.5 + 2), (j * 0.5 + 5 + 0.5, -i * 0.5 + 2 - 0.5), fill: colors.at(idx), stroke: black) + idx += 1 + } + } + + content((7, 3), [#text(8pt)[chunk 3]]) + idx = 9 + for j in range(3) { + rect((j * 0.5 + 7, 2), (j * 0.5 + 7 + 0.5, 1.5), fill: colors.at(idx), stroke: black) + idx += 1 + } + }), + caption: [torch.split: divide il tensore in parti di dimensioni specificate], +) + +#informalmente()[ + *Quando usare chunk vs split*: + - `chunk`: quando vuoi dividere in un *numero fisso di parti* (es. dividere un batch in 4 parti uguali) + - `split`: quando vuoi *controllare la dimensione* di ogni parte (es. train/validation split con proporzioni specifiche) +]

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