A vector space is an object between a field and a group; it has two ops—addition and scalar multiplication. Its not quite a field and its more than a group.

constituents

• A set \(V\)
• An addition on \(V\)
• An scalar multiplication on \(V\)

such that…

requirements

• commutativity in add.: \(u+v=v+u\)
• associativity in add. and mult.: \((u+v)+w=u+(v+w)\); \((ab)v=a(bv)\): \(\forall u,v,w \in V\) and \(a,b \in \mathbb{F}\)
• distributivity: goes both ways \(a(u+v) = au+av\) AND!! \((a+b)v=av+bv\): \(\forall a,b \in \mathbb{F}\) and \(u,v \in V\)
• additive identity: \(\exists 0 \in V: v+0=v \forall v \in V\)
• additive inverse: \(\forall v \in V, \exists w \in V: v+w=0\)
• multiplicative identity: \(1v=v \forall v \in V\)

additional information

• Elements of a vector space are called vectors or points.

vector space “over” fields

Scalar multiplication is not in the set \(V\); instead, “scalars” \(\lambda\) come from this magic faraway land called \(\mathbb{F}\). The choice of \(\mathbb{F}\) for each vector space makes it different; so, when precision is needed, we can say that a vector space is “over” some \(\mathbb{F}\) which contributes its scalars.

this is worse

VGGish is VGG, ish. VGGish is a network based on VGG which is pretrained on the audio-feature-extraction task.

visual prediction task

Given \(n\) frames of video \(x_{1}, …, x_{n}\), predict \(T\) subsequent frames \(x_{n+1}… x_{n+T}\).

Action Conditioned Prediction Network

Visual prediction task. Two inputs:

1. taken action
2. image

Predict the next \(t\) frame. Challenge! What is the action? Predicting the action is not super easy.

RAFI

Conditional Flow Matching with Video Generation.

https://arxiv.org/pdf/2406.14436