In this section, we mainly introduce the axiomatic properties of Shapley value. Weber et al. [17] have proved that Shapley value is the unique metric that satisfies the following axioms: Linearity, Symmetry, Dummy, and Efficiency. If two independent games u and v can be linearly merged into one game w(S) = u(S)+v(S), then the Shapley value of each player i N in the new game w is the sum of Shapley values of the player i in the game uand v, which can be formulated as: ϕw(i|N) = ϕu(i|N)+ϕv(i|N) (1) Symmetry Axiom. Considering two players i and j in a game v, if they satisfy: S N \{i,j},v(S {i}) = v(S {j}) (2) then ϕv(i|N) = ϕv(j|N). The dummy player is defined as the player that has no interaction with other players. Formally, if a player i in a game v satisfies: S N \{i},v(S {i}) = v(S)+v({i}) (3) then this player is defined as the dummy player.

Proof of Thm. 2. We want to show M G(hx)= hM G(x) for all x 2X and h 2 G. From the definition of M G in equation 4, we have M G(hx)= 1P Similar to Yarotsky (2022), we first define Ksym = S g2G gK. Note that Ksym is also a compact set and Ksym X . We want to show that M G,equi(gx)= gM G,equi(x). Hence, ( h(gx) 1gx) is invariant to actions of G. The proof for invariance of M G,inv(x) follows similarly. In addition to properties discussed in section 3.3, here we show that equizero models have autoregressive and invertibility properties. These properties have not been used in the main paper, but we believe they could be of use for future work in this area.

Document intelligence automates the extraction of information from documents and supports many business applications. Recent self-supervised learning methods on large-scale unlabeled document datasets have opened up promising directions towards reducing annotation efforts by training models with self-supervised objectives. However, most of the existing document pretraining methods are still language-dominated.

We study the problem of synthesizing a number of likely future frames from a single input image. In contrast to traditional methods, which have tackled this problem in a deterministic or non-parametric way, we propose to model future frames in a probabilistic manner. Our probabilistic model makes it possible for us to sample and synthesize many possible future frames from a single input image. To synthesize realistic movement of objects, we propose a novel network structure, namely a Cross Convolutional Network; this network encodes image and motion information as feature maps and convolutional kernels, respectively. In experiments, our model performs well on synthetic data, such as 2D shapes and animated game sprites, as well as on real-world video frames. We also show that our model can be applied to visual analogy-making, and present an analysis of the learned network representations.

Moreover, the volume discrepancy between 3D and 2D pretraining datasets highlights the need for effective strategies to transfer the representational abilities of VLMs to 3D learning.