Datasets:

zai-org
/

RPC-Bench

@@ -41,7 +41,7 @@ In order for modality gap to exist, the embeddings from a encoder should be conc
 In the literature, the cone effect has been observed in the language representations from language models (e.g., BERT) [12]. A common explanation is that the unbalanced distribution of word frequencies biased the optimization [15, 33]. However, we found that the cone effect still exists in models with random weights (Figure 2 (c)). In fact, the average cosine similarity there is even higher than in trained models. For example, any two embeddings from a randomly initialized ResNet have on average an almost perfect (0.99) cosine similarity. Interestingly, the cone effect still holds when the input data is random noise3, indicating that unbalanced data distribution suggested in previous works is not necessary for the cone effect. Together these experiments suggest that the cone effect reflects a more general inductive bias of deep networks than might be previously appreciated.
-How narrow is the cone in 512-dim representation space? We clarify that a cosine similarity with 0.56 already indicates that the embedding space is actually an extremely narrow cone in the 512-dimensional feature space. Consider the fraction of surface area in a unit hypersphere: In 2D, arccos $( 0 . 5 6 ) { = } 5 5 . 9 4 ^ { \circ }$ , indicating that a cosine similarity of 0.56 can “occupy” $5 5 . 9 4 ^ { \circ } / 3 6 0 ^ { \circ } = 1 5 . 5 3 \%$ of the 2D unit circle. In 3D, a cosine similarity of 0.56 can “occupy” 2⇡r2(1cos 55.94°2 )4⇡r2 of the 3D unit sphere. In 512D, a cosine similarity of 0.56 can “occupy” less than $\frac { 1 } { 2 ^ { 5 1 2 } }$ fraction of the surface area in a unit 512D hypersphere. These evidences show that the effective embedding space is restricted to an extremely narrow cone.
 # 2.2 The effects of non-linear activation on cone effect

 In the literature, the cone effect has been observed in the language representations from language models (e.g., BERT) [12]. A common explanation is that the unbalanced distribution of word frequencies biased the optimization [15, 33]. However, we found that the cone effect still exists in models with random weights (Figure 2 (c)). In fact, the average cosine similarity there is even higher than in trained models. For example, any two embeddings from a randomly initialized ResNet have on average an almost perfect (0.99) cosine similarity. Interestingly, the cone effect still holds when the input data is random noise3, indicating that unbalanced data distribution suggested in previous works is not necessary for the cone effect. Together these experiments suggest that the cone effect reflects a more general inductive bias of deep networks than might be previously appreciated.
+How narrow is the cone in 512-dim representation space? We clarify that a cosine similarity with 0.56 already indicates that the embedding space is actually an extremely narrow cone in the 512-dimensional feature space. Consider the fraction of surface area in a unit hypersphere: In 2D, arccos $( 0 . 5 6 ) { = } 5 5 . 9 4 ^ { \circ }$ , indicating that a cosine similarity of 0.56 can “occupy” $5 5 . 9 4 ^ { \circ } / 3 6 0 ^ { \circ } = 1 5 . 5 3 \%$ of the 2D unit circle. In 3D, a cosine similarity of 0.56 can “occupy” 2⇡r2(1cos 55.94°2 )4⇡r2 of the 3D unit sphere. In 512D, a cosine similarity of 0.56 can “occupy” less than $\frac { 1 } { 2 ^ { 5 1 2 } }$ fraction of the surface area in a unit 512D hypersphere. These evidences show that the effective embedding space is restricted to an extremely narrow cone.
 # 2.2 The effects of non-linear activation on cone effect