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parse/train/-_FVMKvxVCQo1/-_FVMKvxVCQo1.md

@@ -230,7 +230,7 @@ where $T _ { \mathrm { m i n } }$ was set to a constant 50 to avoid stopping too
 12 $( \alpha _ { \bullet } , \mathbf { v _ { \bullet } } , \varphi _ { \bullet } ( \cdot ) )  \mathrm { B A S E } ( \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \mathbf { W _ { \bullet } } )$   
 13 $\mathrm { I N S E R T } \big ( S , \big ( \mathbf { v _ { * } } , \varphi _ { \bullet } ( \cdot ) , \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \bullet , j \big ) , \gamma ( \mathbf { v _ { * } } , \varphi _ { \bullet } , \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \mathbf { W _ { \bullet } } ) - \gamma ( \mathbf { v } _ { j } , \varphi _ { j } , \mathbf { X _ { \bullet } } , \mathbf { Y _ { \bullet } } , \mathbf { W _ { \bullet } } ) \big )$ . key = edge improvement over parent edge 1 1 + γ(h1, W)   
 14 α = 2 log 1 − γ(h1, W) . standard coefficient of the full tree classifier ${ \mathfrak { h } } _ { 1 }$ (14) 2   
-15 return α, H, l, r

+15 return α, H, l, r

 
 Figure 2. The pseudocode of the Hamming tree base learner. $N$ is the number of inner nodes. The algorithm returns a list of base classifiers ${ \mathfrak H }$ , two index lists l and $\mathfrak { r }$ , and the base coefficient $\alpha$ . The tree classifier is then defined by (14).   
 Table 1. Test error percentages on mid-sized benchmark data sets.   

parse/train/Sy-tszZRZ/Sy-tszZRZ.md

@@ -136,7 +136,7 @@ Proof. $\dim ( { \mathcal S } ) \ = \ \operatorname { r a n k } ( \sigma _ { S ^
 
 In the remainder of the proof of Theorem 1, we combine Lemmas 3 and 4 to construct a recurrence $R ( l , d )$ that bounds the number of regions within a given region of dimension $d$ . Simplifying this recurrence yields the expression in Theorem 1. We formalize this idea and complete the proof of Theorem 1 in Appendix D.
 
-As a side note, Theorem 1 can be further tightened if the weight matrices are known to have small rank. The bound from Lemma 3 can be rewritten as j=0 Pmin{rank(W l),dim(S)} nl
 if we do not relax $\mathrm { r a n k } ( W ^ { l } )$ to $n _ { l }$ in the proof. The term $\mathrm { r a n k } ( W ^ { l } )$ follows through the proof of Theorem 1 and the index set $J$ in the theorem becomes $\{ ( j _ { 1 } , \dotsc , j _ { L } ) \in \mathbb { Z } ^ { L } : 0 \leq j _ { l } \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \dotsc , n _ { l - 1 } -$ $j _ { l - 1 } , \mathrm { r a n k } ( W ^ { l } ) \} \forall l \geq 1 \}$ .
+As a side note, Theorem 1 can be further tightened if the weight matrices are known to have small rank. The bound from Lemma 3 can be rewritten as j=0 Pmin{rank(W l),dim(S)} nl
 if we do not relax $\mathrm { r a n k } ( W ^ { l } )$ to $n _ { l }$ in the proof. The term $\mathrm { r a n k } ( W ^ { l } )$ follows through the proof of Theorem 1 and the index set $J$ in the theorem becomes $\{ ( j _ { 1 } , \dotsc , j _ { L } ) \in \mathbb { Z } ^ { L } : 0 \leq j _ { l } \leq \operatorname* { m i n } \{ n _ { 0 } , n _ { 1 } - j _ { 1 } , \dotsc , n _ { l - 1 } -$ $j _ { l - 1 } , \mathrm { r a n k } ( W ^ { l } ) \} \forall l \geq 1 \}$ .
 
 A key insight from Lemmas 3 and 4 is that the dimensions of the regions are non-increasing as we move through the layers partitioning it. In other words, if at any layer the dimension of a region becomes small, then that region will not be able to be further partitioned into a large number of regions. For instance, if the dimension of a region falls to zero, then that region will never be further partitioned. This suggests that if we want to have many regions, we need to keep dimensions high. We use this idea in the next section to construct a DNN with many regions.
 
@@ -419,7 +419,7 @@ Proof. As illustrated in Figure 1, the partitioning can be viewed as a sequentia
 
 With this process in mind, we recursively bound the number of subregions within a region. More precisely, we construct a recurrence $R ( l , d )$ to be an upper bound to the maximal number of regions obtained from partitioning a region of dimension $d$ with layers $l , l + 1 , \ldots , L$ . The base case of the recurrence is given by Lemma 3: $\begin{array} { r } { R ( L , d ) = \sum _ { j = 0 } ^ { \operatorname* { m i n } \{ n _ { L } , d \} } \binom { n _ { L } } { j } } \end{array}$ . Based on Lemma 4, we can write the recurrence by grouping together regions with the same activation set size $\begin{array} { r } { R ( l , d ) = \sum _ { j = 0 } ^ { n _ { l } } { N _ { n _ { l } , d , j } \bar { R } ( l ^ { ' } + 1 , \bar { \operatorname* { m i n } } \{ j , d \} ) } } \end{array}$ for all $l = 1 , \ldots , L - 1$ . Here, $N _ { n _ { l } , d , j }$ $| S ^ { l } |$ represents the , as follows: maximum number of regions with $| S ^ { l } | = j$ obtained by partitioning a space of dimension $d$ with $n _ { l }$ hyperplanes. We bound this value next.
 
-For each $j$ , there are at most $\binom { n _ { l } } { j }$ regions with $| S ^ { l } | = j$ , as they can be viewed as subsets of $n _ { l }$ neurons of size $j$ . In total, Lemma 3 states that there are at most Pmin{nl,d}j=0 nlj 
 regions. If we allow these regions to have the highest $| S ^ { l } |$ possible, for each $j$ from 0 to $\operatorname* { m i n } \{ n _ { l } , d \}$ we have at most $\textstyle { \binom { n _ { l } } { n _ { l } - j } } = { \binom { n _ { l } } { j } }$ regions with $\vert S ^ { l } \vert = n _ { l } - j$ .
+For each $j$ , there are at most $\binom { n _ { l } } { j }$ regions with $| S ^ { l } | = j$ , as they can be viewed as subsets of $n _ { l }$ neurons of size $j$ . In total, Lemma 3 states that there are at most Pmin{nl,d}j=0 nlj 
 regions. If we allow these regions to have the highest $| S ^ { l } |$ possible, for each $j$ from 0 to $\operatorname* { m i n } \{ n _ { l } , d \}$ we have at most $\textstyle { \binom { n _ { l } } { n _ { l } - j } } = { \binom { n _ { l } } { j } }$ regions with $\vert S ^ { l } \vert = n _ { l } - j$ .
 
 Therefore, we can write the recurrence as