Aron Culotta, Jeffrey Sorensen: Dependency Tree Kernels for Relation Extraction, ACL 2004.

1. Given Figure 1, what is the smallest common subtree that includes both t1 (Troops) and t2 (near)?
2. Section 5: “Therefore, d(a)=l(a).” When is this true and why? (Assume this holds for the following questions.)
3. Let <latex>\phi_m</latex> = {general-pos-tag, entity-type, relation-arguments} (in accordance with the paper). Let <latex>\phi_s = \phi_m</latex> (unlike in the paper). Based on Figure 2 and Section 5, compute the following matching functions and similarity functions:
   • m(t0,u0)=? m(t1,u1)=? m(t2,u2)=?
   • s(t0,u0)=? s(t1,u1)=? s(t2,u2)=?
4. Let <latex>\lambda=0.5</latex>. Compute the contiguous kernel for the two trees in Figure 2: <latex>K_1(T,U)=?</latex>. Provide the final number and some counts along the way, so its clear how you got the number. Optionally, compute also the sparse kernel <latex>K_0(T,U)=?</latex>.
5. Let DT be a function that assigns the correct augmented dependency tree to a sentence. Compute (estimate) contiguous kernel and bag-of-words kernel for the following sentences:
   • <latex>K_1</latex>(DT(“Peter sleeps”), DT(“Bob runs”))=?
   • <latex>K_2</latex>(DT(“Peter sleeps”), DT(“Bob runs”))=?
6. Lets have a pair of sentences:
   • “Bob saw US troops that moved towards Baghdad”
   • “US troops that moved towards Baghdad were seen by Bob”

You want to check the relation between entities “US” and “Baghdad”. Compute (estimate) <latex>K_1</latex> and <latex>K_2</latex>.

- Depends on the exact definition of smallest common subtree, but keep in mind you need at least some non-trivial "context". The definition should be such that contignous and sparse kernels will effecively be different things. The whole subtree is probably the right answer here.
- d(a) is defined as the last member of the sequence - the first member + 1. If the sequence is contignous (no missing indices) it can be shown (eg. by induction) that the equation holds, unless some of the indices is repeated. Strictly speaking (1,1,1) is a valid sequence.
- Depends on how you treat "N/A" values, by the definition you should sum values that are "the same/compatible" (disregarding the "type" of the feature).
  * ''m(t0,u0)=1     m(t1,u1)=1      m(t2,u2)=0''
  * ''s(t0,u0)=5     s(t1,u1)=3      s(t2,u2)=0 (or not defined)''
- First this depends on the previous one (the "N/A" values) and second the paper doesn't say how to compute <latex>K_0(t_i[A],t_j[B])</latex>, where A,B are sequences with more than one member. One proposed solution was to use <latex>K_0(t_i[A],t_j[B])=\sum_{s=0..l(A)}K_0(t_i[a_s],t_j[b_s])</latex>
  * <latex>K_0(T,U)=s(t_0,u_0)+\lambda^2K_0(t_1,u_1)+\lambda^2K_0(t_1,u_2)+\lambda^2K_0(t_2,u_1)+\lambda^2K_0(t_2,u_2)+\lambda^2K_0(t_3,u_1)+\lambda^2K_0(t_3,u_2)+\lambda^4K_0({t_1,t_2},{u_1,u_2})+\lambda^4K_0({t_2,t_3},{u_1,u_2})+\lambda^5K_0({t_1,t_3},{u_1,u_2})= s(t_0,u_0)+\lambda^2s(t_1,u_1)+\lambda^2s(t_3,u_2)+\lambda^4s(t_4,u_3)+\lambda^4s(t_1,u_1)+\lambda^4s(t_3,u_2)+\lambda^6s(t_4,u_3)+\lambda^5s(t_1,u_1)+\lambda^5s(t_3,u_2)+\lambda^7s(t_4,u_3)</latex>   
  * When counting K_1 you leave out the <latex>({t_1,t_3},{u_1,u_2})</latex> part
- When you regard bag-of-words kernel as number of matching forms then K_2 is zero whereas K_1 is (very likely) positive
- It was argued that we'll probably end up with different relation-args, thus there will be no match

- There was some discussion what are the features for bag-of-words kernel (just presence of a word in sentence?)
- Feature selection, mainly the relation-args feature
- "Two level" classification, why it might be a good idea