feat: add implementation details page

pages/losses.py

@@ -0,0 +1,39 @@
+import streamlit as st
+from st_pages import add_indentation
+
+add_indentation()
+
+st.title('Loss functions')
+st.subheader('SDM Loss')
+st.markdown('''
+            The similarity distribution matching (SDM) loss, which is the KL divergence
+            of the image to text and text to image to the label distribution.
+            
+            We define $f^v$ and $f^t$ to be the global representation of the visual and textual features respectively.
+            The cosine similarity $sim(u, v) = \\frac{u \\cdot v}{|u||v|}$ will be used to compute the probability of the labels.
+            
+            We define $y_{i, j}=1$ if the visual feature $f^v_i$ matches the textual feature $f^t_j$, else $y_{i, j}=0$.
+            The predicted label distribution can be formulated by''')
+st.latex(r'''
+p_{i} = \sigma(sim(f^v_i, f^t))
+''')
+
+st.markdown('''
+We can define the image to text loss as
+''')
+
+st.latex(r'''
+\mathcal{L}_{i2t} = KL(\mathbf{p_i} || \mathbf{q_i})
+''')
+
+st.markdown('Where $\\mathbf{q_i}$, the true probability distribution, is defined as')
+
+st.latex(r'''
+q_{i, j} = \frac{y_{i, j}}{\sum_{k=1}^{N} y_{i, k}}
+''')
+
+st.markdown('It should be noted that the reason this computation is needed is because there could be multiple correct labels.')
+
+
+st.subheader('IRR (MLM) Loss')
+st.subheader('ID Loss')