import streamlit as st
from st_pages import add_indentation

add_indentation()

st.title('Loss functions')
st.markdown('In order to align textual and visual features, multiple loss functions are employed. '
            'The most notable loss function was proposed in [arXiv: Cross-Modal Implicit Relation Reasoning and Aligning for Text-to-Image Person Retrieval](https://arxiv.org/abs/2303.12501) '
            'with the introduction of the SDM loss and the usage of the IRR (Implicit Reason Relations) loss.')
with st.expander('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.markdown('The SDM loss can be formulated as')
    st.latex(r'''
    \mathcal{L}_{sdm} = \mathcal{L}_{i2t} + \mathcal{L}_{t2i}
    ''')

with st.expander('IRR (MLM) Loss'):
    ...
with st.expander('ID Loss'):
    ...