What's wrong with my derivation of the gradient of KL divergence?

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I'm reading the paper Visualizing Data using t-SNE, and I get stuck on the gradient of KL divergence.

In the paper, it defines the similarity between datapoints, $p_j$, as follows

$$p_j=x_i-x_kright$$

the similarity between map points is given by

$$q_j=^2)over sum_kne iexp(-left$$

and at last, the cost function, which is the sum of KL divergences over all datapoints is defined by
$$C=sum_iKL(P_ivert|Q_i)=sum_isum_jp_jlogp_jover q_j$$

Then it says the gradient is

$$partial Cover partial y_i=2sum_j(p_j-q_j+p_j-q_j)(y_i-y_j)$$

However, I derive a different result

$$beginalign
partial Cover partial y_i&=-sum_j p_j1 over q_jnabla_y_iq_j + p_j1over q_jnabla_y_iq_j
endalign$$
and
$$beginalign
nabla_y_iq_j&=q_jleft(2(y_j-y_i)-2^2)right)\
nabla_y_iq_j&=q_j(2(y_j-y_i)-2q_j(y_j-y_i))
endalign$$
so
$$partial Coverpartial y_i=2sum_j(y_i-y_j)left(p_j+p_j-q_j-^2)right)$$

I've checked my derivation several times, but still have no clue about where the wrong is. Could someone help me check it? I'll be very grateful for that. Thanks.

derivatives

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edited Jul 17 at 3:01

asked Jul 16 at 4:56

Sherwin Chen

536

add a comment | 

up vote
0
down vote

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I'm reading the paper Visualizing Data using t-SNE, and I get stuck on the gradient of KL divergence.

In the paper, it defines the similarity between datapoints, $p_j$, as follows

$$p_j=x_i-x_kright$$

the similarity between map points is given by

$$q_j=^2)over sum_kne iexp(-left$$

and at last, the cost function, which is the sum of KL divergences over all datapoints is defined by
$$C=sum_iKL(P_ivert|Q_i)=sum_isum_jp_jlogp_jover q_j$$

Then it says the gradient is

$$partial Cover partial y_i=2sum_j(p_j-q_j+p_j-q_j)(y_i-y_j)$$

However, I derive a different result

$$beginalign
partial Cover partial y_i&=-sum_j p_j1 over q_jnabla_y_iq_j + p_j1over q_jnabla_y_iq_j
endalign$$
and
$$beginalign
nabla_y_iq_j&=q_jleft(2(y_j-y_i)-2^2)right)\
nabla_y_iq_j&=q_j(2(y_j-y_i)-2q_j(y_j-y_i))
endalign$$
so
$$partial Coverpartial y_i=2sum_j(y_i-y_j)left(p_j+p_j-q_j-^2)right)$$

I've checked my derivation several times, but still have no clue about where the wrong is. Could someone help me check it? I'll be very grateful for that. Thanks.

derivatives

share|cite|improve this question

edited Jul 17 at 3:01

asked Jul 16 at 4:56

Sherwin Chen

536

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I'm reading the paper Visualizing Data using t-SNE, and I get stuck on the gradient of KL divergence.

In the paper, it defines the similarity between datapoints, $p_j$, as follows

$$p_j=x_i-x_kright$$

the similarity between map points is given by

$$q_j=^2)over sum_kne iexp(-left$$

and at last, the cost function, which is the sum of KL divergences over all datapoints is defined by
$$C=sum_iKL(P_ivert|Q_i)=sum_isum_jp_jlogp_jover q_j$$

Then it says the gradient is

$$partial Cover partial y_i=2sum_j(p_j-q_j+p_j-q_j)(y_i-y_j)$$

However, I derive a different result

$$beginalign
partial Cover partial y_i&=-sum_j p_j1 over q_jnabla_y_iq_j + p_j1over q_jnabla_y_iq_j
endalign$$
and
$$beginalign
nabla_y_iq_j&=q_jleft(2(y_j-y_i)-2^2)right)\
nabla_y_iq_j&=q_j(2(y_j-y_i)-2q_j(y_j-y_i))
endalign$$
so
$$partial Coverpartial y_i=2sum_j(y_i-y_j)left(p_j+p_j-q_j-^2)right)$$

I've checked my derivation several times, but still have no clue about where the wrong is. Could someone help me check it? I'll be very grateful for that. Thanks.

derivatives

share|cite|improve this question

edited Jul 17 at 3:01

asked Jul 16 at 4:56

Sherwin Chen

536

I'm reading the paper Visualizing Data using t-SNE, and I get stuck on the gradient of KL divergence.

In the paper, it defines the similarity between datapoints, $p_j$, as follows

$$p_j=x_i-x_kright$$

the similarity between map points is given by

$$q_j=^2)over sum_kne iexp(-left$$

and at last, the cost function, which is the sum of KL divergences over all datapoints is defined by
$$C=sum_iKL(P_ivert|Q_i)=sum_isum_jp_jlogp_jover q_j$$

Then it says the gradient is

$$partial Cover partial y_i=2sum_j(p_j-q_j+p_j-q_j)(y_i-y_j)$$

However, I derive a different result

$$beginalign
partial Cover partial y_i&=-sum_j p_j1 over q_jnabla_y_iq_j + p_j1over q_jnabla_y_iq_j
endalign$$
and
$$beginalign
nabla_y_iq_j&=q_jleft(2(y_j-y_i)-2^2)right)\
nabla_y_iq_j&=q_j(2(y_j-y_i)-2q_j(y_j-y_i))
endalign$$
so
$$partial Coverpartial y_i=2sum_j(y_i-y_j)left(p_j+p_j-q_j-^2)right)$$

I've checked my derivation several times, but still have no clue about where the wrong is. Could someone help me check it? I'll be very grateful for that. Thanks.

derivatives

share|cite|improve this question

edited Jul 17 at 3:01

asked Jul 16 at 4:56

Sherwin Chen

536

share|cite|improve this question

share|cite|improve this question

edited Jul 17 at 3:01

edited Jul 17 at 3:01

edited Jul 17 at 3:01

asked Jul 16 at 4:56

Sherwin Chen

536

asked Jul 16 at 4:56

Sherwin Chen

536

asked Jul 16 at 4:56

Sherwin Chen

536

536

add a comment | 

add a comment | 

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