Mathematical Foundations of Generative AI · End Term · May 2026 · AN

Which of the following statements correctly describe the learned reverse process in a DDPM?

You are training a Conditional GAN to generate labelled chest X-ray images. The dataset

is small, where

is a one-hot class label (e.g., "pneumonia'' vs. "normal''). Both the generator

and discriminator

are conditioned on

. To improve training, you consider applying the following image augmentation operations to real samples before feeding them to the discriminator: 1. **Random horizontal flip** of the X-ray image 2. **Random Gaussian noise** added to pixel intensities 3. **Global contrast inversion** (replace each pixel

with

) 4. **Random crop and resize** to the original resolution The goal is to augment the training distribution of

without distorting the class-conditional structure

that the discriminator must learn to classify. Which augmentation operations are appropriate for this setting? Select **all** that apply.

Trajectories are collected under

. The importance-sampling surrogate and PPO clipped objectives are:

where

. For a single step:

,

,

,

. What is the effect of this clip on the gradient update to

? Select all that apply.

Which among the following about DDPM and DDIM are correct?

In classifier-free guidance, the conditional score is approximated as:

where

is a hyperparameter. Which option correctly identifies the key advantage of this approach over classifier guidance?

A Transformer uses sinusoidal positional encodings instead of learned positional embeddings. Which statement best explains why sinusoidal encodings can be used for sequence lengths not seen during training?

A practitioner wants to use a trained GAN - consisting of a generator

and an encoder (invertor)

- to edit a real image

by manipulating its latent code. Which sequence of steps correctly implements this editing pipeline?

A transformer block has model dimension

and

attention heads with per-head dimension

. Each head

has projection matrices

. In a single-head setup (

,

), there is one set

. Excluding biases, which of the following is correct?

A single transformer layer has model dimension

,

attention heads with

. The layer includes all query, key, and value projection matrices

for

, an output projection

, and a two-layer position-wise fully connected layer (FCL) with weights

and

. All biases are ignored. The total number of learnable parameters in this layer is

. Compute the FCL hidden dimension

.

Let

be a value function,

be a

-function and

be the advantage function for state

and action

. Then, which of the following are correct?

A language model policy generates a response of length 4. A single trajectory yields the data below. The discount factor is

.

Based on the above data, answer the given subquestions.

Using the backwards recursion

, compute

, where

denotes the reward-to- go. Enter the answer correct to two decimal places.

A language model policy generates a response of length 4. A single trajectory yields the data below. The discount factor is

.

Based on the above data, answer the given subquestions.

Compute

for each

, then compute the policy gradient estimate . Enter the answer correct to three decimal places.

A transformer processes a sequence of

tokens with model dimension

using

attention heads, each with

. The input embedding matrix is:

Head 1 uses projections:

Head 2 uses projections:

Use

for scaling. All biases are zero. Based on the above data, answer the given subquestions.

Compute

,

,

for Head 1 and the scaled score matrix . Enter the value of the

entry of

. Enter the answer correct to two decimal places.

A transformer processes a sequence of

tokens with model dimension

using

attention heads, each with

. The input embedding matrix is:

Head 1 uses projections:

Head 2 uses projections:

Use

for scaling. All biases are zero. Based on the above data, answer the given subquestions.

Causal masking is applied on

. Compute the attention weights for the third query, and enter the attention weight assigned to the third position, correct to two decimal places.

A transformer processes a sequence of

tokens with model dimension

using

attention heads, each with

. The input embedding matrix is:

Head 1 uses projections:

Head 2 uses projections:

Use

for scaling. All biases are zero. Based on the above data, answer the given subquestions.

Using causal masking, compute the attention outputs for the third token from both heads and concatenate them as

Enter the fourth entry of the third row of

. Enter the answer correct to two decimal places.

A DDPM is trained with the

-prediction parameterization. The forward process corrupts data as:

The model learns a noise predictor

, which is equivalent to predicting the score function of the noisy marginal

via :

Following are the values given:

, true data point

, sampled noise

. The trained model predicts

. Based on the above data, answer the given subquestions.

Compute the noisy observation

produced by the forward process.

A DDPM is trained with the

-prediction parameterization. The forward process corrupts data as:

The model learns a noise predictor

, which is equivalent to predicting the score function of the noisy marginal

via :

Following are the values given:

, true data point

, sampled noise

. The trained model predicts

. Based on the above data, answer the given subquestions.

Compute the model's predicted clean data point

. Enter the answer correct to two decimal places.

A DDPM is trained with the

-prediction parameterization. The forward process corrupts data as:

The model learns a noise predictor

, which is equivalent to predicting the score function of the noisy marginal

via :

Following are the values given:

, true data point

, sampled noise

. The trained model predicts

. Based on the above data, answer the given subquestions.

Compute both the noise-prediction error

and the data-prediction error

, and compute their ratio:

Express the answer correct to two decimal places.

A model is trained with

diffusion steps. Using DDIM, samples are generated along a sub-sequence

of length

(each step is one network evaluation). How many network evaluations does one DDIM sample require?

In a DDPM, the forward process is

. At a particular timestep,

. Which of the following statements are correct?