k-Multisection Neuron Coverage (KMNC)

• k-Multisection Neuron Coverage (KMNC) is a testing metric that partitions each neuron's activation range into k equal sections to assess test input diversity.
• It computes coverage by profiling neuron activations to determine bounds and then mapping test inputs to the corresponding activation sections.
• KMNC helps identify under-tested activation ranges and guides parameter selection, while its effectiveness depends on accurate profiling and appropriate k values.

k-Multisection Neuron Coverage (kMNC) is a structural testing metric designed to quantify how thoroughly a test suite exercises the internal activation space of a deep neural network (DNN). Unlike plain neuron coverage, which measures whether a neuron is ever activated, kMNC evaluates how extensively test inputs explore the range of activations for each neuron by partitioning that range into $k$ discrete sections and tracking coverage of each. This metric is applicable to the hidden layers of feed-forward DNNs and provides a more fine-grained view of test adequacy (Usman et al., 2022).

1. Formal Definition

Let $N$ denote the total number of neurons across all hidden layers of a DNN (excluding input and output layers by convention). For each neuron $i$, a profiling set (often the training set) is used to empirically determine the closed activation interval $[\mathrm{lb}_i, \mathrm{ub}_i]$, where

$$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$

This interval is divided into $k$ equal-length sections per neuron. For $j = 1$ to $k$, the $j$th section for neuron $i$ is defined as

$N$0

with the last section typically closed on the right. For a given test input $N$1, section $N$2 is covered if $N$3.

2. Computation Procedure

The computation of kMNC proceeds in the following steps:

1. Profiling activations: For each neuron $N$4, collect $N$5 and $N$6 using the profiling set.
2. Sectioning per neuron: Compute $N$7 sections $N$8 for each neuron.
3. Initialization: Set up a Boolean table $N$9 for each neuron-section pair.
4. Test coverage evaluation: For each test input $i$0 in the test suite $i$1 and for each neuron $i$2:
   1. Compute the activation $i$3.
   2. Compute the bin index:

   $i$4

   (clamped to $i$5). 3. Mark $i$6.

5. Aggregating coverage: After all test inputs, count the number of $i$7 entries in the $i$8 matrix.

3. Metric Formula

The overall kMNC score is computed as: $i$9 where

$[\mathrm{lb}_i, \mathrm{ub}_i]$0

Here, $[\mathrm{lb}_i, \mathrm{ub}_i]$1 is the neuron count (in hidden layers), $[\mathrm{lb}_i, \mathrm{ub}_i]$2 is the number of sections per neuron, $[\mathrm{lb}_i, \mathrm{ub}_i]$3 is the $[\mathrm{lb}_i, \mathrm{ub}_i]$4-th activation interval of neuron $[\mathrm{lb}_i, \mathrm{ub}_i]$5, and $[\mathrm{lb}_i, \mathrm{ub}_i]$6 is the test set. The metric thus represents the fraction of all neuron-section pairs that are exercised by at least one test input (Usman et al., 2022).

4. Illustrative Example

Consider a scenario with two neurons ($[\mathrm{lb}_i, \mathrm{ub}_i]$7) each divided into $[\mathrm{lb}_i, \mathrm{ub}_i]$8 sections:

• Profiling yields: $[\mathrm{lb}_i, \mathrm{ub}_i]$9; $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$0.
• Each section has width $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$1 for both neurons.

| Neuron | Section 1 | Section 2 | Section 3 | Section 4 | |--------|----------------|--------------|--------------|-------------| | 1 | [0,2) | [2,4) | [4,6) | [6,8] | | 2 | [2,4) | [4,6) | [6,8) | [8,10] |

Suppose test suite $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$2 produces the following activations:

• $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$3: $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$4 (bin 1), $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$5 (bin 2)
• $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$6: $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$7 (bin 3), $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$8 (bin 4)
• $$\mathrm{lb}_i = \min_{x \in \mathit{Profile}} a_i(x), \quad \mathrm{ub}_i = \max_{x \in \mathit{Profile}} a_i(x).$$9: $k$0 (bin 4), $k$1 (bin 1)

Coverage summary:

• Neuron 1: bins 1, 3, 4 covered
• Neuron 2: bins 1, 2, 4 covered

Out of $k$2 bins, $k$3 are covered, yielding $k$4 (Usman et al., 2022).

5. Advantages and Limitations

Benefits

• More fine-grained than plain neuron coverage, enabling detection of test-set inadequacy that broad coverage metrics may miss.
• Explicitly measures the extent to which the test suite exercises the full activation range of each neuron.
• Can identify “dead” sections of activation range even when a neuron is sometimes activated.
• Straightforward to implement and scales well to large networks (Usman et al., 2022).

Limitations

• Strong dependence on quality profiling: if $k$5 is misestimated, section boundaries may become uninformative.
• Purely structural—does not account for semantic relationships or clusters in the input space.
• Sensitivity to parameter $k$6: small $k$7 leads to coarse, easily saturated coverage; large $k$8 may result in bins that are rarely or never covered without a very large test suite.
• Does not account for interactions or dependencies between neurons; it only analyzes 1-D activation axes (Usman et al., 2022).

6. Parameter Selection and Practical Guidance

Practical guidance for selecting $k$9 includes:

• For typical convolutional architectures, an initial $j = 1$0 balances resolution with tractability.
• If kMNC rapidly saturates, increasing $j = 1$1 may reveal additional inadequacies in the test set.
• If coverage remains low across many tests, consider reducing $j = 1$2.
• Monitoring the slope of kMNC as new tests are added can inform about diminishing returns in test adequacy.

In operational use, kMNC is often computed alongside other neuron-level and decision-structure coverage criteria, such as top-$j = 1$3 neuron coverage, neuron boundary coverage, and MC/DC variants. Combining these criteria supports comprehensive test generation and test-suite minimization (Usman et al., 2022).